A.M. was supported by IKERBASQUE, by the project PID2024-157724NB-I00, funded by MICIU/AEI/10.13039/501100011033/FEDER, EU, and by the Europa Excelencia grant EUR2025-165042, funded by the Ministerio de Ciencia, Innovación y Universidades and Agencia Estatal de Investigación, and by IT1875-26 (Basque Government). M.M. was supported by IKERBASQUE and partially supported by the project PID2024-157724NB-I00, funded by MICIU/AEI/10.13039/501100011033/FEDER, EU, and by IT-1615-22 and IT1875-26 (Basque Government). C.P. was supported by the Ramón y Cajal fellowship grant RYC2023-045598-I (MICIU, Spain), and partially supported by the grants PID2023-150984NB-I00 and PID2024-155320NB-I00 funded by MICIU/AEI/10.13039/501100011033/FEDER, EU.
Introduction
A central line of research in harmonic analysis aims to determine sufficient conditions for the boundedness of singular integral operators (SIOs) in terms of their action on suitable testing functions, and has given rise to so-called T 1 T1 T 1 T b Tb T b T b Tb T b L 2 L^2 L 2 b B b_B b B b B ∗ b_B^* b B ∗ B B B L p L^p L p L 2 L^2 L 2 [31 Section 3.3.1] or the survey by S. Hofmann and A. McIntosh [36 Section 5.1.2] . It turned out to be a delicate question, even in the case of SIOs associated with the Lebesgue measure on Euclidean spaces, and several advances have appeared in the recent literature in non-homogeneous harmonic analysis, as we discuss later in this introduction (see § 1.1 ).
Our main result, Theorem 1.3 , is a local T b Tb T b
The testing assumptions are formulated on suitable families of doubling balls with small boundary, and they do not refer to any dyadic lattice. Passing from these assumptions to a dyadic testing system is delicate, since the accretivity on a ball need not persist on its dyadic subcells, and the testing balls associated with nested cells need not be nested. Our proof merges this geometry of balls with two independent random dyadic systems and a fixed David-Mattila lattice, by means of a mixed corona decomposition and two-measure estimates for suppressed operators.
The testing measures satisfy a quantitative small-set condition, and they need not be absolutely continuous with respect to the underlying measure. In the operator-testing alternative, only a one-level L 0 L^0 L 0 1
For testing functions with L p L^p L p L q L^q L q T b Tb T b 1 < p , q ≤ ∞ 1 < p, q \leq\infty 1 < p , q ≤ ∞ [60 ] , it also gives buffered and unbuffered criteria for fixed truncations, in the corresponding ranges of exponents, for kernels that are not necessarily antisymmetric and with testing conditions on metric balls, see Corollary 1.5.
Following the terminology in [41 ] , we say that a metric space ( X , d ) (X,d) ( X , d ) geometrically doubling if there exists D ∈ N D \in\mathbb{N} D ∈ N x ∈ X x \in X x ∈ X r > 0 r > 0 r > 0 B ( x , r ) : = { y ∈ X : d ( x , y ) < r } B(x,r) := \{y \in X : d(x,y) < r\} B ( x , r ) := { y ∈ X : d ( x , y ) < r } D D D B ( x j , r / 2 ) B(x_j,r/2) B ( x j , r /2 ) x j ∈ X x_j \in X x j ∈ X B ⊂ X B \subset X B ⊂ X r ( B ) r(B) r ( B ) B = B ( x , r ) B = B(x,r) B = B ( x , r ) λ > 0 \lambda> 0 λ > 0 λ B : = B ( x , λ r ) \lambda B := B(x,\lambda r) λ B := B ( x , λ r )
Definition 1.1. We say that K : ( X × X ) ∖ { ( x , x ) : x ∈ X } → C K : (X \times X) \setminus\{(x,x) : x \in X\} \to\mathbb{C} K : ( X × X ) ∖ {( x , x ) : x ∈ X } → C n n n C > 0 C > 0 C > 0 γ ∈ ( 0 , 1 ] \gamma\in(0,1] γ ∈ ( 0 , 1 ]
(1) For all x , y ∈ X x,y \in X x , y ∈ X x ≠ y x \ne y x = y
∣ K ( x , y ) ∣ ≤ C d ( x , y ) n |K(x,y)| \leq\frac{C}{d(x,y)^n} ∣ K ( x , y ) ∣ ≤ d ( x , y ) n C (2) For all x , x ′ , y ∈ X x,x',y \in X x , x ′ , y ∈ X d ( x , y ) ≥ 2 d ( x , x ′ ) d(x,y) \geq2d(x,x') d ( x , y ) ≥ 2 d ( x , x ′ )
∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ C d ( x , x ′ ) γ d ( x , y ) n + γ . |K(x,y)-K(x',y)| + |K(y,x)-K(y,x')| \leq C\frac{d(x,x')^\gamma}{d(x,y)^{n+\gamma}}. ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ C d ( x , y ) n + γ d ( x , x ′ ) γ . Given a Calderón-Zygmund kernel K K K μ \mu μ X X X ε > 0 \varepsilon> 0 ε > 0 singular integral operator (SIO) as
T μ f ( x ) : = ∫ K ( x , y ) f ( y ) d μ ( y ) for f ∈ L 1 ( μ ) and x ∉ supp ( f μ ) T_\mu f(x) := \int K(x,y)f(y)\,d\mu(y) \qquad\text{for } f \in L^1(\mu) \text{ and } x \notin\operatorname{supp}(f\mu) T μ f ( x ) := ∫ K ( x , y ) f ( y ) d μ ( y ) for f ∈ L 1 ( μ ) and x ∈ / supp ( f μ ) and its associated ε \varepsilon ε
T μ , ε f ( x ) = ∫ d ( x , y ) > ε K ( x , y ) f ( y ) d μ ( y ) . T_{\mu,\varepsilon}f(x)=\int_{d(x,y)>\varepsilon}K(x,y)f(y)\,d\mu(y). T μ , ε f ( x ) = ∫ d ( x , y ) > ε K ( x , y ) f ( y ) d μ ( y ) . These integrals are absolutely convergent for f ∈ L 1 ( μ ) f\in L^1(\mu) f ∈ L 1 ( μ ) μ \mu μ (1) below, also for f ∈ L p ( μ ) f\in L^p(\mu) f ∈ L p ( μ ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ T μ T_\mu T μ L p ( μ ) L^p(\mu) L p ( μ ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ L 1 ( μ ) L^1(\mu) L 1 ( μ ) L 1 , ∞ ( μ ) L^{1,\infty}(\mu) L 1 , ∞ ( μ ) T μ , ε T_{\mu,\varepsilon} T μ , ε L p ( μ ) L^p(\mu) L p ( μ ) L 1 ( μ ) L^1(\mu) L 1 ( μ ) L 1 , ∞ ( μ ) L^{1,\infty}(\mu) L 1 , ∞ ( μ ) ε > 0 \varepsilon>0 ε > 0
∥ T μ ∥ L p ( μ ) → L p ( μ ) : = sup ε > 0 ∥ T μ , ε ∥ L p ( μ ) → L p ( μ ) . \lVert T_\mu\rVert_{L^p(\mu)\to L^p(\mu)}:=\sup_{\varepsilon>0}\lVert T_{\mu,\varepsilon}\rVert_{L^p(\mu)\to L^p(\mu)}. ∥ T μ ∥ L p ( μ ) → L p ( μ ) := ε > 0 sup ∥ T μ , ε ∥ L p ( μ ) → L p ( μ ) . We also define the maximal singular integral as
T μ , ♯ f ( x ) = sup ε > 0 ∣ T μ , ε f ( x ) ∣ . T_{\mu,\sharp}f(x)=\sup_{\varepsilon>0}\lvert T_{\mu,\varepsilon}f(x)\rvert. T μ , ♯ f ( x ) = ε > 0 sup ∣ T μ , ε f ( x )∣ . We further understand that T ε μ ( x ) : = T μ , ε 1 ( x ) T_\varepsilon\mu(x):=T_{\mu,\varepsilon}1(x) T ε μ ( x ) := T μ , ε 1 ( x ) μ \mu μ T ♯ μ ( x ) T_\sharp\mu(x) T ♯ μ ( x ) T T T K ( x , y ) K(x,y) K ( x , y ) T ∗ T^* T ∗ K ( y , x ) K(y,x) K ( y , x )
For n > 0 n>0 n > 0 μ ∈ M + ( X ) \mu\in\mathcal{M}_+(X) μ ∈ M + ( X ) growth of degree n n n n n n growth with constant C 0 > 0 C_0>0 C 0 > 0
μ ( B ( x , r ) ) ≤ C 0 r n , for all x ∈ X and all r > 0. (1) \mu(B(x,r))\leq C_0r^n,\qquad\text{for all }x\in X\text{ and all }r>0. \tag*{(1)} μ ( B ( x , r )) ≤ C 0 r n , for all x ∈ X and all r > 0. (1) Before stating the precise testing assumptions, we briefly explain their form. We allow finite complex measures in place of testing functions, with a total variation bound and a small-set condition requiring that sets of small relative μ \mu μ [62 Theorem 1.6] . On each testing ball, we allow two alternatives: either the maximal truncations are bounded on a contracted ball outside a set of small relative μ \mu μ
Definition 1.2 (Accretive measure testing system). Let B \mathcal{B} B C 1 ≥ 1 \mathcal{C}_1\geq1 C 1 ≥ 1 ε 0 ∈ ( 0 , 1 ) \varepsilon_0\in(0,1) ε 0 ∈ ( 0 , 1 ) τ , ζ ∈ ( 0 , 1 ) \tau,\zeta\in(0,1) τ , ζ ∈ ( 0 , 1 ) T T T { ν ^ B } B ∈ B \{\widehat{\nu}_B\}_{B\in\mathcal{B}} { ν B } B ∈ B
∣ ν ^ B ∣ ( X ∖ B ) = 0 , ν ^ B ( B ) = μ ( B ) , ∥ ν ^ B ∥ ≤ C 1 μ ( B ) . (2) \lvert\widehat{\nu}_B\rvert(X\setminus B)=0,\qquad\widehat{\nu}_B(B)=\mu(B),\qquad\lVert\widehat{\nu}_B\rVert\leq\mathcal{C}_1\mu(B). \tag*{(2)} ∣ ν B ∣ ( X ∖ B ) = 0 , ν B ( B ) = μ ( B ) , ∥ ν B ∥ ≤ C 1 μ ( B ) . (2) For every Borel set A ⊂ B A\subset B A ⊂ B
If μ ( A ) ≤ ε 0 μ ( B ) , then ∣ ν ^ B ∣ ( A ) ≤ ∥ ν ^ B ∥ 200 C 1 . (3) \text{If }\mu(A)\leq\varepsilon_0\mu(B),\qquad\text{then}\qquad\lvert\widehat{\nu}_B\rvert(A)\leq\frac{\lVert\widehat{\nu}_B\rVert}{200\mathcal{C}_1}. \tag*{(3)} If μ ( A ) ≤ ε 0 μ ( B ) , then ∣ ν B ∣ ( A ) ≤ 200 C 1 ∥ ν B ∥ . (3) Moreover, for every η ∈ ( 0 , 1 / 10 ) \eta\in(0,1/10) η ∈ ( 0 , 1/10 ) C 2 ( η ) > 0 \mathcal{C}_2(\eta)>0 C 2 ( η ) > 0 B ∈ B B\in\mathcal{B} B ∈ B U B = U B ( η ) ⊂ B U_B=U_B(\eta)\subset B U B = U B ( η ) ⊂ B
(i) There is a Borel set V B ⊂ U B V_B\subset U_B V B ⊂ U B
diam ( V B ) ≤ η r ( B ) and μ ( V B ) ≥ ζ 2 μ ( B ) . (4) \operatorname{diam}(V_B)\leq\eta r(B)\qquad\text{and}\qquad\mu(V_B)\geq\frac{\zeta}{2}\mu(B). \tag*{(4)} diam ( V B ) ≤ η r ( B ) and μ ( V B ) ≥ 2 ζ μ ( B ) . (4) (ii) We have
μ ( U B ) < ζ μ ( B ) , T ♯ ν ^ B ( x ) ≤ C 2 ( η ) for every x ∈ ( 1 − τ ) B ∖ U B . (5) \mu(U_B)<\zeta\mu(B),\qquad T_\sharp\widehat{\nu}_B(x)\leq\mathcal{C}_2(\eta)\quad\text{for every }x\in(1-\tau)B\setminus U_B. \tag*{(5)} μ ( U B ) < ζ μ ( B ) , T ♯ ν B ( x ) ≤ C 2 ( η ) for every x ∈ ( 1 − τ ) B ∖ U B . (5) An accretive measure testing system for T ∗ T^* T ∗ T ∗ T^* T ∗ T T T T T T T ∗ T^* T ∗ C 1 \mathcal{C}_1 C 1 ε 0 \varepsilon_0 ε 0 τ \tau τ ζ \zeta ζ C 2 ( η ) \mathcal{C}_2(\eta) C 2 ( η )
Alternative (5) is equivalent to the one-level L 0 L^0 L 0
μ ( { x ∈ ( 1 − τ ) B : T ♯ ν ^ B ( x ) > C 2 ( η ) } ) < ζ μ ( B ) , (6) \mu\left(\left\{x\in(1-\tau)B:T^\sharp\widehat{\nu}_B(x)>\mathcal{C}_2(\eta)\right\}\right)<\zeta\mu(B), \tag*{(6)} μ ( { x ∈ ( 1 − τ ) B : T ♯ ν B ( x ) > C 2 ( η ) } ) < ζ μ ( B ) , (6) since one can take U B U_B U B T ♯ ν ^ B T^\sharp\widehat{\nu}_B T ♯ ν B C 2 ( η ) \mathcal{C}_2(\eta) C 2 ( η ) (3) does not imply that ν ^ B \widehat{\nu}_B ν B μ \mu μ
We now state the main result of the paper.
Theorem 1.3. Let ( X , d ) (X,d) ( X , d ) D > 1 D>1 D > 1 μ \mu μ n n n C 0 \mathcal{C}_0 C 0 n ≥ 1 n\geq1 n ≥ 1 T T T n n n C C C γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] Λ ≥ 100 \Lambda\geq100 Λ ≥ 100 β 0 = β 0 ( D , n , δ , Λ ) > 1 \beta_0=\beta_0(D,n,\delta,\Lambda)>1 β 0 = β 0 ( D , n , δ , Λ ) > 1 β ≥ β 0 \beta\geq\beta_0 β ≥ β 0 C 3 ≥ 220 \mathcal{C}_3\geq220 C 3 ≥ 220
Let B \mathcal{B} B B = B ( x , r ) B=B(x,r) B = B ( x , r ) x ∈ supp μ x\in\operatorname{supp}\mu x ∈ supp μ 0 < r ≤ diam ( supp μ ) 0<r\leq\operatorname{diam}(\operatorname{supp}\mu) 0 < r ≤ diam ( supp μ ) C 3 \mathcal{C}_3 C 3 μ \mu μ
μ ( Λ B ) ≤ β μ ( δ B ) . (7) \mu(\Lambda B)\leq\beta\mu(\delta B). \tag*{(7)} μ ( Λ B ) ≤ β μ ( δ B ) . (7) We fix C 1 ≥ 1 \mathcal{C}_1\geq1 C 1 ≥ 1 ε 0 ∈ ( 0 , 1 ) \varepsilon_0\in(0,1) ε 0 ∈ ( 0 , 1 )
τ 0 = min { 1 100 , ε 0 4 β C 3 } and ζ 0 = min { 1 100 , ε 0 8 } . (8) \tau_0=\min\left\{\frac{1}{100},\frac{\varepsilon_0}{4\beta\mathcal{C}_3}\right\}
\qquad\text{and}\qquad
\zeta_0=\min\left\{\frac{1}{100},\frac{\varepsilon_0}{8}\right\}. \tag*{(8)} τ 0 = min { 100 1 , 4 β C 3 ε 0 } and ζ 0 = min { 100 1 , 8 ε 0 } . (8) Suppose that, for 0 < τ < τ 0 0<\tau<\tau_0 0 < τ < τ 0 0 < ζ < ζ 0 0<\zeta<\zeta_0 0 < ζ < ζ 0 T T T T ∗ T^* T ∗ B \mathcal{B} B T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ ) D D D n n n C 0 \mathcal{C}_0 C 0 C C C γ \gamma γ δ \delta δ Λ \Lambda Λ β \beta β C 3 \mathcal{C}_3 C 3 C 1 \mathcal{C}_1 C 1 ε 0 \varepsilon_0 ε 0 ζ \zeta ζ C 2 ( η ∗ ) \mathcal{C}_2(\eta_*) C 2 ( η ∗ ) η ∗ \eta_* η ∗ (101) . In particular, the thresholds (8) are independent of C 2 ( η ∗ ) \mathcal{C}_2(\eta_*) C 2 ( η ∗ )
Note that the thresholds (8) give 2 ζ + τ β C 3 ≤ ε 0 / 2 2\zeta+\tau\beta\mathcal{C}_3\leq\varepsilon_0/2 2 ζ + τ β C 3 ≤ ε 0 /2 (5) on B \mathcal{B} B (7) and the small boundary property (18) , the union of the sets U B U_B U B B ∖ ( 1 − τ ) B B\setminus(1-\tau)B B ∖ ( 1 − τ ) B ε 0 μ ( B ) / 2 \varepsilon_0\mu(B)/2 ε 0 μ ( B ) /2 (3) , this union carries at most a fraction ( 200 C 1 ) − 1 (200\mathcal{C}_1)^{-1} ( 200 C 1 ) − 1 B B B
For an explicit expression of β 0 \beta_0 β 0
Theorem 1.3 applies, in particular, to testing functions with L p L^p L p L q L^q L q 1 < p , q ≤ ∞ 1<p,q\leq\infty 1 < p , q ≤ ∞ B B B b B ∈ L p ( μ ) b_B\in L^p(\mu) b B ∈ L p ( μ ) b B ∗ ∈ L q ( μ ) b_B^*\in L^q(\mu) b B ∗ ∈ L q ( μ ) B B B
∣ ∫ B b B d μ ∣ = ∣ ∫ B b B ∗ d μ ∣ = μ ( B ) . \left|\int_B b_B\,d\mu\right|=\left|\int_B b_B^*\,d\mu\right|=\mu(B). ∫ B b B d μ = ∫ B b B ∗ d μ = μ ( B ) . Multiplying the testing functions by constants of modulus one, we may assume that ∫ B b B d μ = ∫ B b B ∗ d μ = μ ( B ) \int_B b_B\,d\mu=\int_B b_B^*\,d\mu=\mu(B) ∫ B b B d μ = ∫ B b B ∗ d μ = μ ( B ) A ≥ 1 A\geq1 A ≥ 1 ∥ b B ∥ L p ( μ ) ≤ A μ ( B ) 1 / p \lVert b_B\rVert_{L^p(\mu)}\leq A\mu(B)^{1/p} ∥ b B ∥ L p ( μ ) ≤ A μ ( B ) 1/ p ∥ b B ∗ ∥ L q ( μ ) ≤ A μ ( B ) 1 / q \lVert b_B^*\rVert_{L^q(\mu)}\leq A\mu(B)^{1/q} ∥ b B ∗ ∥ L q ( μ ) ≤ A μ ( B ) 1/ q E ⊂ B E\subset B E ⊂ B
∫ E ∣ b B ∣ d μ ≤ A μ ( B ) ( μ ( E ) μ ( B ) ) 1 / p ′ , \int_E |b_B|\,d\mu\leq A\mu(B)\left(\frac{\mu(E)}{\mu(B)}\right)^{1/p'}, ∫ E ∣ b B ∣ d μ ≤ A μ ( B ) ( μ ( B ) μ ( E ) ) 1/ p ′ , and, moreover, μ ( B ) ≤ ∥ b B μ ∥ ≤ A μ ( B ) \mu(B)\leq\lVert b_B\mu\rVert\leq A\mu(B) μ ( B ) ≤ ∥ b B μ ∥ ≤ A μ ( B ) b B ∗ b_B^* b B ∗ q q q p p p ν ^ B : = b B μ \widehat{\nu}_B:=b_B\mu ν B := b B μ ν ^ B ∗ : = b B ∗ μ \widehat{\nu}_B^*:=b_B^*\mu ν B ∗ := b B ∗ μ (2) and (3) , with the constants of Definition 1.2 given by
C 1 = A and ε 0 = ( 200 A 2 ) − max { p ′ , q ′ } , (9) \mathcal{C}_1=A\qquad\text{and}\qquad\varepsilon_0=(200A^2)^{-\max\{p',q'\}}, \tag*{(9)} C 1 = A and ε 0 = ( 200 A 2 ) − m a x { p ′ , q ′ } , (9) where we understand that ∞ ′ = 1 \infty'=1 ∞ ′ = 1 s , L > 0 s,L>0 s , L > 0
sup t > 0 t s μ ( { x ∈ ( 1 − τ ) B : T μ , # ( b B μ ) ( x ) > t } ) ≤ L μ ( B ) , \sup_{t>0}t^s\mu\left(\left\{x\in(1-\tau)B:T_{\mu,\#}(b_B\mu)(x)>t\right\}\right)\leq L\mu(B), t > 0 sup t s μ ( { x ∈ ( 1 − τ ) B : T μ , # ( b B μ ) ( x ) > t } ) ≤ Lμ ( B ) , then (1.6) holds, for every η \eta η ( 2 L / ζ ) 1 / s (2L/\zeta)^{1/s} ( 2 L / ζ ) 1/ s b B ∗ b_B^* b B ∗ T ∗ T^* T ∗
By Chebyshev’s inequality, for any p , q ∈ ( 1 , ∞ ] p,q\in(1,\infty] p , q ∈ ( 1 , ∞ ] Theorem 1.3 gives the L 2 ( μ ) L^2(\mu) L 2 ( μ ) T μ T_\mu T μ B ∈ B B\in\mathcal B B ∈ B b B b_B b B b B ∗ b_B^* b B ∗
∫ B ( T μ , # b B ) q ′ d μ + ∫ B ( T μ , # ∗ b B ∗ ) p ′ d μ ≲ μ ( B ) . \int_B(T_{\mu,\#}b_B)^{q'}\,d\mu+\int_B(T_{\mu,\#}^*b_B^*)^{p'}\,d\mu\lesssim\mu(B). ∫ B ( T μ , # b B ) q ′ d μ + ∫ B ( T μ , # ∗ b B ∗ ) p ′ d μ ≲ μ ( B ) . More precisely, we obtain the following corollary.
Corollary 1.4. Let ( X , d ) (X,d) ( X , d ) D > 1 D>1 D > 1 μ \mu μ X X X n n n C 0 > 0 \mathcal C_0>0 C 0 > 0 n ≥ 1 n\geq1 n ≥ 1 T μ T_\mu T μ n n n C > 0 \mathcal C>0 C > 0 γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) 1 < p , q ≤ ∞ 1<p,q\leq\infty 1 < p , q ≤ ∞ B \mathcal B B Theorem 1.3 , for some δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] Λ ≥ 100 \Lambda\geq100 Λ ≥ 100 β ≥ β 0 \beta\geq\beta_0 β ≥ β 0 C 3 ≥ 220 \mathcal C_3\geq220 C 3 ≥ 220 C 1 , C 2 > 0 \mathcal C_1,\mathcal C_2>0 C 1 , C 2 > 0 B ∈ B B\in\mathcal B B ∈ B b B ∈ L p ( μ ) b_B\in L^p(\mu) b B ∈ L p ( μ ) b B ∗ ∈ L q ( μ ) b_B^*\in L^q(\mu) b B ∗ ∈ L q ( μ ) B B B
∣ ∫ B b B d μ ∣ = μ ( B ) , ∣ ∫ B b B ∗ d μ ∣ = μ ( B ) , \left|\int_B b_B\,d\mu\right|=\mu(B),\qquad\left|\int_B b_B^*\,d\mu\right|=\mu(B), ∫ B b B d μ = μ ( B ) , ∫ B b B ∗ d μ = μ ( B ) , ∥ b B ∥ L p ( μ ) ≤ C 1 μ ( B ) 1 / p , ∥ b B ∗ ∥ L q ( μ ) ≤ C 1 μ ( B ) 1 / q , \|b_B\|_{L^p(\mu)}\leq\mathcal C_1\mu(B)^{1/p},\qquad\|b_B^*\|_{L^q(\mu)}\leq\mathcal C_1\mu(B)^{1/q}, ∥ b B ∥ L p ( μ ) ≤ C 1 μ ( B ) 1/ p , ∥ b B ∗ ∥ L q ( μ ) ≤ C 1 μ ( B ) 1/ q , ∫ B ( T μ , # b B ) q ′ d μ ≤ C 2 μ ( B ) , ∫ B ( T μ , # ∗ b B ∗ ) p ′ d μ ≤ C 2 μ ( B ) , \int_B(T_{\mu,\#}b_B)^{q'}\,d\mu\leq\mathcal C_2\mu(B),\qquad\int_B(T_{\mu,\#}^*b_B^*)^{p'}\,d\mu\leq\mathcal C_2\mu(B), ∫ B ( T μ , # b B ) q ′ d μ ≤ C 2 μ ( B ) , ∫ B ( T μ , # ∗ b B ∗ ) p ′ d μ ≤ C 2 μ ( B ) , with the convention ∞ ′ = 1 \infty'=1 ∞ ′ = 1 T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ )
Combined with the Cotlar-type inequality of H. Martikainen, X. Tolsa, and the second named author [60 Corollary 3.6] , Theorem 1.3 also yields the following result on Hofmann’s problem, see § 1.1 , for kernels that are not necessarily antisymmetric.
Corollary 1.5. Let ( X , d ) (X,d) ( X , d ) D > 1 D>1 D > 1 μ \mu μ X X X n n n C 0 > 0 \mathcal C_0>0 C 0 > 0 n ≥ 1 n\geq1 n ≥ 1 T μ T_\mu T μ n n n C > 0 \mathcal C>0 C > 0 γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) p , q ∈ ( 1 , 2 ] p,q\in(1,2] p , q ∈ ( 1 , 2 ] δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] Λ ≥ 100 \Lambda\geq100 Λ ≥ 100 β ≥ β 0 \beta\geq\beta_0 β ≥ β 0 C 3 ≥ 220 \mathcal C_3\geq220 C 3 ≥ 220 D D D n n n δ \delta δ Λ \Lambda Λ B \mathcal B B Theorem 1.3 . Suppose that there are constants C 1 , C 2 > 0 \mathcal C_1,\mathcal C_2>0 C 1 , C 2 > 0 B ∈ B B\in\mathcal B B ∈ B b B b_B b B b B ∗ b_B^* b B ∗ B B B
∫ B b B d μ = μ ( B ) , ∫ B b B ∗ d μ = μ ( B ) , \int_B b_B\,d\mu=\mu(B),\qquad\int_B b_B^*\,d\mu=\mu(B), ∫ B b B d μ = μ ( B ) , ∫ B b B ∗ d μ = μ ( B ) , ∫ B ∣ b B ∣ p d μ ≤ C 1 μ ( B ) , ∫ B ∣ b B ∗ ∣ q d μ ≤ C 1 μ ( B ) , \int_B|b_B|^p\,d\mu\leq\mathcal C_1\mu(B),\qquad\int_B|b_B^*|^q\,d\mu\leq\mathcal C_1\mu(B), ∫ B ∣ b B ∣ p d μ ≤ C 1 μ ( B ) , ∫ B ∣ b B ∗ ∣ q d μ ≤ C 1 μ ( B ) , sup ε > 0 ∫ B 1 ∣ T μ , ε b B ∣ q ′ d μ ≤ C 2 μ ( B ) , sup ε > 0 ∫ B 1 ∣ T μ , ε ∗ b B ∗ ∣ p ′ d μ ≤ C 2 μ ( B ) , \sup_{\varepsilon>0}\int_{B_1}|T_{\mu,\varepsilon}b_B|^{q'}\,d\mu\leq\mathcal C_2\mu(B),\qquad\sup_{\varepsilon>0}\int_{B_1}|T_{\mu,\varepsilon}^*b_B^*|^{p'}\,d\mu\leq\mathcal C_2\mu(B), ε > 0 sup ∫ B 1 ∣ T μ , ε b B ∣ q ′ d μ ≤ C 2 μ ( B ) , ε > 0 sup ∫ B 1 ∣ T μ , ε ∗ b B ∗ ∣ p ′ d μ ≤ C 2 μ ( B ) , where
B 1 : = { B if 1 p + 1 q < 1 + 1 n max { p , q } , 2 B otherwise . B_1:=\begin{cases}
B & \text{if }\dfrac{1}{p}+\dfrac{1}{q}<1+\dfrac{1}{n\max\{p,q\}},\\
2B & \text{otherwise}.
\end{cases} B 1 := ⎩ ⎨ ⎧ B 2 B if p 1 + q 1 < 1 + n max { p , q } 1 , otherwise . Then, the operator T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ )
In particular, the unbuffered testing conditions, with B 1 = B B_1=B B 1 = B p , q ∈ ( 1 , 2 ] p,q\in(1,2] p , q ∈ ( 1 , 2 ] n = 1 n=1 n = 1 p , q ∈ [ 2 − ε ( n ) , 2 ] p,q\in[2-\varepsilon(n),2] p , q ∈ [ 2 − ε ( n ) , 2 ] ε ( n ) > 0 \varepsilon(n)>0 ε ( n ) > 0 B 1 = 2 B B_1=2B B 1 = 2 B p , q ∈ ( 1 , 2 ] p,q\in(1,2] p , q ∈ ( 1 , 2 ] [60 ] for measures with n n n R d \mathbb{R}^d R d
As in [60 ] , Corollary 1.5 1.5 1.5 [60 Corollary 3.6] , whose proof does not use the antisymmetry of the kernel. The argument there extends to balls in geometrically doubling metric spaces, with the dilated Hardy–Littlewood maximal function (21) in place of the Euclidean maximal functions, and gives, if β \beta β E 3 \mathcal{E}_3 E 3
∫ ( 1 − τ ) B ( T μ , ♯ b B + T μ , ♯ b ∗ B ∗ ) d μ ≲ τ μ ( B ) \int_{(1-\tau)B}\left(T_{\mu,\sharp bB}+T_{\mu,\sharp b^*B}^{*}\right)\,d\mu\lesssim_{\tau}\mu(B) ∫ ( 1 − τ ) B ( T μ , ♯ b B + T μ , ♯ b ∗ B ∗ ) d μ ≲ τ μ ( B ) for every B ∈ B B\in\mathcal{B} B ∈ B τ ∈ ( 0 , 1 ) \tau\in(0,1) τ ∈ ( 0 , 1 ) T b Tb T b 1.3 1.3 1.3 (9) and Chebyshev’s inequality.
We stress that neither the doubling property of μ \mu μ K K K 1.3 1.3 1.3 1.5 1.5 1.5 § 1.3 .
Remark 1.6. We state Theorem 1.3 1.3 1.3 1.5 1.5 1.5 n ≥ 1 n\geq1 n ≥ 1 1.9 1.9 1.9 n ≥ 1 n\geq1 n ≥ 1 0 < n < 1 0<n<1 0 < n < 1 n n n (68) by 1 1 1 3.13 3.13 3.13 M = 1 M=1 M = 1
Remark 1.7. A natural extension of Theorem 1.3 1.3 1.3 upper doubling condition
μ ( B ( x , r ) ) ≤ λ ( x , r ) , for all x ∈ X and all r > 0 , (10) \mu(B(x,r))\leq\lambda(x,r),\qquad\text{for all }x\in X\text{ and all }r>0, \tag*{(10)} μ ( B ( x , r )) ≤ λ ( x , r ) , for all x ∈ X and all r > 0 , (10) where λ ( ⋅ , ⋅ ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) \lambda(\cdot,\cdot):X\times(0,\infty)\to(0,\infty) λ ( ⋅ , ⋅ ) : X × ( 0 , ∞ ) → ( 0 , ∞ )
(i) λ ( x , ⋅ ) \lambda(x,\cdot) λ ( x , ⋅ ) x ∈ X x\in X x ∈ X
(ii) λ ( x , 2 r ) ≤ C λ λ ( x , r ) \lambda(x,2r)\leq C_{\lambda}\lambda(x,r) λ ( x , 2 r ) ≤ C λ λ ( x , r ) x ∈ X , r > 0 x\in X,r>0 x ∈ X , r > 0
and the associated Calderón–Zygmund kernel K : ( X × X ) ∖ { ( x , x ) : x ∈ X } → C K:(X\times X)\setminus\{(x,x):x\in X\}\to\mathbb{C} K : ( X × X ) ∖ {( x , x ) : x ∈ X } → C γ ∈ ( 0 , 1 ] \gamma\in(0,1] γ ∈ ( 0 , 1 ] C > 0 C>0 C > 0
For all x , y ∈ X x,y\in X x , y ∈ X x ≠ y x\neq y x = y
∣ K ( x , y ) ∣ ≤ C λ ( x , d ( x , y ) ) . |K(x,y)|\leq\frac{C}{\lambda(x,d(x,y))}. ∣ K ( x , y ) ∣ ≤ λ ( x , d ( x , y )) C . For all y , x , x ′ ∈ X y,x,x'\in X y , x , x ′ ∈ X m : = max { d ( y , x ) , d ( y , x ′ ) } m:=\max\{d(y,x),d(y,x')\} m := max { d ( y , x ) , d ( y , x ′ )} 2 d ( x , x ′ ) ≤ m 2d(x,x')\leq m 2 d ( x , x ′ ) ≤ m
∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ C λ ( y , m ) ( d ( x , x ′ ) m ) γ . |K(x,y)-K(x',y)|+|K(y,x)-K(y,x')|
\leq\frac{C}{\lambda(y,m)}\left(\frac{d(x,x')}{m}\right)^{\gamma}. ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ λ ( y , m ) C ( m d ( x , x ′ ) ) γ . The proof in this paper is presented for n n n T b Tb T b
History and motivation
The first version of the T 1 T1 T 1 [18 ] , and the classical reference for the T b Tb T b [19 ] . Their work was further extended in the very influential paper [72 ] , in which F. Nazarov, S. Treil, and A. Volberg proved a (global) T b Tb T b [70 ] and Volberg’s monograph [Vol03].
Nazarov-Treil-Volberg’s articles are central to the literature also because of the techniques pioneered therein, such as, among others, the randomization of dyadic grids and the surgery arguments, the construction of suppressed kernels, and their associated SIOs, which have since become fundamental tools in a wide range of settings and now play a central role in non-homogeneous harmonic analysis and related areas. Their T b Tb T b A 2 A_2 A 2 [38 ] . For other versions of the T b Tb T b [17 ] and G. David-P. Mattila [20 ] . To highlight the role of T 1 / T b T1/Tb T 1/ T b [63 ] , the characterization of Varopoulos’ ε \varepsilon ε [24 ] , and the papers [57 ] , [58 ] , and [HM14]. Further connections can be found in the interplay with PDEs and free-boundary problems such as the rectifiability of harmonic measure in [AHM3TV16] and its generalization to two-phase problems and to elliptic operators ([6 ] , [7 ] , [8 ] , [MT20], [73 ] , [74 ] , [65 ] ).
Moreover, there has been growing interest in extending several of the above results to more general settings, including parabolic and Kolmogorov spaces, sub-Riemannian geometry, and more general metric settings; see, for instance, [11 ] , [12 ] , [27 ] , [HJ25], [56 ] , [64 ] , [66 ] . For example, we note that, in [28 ] , the global T b Tb T b ( − Δ x ) 1 / 2 + ∂ t (-\Delta_x)^{1/2}+\partial_t ( − Δ x ) 1/2 + ∂ t T b Tb T b Theorem 1.3 opens the way to a wide range of potential future applications and investigations, see also § 1.2 .
The first local T b Tb T b [15 Main Theorem 10] in the doubling (homogeneous) setting. Namely, Christ established the L 2 ( μ ) L^2(\mu) L 2 ( μ ) L ∞ ( μ ) L^\infty(\mu) L ∞ ( μ ) b Q ∈ L ∞ ( μ ) b_Q \in L^\infty(\mu) b Q ∈ L ∞ ( μ ) Q ⊂ R d Q \subset\mathbb{R}^d Q ⊂ R d
∣ ∫ Q b Q d μ ∣ ≳ μ ( Q ) , \left|\int_Q b_Q\,d\mu\right|\gtrsim\mu(Q), ∫ Q b Q d μ ≳ μ ( Q ) , together with an analogous pseudo-accretive system for the adjoint operator.
Christ’s result was extended through a delicate use of random dyadic grids to the case of non-doubling measures by Nazarov, Treil, and Volberg in [NTV02], assuming that b Q , b Q ∗ ∈ L ∞ ( μ ) b_Q,b_Q^*\in L^\infty(\mu) b Q , b Q ∗ ∈ L ∞ ( μ ) T b Q , T ∗ b Q ∗ Tb_Q,T^*b_Q^* T b Q , T ∗ b Q ∗ L 2 L^2 L 2 Q Q Q T b Tb T b [41 ] , using a metric variant of dyadic systems and an almost-covering of the space by balls. Both techniques are crucial for formulating Theorem 1.3 in its full generality.
The interest in lowering the integrability assumptions on the testing functions arose with the work of P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele [2 ] , who proved a local T b Tb T b
∫ Q ( ∣ b Q ∣ p + ∣ b Q ∗ ∣ q + ∣ T b Q ∣ q ′ + ∣ T ∗ b Q ∗ ∣ p ′ ) ≲ ∣ Q ∣ , (11) \int_Q \left(|b_Q|^p+|b_Q^*|^q+|Tb_Q|^{q'}+|T^*b_Q^*|^{p'}\right)\lesssim|Q|, \tag*{(11)} ∫ Q ( ∣ b Q ∣ p + ∣ b Q ∗ ∣ q + ∣ T b Q ∣ q ′ + ∣ T ∗ b Q ∗ ∣ p ′ ) ≲ ∣ Q ∣ , (11) in the model case of Calderón-Zygmund operators with perfect cancellation. The case of general SIOs is much more delicate.
S. Hofmann proved in the unpublished work [30 ] that, for the Lebesgue measure on R d \mathbb{R}^{d} R d Definition 1.1 to be bounded on L 2 L^{2} L 2 { b Q } \{b_{Q}\} { b Q } { b Q ∗ } \{b_{Q}^{*}\} { b Q ∗ } b Q b_{Q} b Q b Q ∗ b_{Q}^{*} b Q ∗ Q Q Q Q ⊂ R d Q \subset\mathbb{R}^{d} Q ⊂ R d
∫ Q ∣ b Q ∣ p + ∣ b Q ∗ ∣ p + ∣ T b Q ∣ 2 + ∣ T ∗ b Q ∗ ∣ 2 ≲ ∣ Q ∣ (12) \int_{Q} |b_{Q}|^{p} + |b_{Q}^{*}|^{p} + |Tb_{Q}|^{2} + |T^{*}b_{Q}^{*}|^{2} \lesssim|Q| \tag*{(12)} ∫ Q ∣ b Q ∣ p + ∣ b Q ∗ ∣ p + ∣ T b Q ∣ 2 + ∣ T ∗ b Q ∗ ∣ 2 ≲ ∣ Q ∣ (12) for some p > 2 p > 2 p > 2 L 2 L^{2} L 2 [1 ] .
The question of whether (11) implies L 2 L^{2} L 2 p , q ∈ ( 1 , ∞ ) p,q \in(1,\infty) p , q ∈ ( 1 , ∞ ) T b Tb T b
P. Auscher and Q.X. Yang proved in [4 ] the analogue of [30 ] for p = 2 p=2 p = 2 (12) , by reducing the study to perfect dyadic operators, and hence invoking [AHMTT02]. In spaces of homogeneous type, P. Auscher and E. Routin gave in [3 ] a direct proof of the local T b Tb T b (11) in the case 1 / p + 1 / q ≤ 1 1/p+1/q\leq1 1/ p + 1/ q ≤ 1 1 / p + 1 / q > 1 1/p+1/q>1 1/ p + 1/ q > 1 twisted martingale transform . Establishing a suitable variant of these transforms played a crucial role in the paper [52 ] , where M. Lacey and H. Martikainen proved, in the non-homogeneous setting, a local T b Tb T b p = q = 2 p=q=2 p = q = 2
Hofmann’s problem has been solved by T. Hytönen and F. Nazarov in [43 ] for p , q ∈ ( 1 , ∞ ) p,q \in(1,\infty) p , q ∈ ( 1 , ∞ ) Q ⊂ R d Q \subset\mathbb{R}^{d} Q ⊂ R d
∫ Q ∣ b Q ∣ p + ∣ b Q ∗ ∣ q ≲ ∣ Q ∣ and ∫ 2 Q ∣ T b Q ∣ q ′ + ∣ T ∗ b Q ∗ ∣ p ′ ≲ ∣ Q ∣ . \int_{Q} |b_{Q}|^{p} + |b_{Q}^{*}|^{q} \lesssim|Q|
\qquad\text{and}\qquad
\int_{2Q} |Tb_{Q}|^{q'} + |T^{*}b_{Q}^{*}|^{p'} \lesssim|Q|. ∫ Q ∣ b Q ∣ p + ∣ b Q ∗ ∣ q ≲ ∣ Q ∣ and ∫ 2 Q ∣ T b Q ∣ q ′ + ∣ T ∗ b Q ∗ ∣ p ′ ≲ ∣ Q ∣. which, combined with a Cotlar-type lemma for “buffered” accretive systems (see [43 Lemma 3.2] ), made it possible to control maximal truncations of the SIO. We remark that, by Hardy’s inequalities, in the case of the Lebesgue measure and if 1 / p + 1 / q ≤ 1 1/p+1/q\leq1 1/ p + 1/ q ≤ 1 (11) imply the buffered ones. The buffered theorem in [43 ] follows from their local T b Tb T b [43 Theorem 2.13] , which allows uniform L ∞ L^{\infty} L ∞ [43 Section 6] .
A different Cotlar-type approach was combined with the good lambda inequality by H. Martikainen, X. Tolsa, and the second named author in [60 ] to fully solve Hofmann’s problem in R \mathbb{R} R p , q ∈ ( 1 , 2 ] p,q \in(1,2] p , q ∈ ( 1 , 2 ] R d \mathbb{R}^{d} R d d > 1 d>1 d > 1 p , q ∈ [ 2 − ε ( d ) , 2 ] p,q \in[2-\varepsilon(d),2] p , q ∈ [ 2 − ε ( d ) , 2 ] ε ( d ) > 0 \varepsilon(d)>0 ε ( d ) > 0 Corollary 1.5 extends the results of [60 ] for measures with n n n R d \mathbb{R}^{d} R d L 2 L^{2} L 2 [60 ] , and the buffered ones for all p , q ∈ ( 1 , 2 ] p,q \in(1,2] p , q ∈ ( 1 , 2 ] Corollary 1.5 extends the fixed-truncation results of [60 ] to kernels that need not be antisymmetric and to geometrically doubling metric spaces, with testing on balls. We do not know whether the unbuffered fixed-truncation theorem holds in this generality for the full range 1 < p , q ≤ ∞ 1<p,q\leq\infty 1 < p , q ≤ ∞
Finally, we highlight that local T b Tb T b [26 ] , [53 ] , [59 ] , [61 ] , [62 ] , and the discussion therein. For other related works, we also refer the reader to [44 ] , [54 ] , [10 ] , [Viz24], and the references therein. For further expositions on the methods in [72 ] , we also refer to X. Tolsa’s monograph [Tol14, Chapter 5].
Applications to the heat equation
Our main application concerns absolute continuity of caloric measure and gradients of single layer potentials for the heat equation.
We denote by P n : = R n × R \mathbb{P}^n := \mathbb{R}^n \times\mathbb{R} P n := R n × R X = ( X , t ) \mathbf{X} = (X,t) X = ( X , t ) ∥ X ∥ : = max { ∣ X ∣ , ∣ t ∣ 1 / 2 } \lVert\mathbf{X}\rVert:= \max\{|X|,|t|^{1/2}\} ∥ X ∥ := max { ∣ X ∣ , ∣ t ∣ 1/2 } ∥ ⋅ ∥ \lVert\cdot\rVert ∥ ⋅ ∥ P n \mathbb{P}^n P n ( n + 2 ) (n+2) ( n + 2 )
B ( X , r ) = { ( Y , s ) : ∣ Y − X ∣ < r , ∣ s − t ∣ < r 2 } , X ∈ P n , r > 0. B(\mathbf{X},r) = \{(Y,s): |Y-X|<r,\ |s-t|<r^2\}, \qquad\mathbf{X}\in\mathbb{P}^n,\ r>0. B ( X , r ) = {( Y , s ) : ∣ Y − X ∣ < r , ∣ s − t ∣ < r 2 } , X ∈ P n , r > 0. For X , Y ∈ P n \mathbf{X},\mathbf{Y}\in\mathbb{P}^n X , Y ∈ P n X ≠ Y \mathbf{X}\ne\mathbf{Y} X = Y
Γ ( X , Y ) : = χ { t > s } ( 4 π ( t − s ) ) n / 2 exp ( − ∣ X − Y ∣ 2 4 ( t − s ) ) . \Gamma(\mathbf{X},\mathbf{Y}) := \frac{\chi_{\{t>s\}}}{(4\pi(t-s))^{n/2}}\exp\left(-\frac{|X-Y|^2}{4(t-s)}\right). Γ ( X , Y ) := ( 4 π ( t − s ) ) n /2 χ { t > s } exp ( − 4 ( t − s ) ∣ X − Y ∣ 2 ) . the fundamental solution of the heat equation ∂ t u − Δ X u = 0 \partial_tu-\Delta_Xu=0 ∂ t u − Δ X u = 0 K ( X , Y ) : = ∇ X Γ ( X , Y ) K(\mathbf{X},\mathbf{Y}):=\nabla_X\Gamma(\mathbf{X},\mathbf{Y}) K ( X , Y ) := ∇ X Γ ( X , Y ) ∇ X \nabla_X ∇ X K K K ( n + 1 ) (n+1) ( n + 1 ) [64 Lemma 5.4] . If σ \sigma σ P n \mathbb{P}^n P n ( n + 1 ) (n+1) ( n + 1 )
T σ f ( X ) = ∫ K ( X , Y ) f ( Y ) d σ ( Y ) , f ∈ L 1 ( σ ) . T_\sigma f(\mathbf{X})=\int K(\mathbf{X},\mathbf{Y})f(Y)\,d\sigma(Y), \qquad f\in L^1(\sigma). T σ f ( X ) = ∫ K ( X , Y ) f ( Y ) d σ ( Y ) , f ∈ L 1 ( σ ) . to be interpreted in the usual sense of truncations, which corresponds to the spatial gradient of the single layer potential for the heat equation.
Let Ω ⊂ P n \Omega\subset\mathbb{P}^n Ω ⊂ P n σ \sigma σ ∂ Ω \partial\Omega ∂ Ω ( n + 1 ) (n+1) ( n + 1 ) P ∈ Ω \mathbf{P}\in\Omega P ∈ Ω ω P \omega^{\mathbf{P}} ω P ω ∗ P \omega_*^{\mathbf{P}} ω ∗ P Ω \Omega Ω P \mathbf{P} P § 8 . We assume the following A ∞ A_\infty A ∞ ω P \omega^{\mathbf{P}} ω P ω ∗ P \omega_*^{\mathbf{P}} ω ∗ P σ \sigma σ
Definition 1.8 (The classes A ∞ + ( σ ; κ , κ ′ ) A^+_\infty(\sigma;\kappa,\kappa') A ∞ + ( σ ; κ , κ ′ ) A ∞ − ( σ ; κ , κ ′ ) A^-_\infty(\sigma;\kappa,\kappa') A ∞ − ( σ ; κ , κ ′ ) δ , ϑ , κ , κ ′ ∈ ( 0 , 1 ) \delta,\vartheta,\kappa,\kappa'\in(0,1) δ , ϑ , κ , κ ′ ∈ ( 0 , 1 ) Λ ≥ 100 \Lambda\ge100 Λ ≥ 100 C 3 ≥ 220 \mathfrak{C}_3\ge220 C 3 ≥ 220 β \beta β
We write ω ∈ A ∞ + ( σ ; κ , κ ′ ) \omega\in A^+_\infty(\sigma;\kappa,\kappa') ω ∈ A ∞ + ( σ ; κ , κ ′ ) B = B ( X 0 , r ) B=B(\mathbf{X}_0,r) B = B ( X 0 , r ) X 0 = ( X 0 , t 0 ) \mathbf{X}_0=(X_0,t_0) X 0 = ( X 0 , t 0 ) supp σ \operatorname{supp}\sigma supp σ 0 < r ≤ diam ( supp σ ) 0<r\le\operatorname{diam}(\operatorname{supp}\sigma) 0 < r ≤ diam ( supp σ ) C 3 \mathfrak{C}_3 C 3 σ ( Λ B ) ≤ β σ ( δ B ) \sigma(\Lambda B)\le\beta\sigma(\delta B) σ ( Λ B ) ≤ β σ ( δ B ) P B + = ( X P + , t P + ) ∈ Ω ∩ δ B \mathbf{P}_B^+=(X_{P^+},t_{P^+})\in\Omega\cap\delta B P B + = ( X P + , t P + ) ∈ Ω ∩ δ B
t 0 + ϑ ( δ r ) 2 ≤ t P + ≤ t 0 + ( δ r ) 2 , (13) t_0+\vartheta(\delta r)^2\le t_{P^+}\le t_0+(\delta r)^2, \tag*{(13)} t 0 + ϑ ( δ r ) 2 ≤ t P + ≤ t 0 + ( δ r ) 2 , (13) such that, for every Borel set E ⊂ B E\subset B E ⊂ B
σ ( E ) ≤ κ σ ( B ) ⟹ ω B P B + ( E ) ≤ κ ′ ω B P B + ( B ) . (14) \sigma(E)\le\kappa\sigma(B) \quad\Longrightarrow\quad\omega_B^{\mathbf{P}_B^+}(E)\le\kappa'\omega_B^{\mathbf{P}_B^+}(B). \tag*{(14)} σ ( E ) ≤ κσ ( B ) ⟹ ω B P B + ( E ) ≤ κ ′ ω B P B + ( B ) . (14) The property ω ∗ ∈ A ∞ − ( σ ; κ , κ ′ ) \omega_*\in A^-_\infty(\sigma;\kappa,\kappa') ω ∗ ∈ A ∞ − ( σ ; κ , κ ′ ) ω ∗ \omega_* ω ∗ ω B P B + \omega_B^{\mathbf{P}_B^+} ω B P B + P B − ∈ Ω ∩ δ B \mathbf{P}_B^- \in\Omega\cap\delta B P B − ∈ Ω ∩ δ B
t 0 − ( δ r ) 2 ≤ t P − ≤ t 0 − ϑ ( δ r ) 2 . (15) t_0-(\delta r)^2\le t_{P^-}\le t_0-\vartheta(\delta r)^2. \tag*{(15)} t 0 − ( δ r ) 2 ≤ t P − ≤ t 0 − ϑ ( δ r ) 2 . (15) We remark that, unlike reverse Hölder conditions, Definition 1.8 does not require ω B P B + ∣ B \omega_B^{\mathbf{P}_B^+}|_B ω B P B + ∣ B ω ∗ B P B − ∣ B \omega_{*B}^{\mathbf{P}_B^-}|_B ω ∗ B P B − ∣ B σ \sigma σ ω B P B + ∣ B ≪ σ ∣ B \omega_B^{\mathbf{P}_B^+}|_B\ll\sigma|_B ω B P B + ∣ B ≪ σ ∣ B k B + k_B^+ k B +
\left(\fint_B (k_B^+)^s\,d\sigma\right)^{1/s}\le C\frac{\omega_B^{\mathbf{P}_B^+}(B)}{\sigma(B)},
for some s > 1 s>1 s > 1 C ≥ 1 C\ge1 C ≥ 1
ω B P B + ( E ) ≤ C ( σ ( E ) σ ( B ) ) 1 / s ′ ω B P B + ( B ) , for every Borel set E ⊂ B . \omega_{B}^{\mathbf{P}_{B}^{+}}(E)\le C\left(\frac{\sigma(E)}{\sigma(B)}\right)^{1/s'}\omega_{B}^{\mathbf{P}_{B}^{+}}(B),
\qquad\text{for every Borel set }E\subset B. ω B P B + ( E ) ≤ C ( σ ( B ) σ ( E ) ) 1/ s ′ ω B P B + ( B ) , for every Borel set E ⊂ B . Hence, (14) holds for every κ ′ ∈ ( 0 , 1 ) \kappa'\in(0,1) κ ′ ∈ ( 0 , 1 ) κ : = ( κ ′ / C ) s ′ \kappa:=(\kappa'/C)^{s'} κ := ( κ ′ / C ) s ′ ω ∗ P B − \omega_{*}^{\mathbf{P}_{B}^{-}} ω ∗ P B −
In corkscrew domains with Ahlfors-regular boundaries, suitable reverse Hölder estimates and A ∞ A_{\infty} A ∞ L p L^{p} L p 1 < p < ∞ 1<p<\infty 1 < p < ∞ L ∞ L^{\infty} L ∞ [68 ] . For the Laplacian, these conditions also imply uniform rectifiability of the boundary; see [32 MT20] and the references therein.
In the parabolic setting, E. Fabes and S. Salsa showed in [22 ] that, in Lipschitz cylinders, the caloric Poisson kernel satisfies a reverse Hölder inequality with exponent 2 2 2 [46 ] that caloric measure may fail to be absolutely continuous with respect to surface measure on the graph of a Lip ( 1 , 1 / 2 ) \operatorname{Lip}(1,1/2) Lip ( 1 , 1/2 ) [55 ] that it is an A ∞ A_{\infty} A ∞ [33 ] . These graphs are the model for the parabolic uniformly rectifiable sets introduced by S. Hofmann, J. L. Lewis, and K. Nyström in [34 , 35 ] . More recently, A. Genschaw and S. Hofmann proved in [25 ] a weak reverse Hölder inequality for caloric measure, equivalent to the L p L^{p} L p [13 , 14 ] that, for domains with time-symmetric Ahlfors regular boundary satisfying an interior corkscrew condition, this property implies parabolic uniform rectifiability.
Definition 1.8 is a local version of these conditions, where the poles P B + \mathbf{P}_{B}^{+} P B + P B − \mathbf{P}_{B}^{-} P B − δ B \delta B δ B B B B ( δ r ( B ) ) 2 (\delta r(B))^{2} ( δ r ( B ) ) 2 σ \sigma σ Ω \Omega Ω
Time-directed reverse Hölder and A ∞ A_{\infty} A ∞ [50 , 51 ] in connection with the doubly nonlinear equation ∂ t ( ∣ u ∣ p − 2 u ) − div ( ∣ ∇ u ∣ p − 2 ∇ u ) = 0 \partial_t(|u|^{p-2}u)-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0 ∂ t ( ∣ u ∣ p − 2 u ) − div ( ∣∇ u ∣ p − 2 ∇ u ) = 0 p = 2 p=2 p = 2 [49 , 47 , 48 ] and also [75 , 9 ] and the references therein. There, w w w P n \mathbb{P}^{n} P n (\fint_{R^{-}}w^{s})^{1/s}\le C\fint_{R^{+}}w controls the integrability of w w w R − R^{-} R − R R R R + R^{+} R + R − R^{-} R − P n \mathbb{P}^{n} P n [50 ] . In Definition 1.8, instead, both sides of (14) involve the same ball B B B P B + \mathbf{P}_{B}^{+} P B + ω B P B + \omega_{B}^{\mathbf{P}_{B}^{+}} ω B P B + A ∞ + ( σ ; κ , κ ′ ) A_{\infty}^{+}(\sigma;\kappa,\kappa') A ∞ + ( σ ; κ , κ ′ ) [49 ] , through Calderón–Zygmund decompositions and chaining arguments, which yield the self-improvement of parabolic reverse Hölder inequalities and several characterizations of the parabolic A ∞ A_{\infty} A ∞ [47 , 48 ] . Since σ \sigma σ ( n + 1 ) (n+1) ( n + 1 ) (14) only on balls with small boundary such that σ ( λ B ) ≤ β σ ( δ B ) \sigma(\lambda B)\le\beta\sigma(\delta B) σ ( λ B ) ≤ β σ ( δ B ) κ \kappa κ κ ′ \kappa' κ ′
Theorem 1.9. Let n ≥ 1 n\ge1 n ≥ 1 Ω ⊂ P n \Omega\subset\mathbb{P}^{n} Ω ⊂ P n σ \sigma σ ∂ Ω \partial\Omega ∂ Ω σ ( B ( X , r ) ) ≤ C 0 r n + 1 \sigma(B(X,r))\le C_{0}r^{n+1} σ ( B ( X , r )) ≤ C 0 r n + 1 X ∈ P n X\in\mathbb{P}^{n} X ∈ P n r > 0 r>0 r > 0
that ω ∈ A ∞ + ( σ ; κ , κ ′ ) \omega\in A^{+}_{\infty}(\sigma;\kappa,\kappa') ω ∈ A ∞ + ( σ ; κ , κ ′ ) ω ∗ ∈ A ∞ − ( σ ; κ , κ ′ ) \omega_* \in A^{-}_{\infty}(\sigma;\kappa,\kappa') ω ∗ ∈ A ∞ − ( σ ; κ , κ ′ ) κ ∈ ( 0 , 1 ) \kappa\in(0,1) κ ∈ ( 0 , 1 ) κ ′ ∈ ( 0 , 1 / 200 ] \kappa' \in(0,1/200] κ ′ ∈ ( 0 , 1/200 ] δ \delta δ n n n ϑ \vartheta ϑ β \beta β n n n δ \delta δ Λ \Lambda Λ T σ T_\sigma T σ L 2 ( σ ) L^2(\sigma) L 2 ( σ )
sup ε > 0 ∥ T σ , ε ∥ L 2 ( σ ) → L 2 ( σ ; R n ) < ∞ . \sup_{\varepsilon>0}\|T_{\sigma,\varepsilon}\|_{L^2(\sigma)\to L^2(\sigma;\mathbb{R}^n)}<\infty. ε > 0 sup ∥ T σ , ε ∥ L 2 ( σ ) → L 2 ( σ ; R n ) < ∞. The bound depends only on n n n C 0 \mathcal{C}_0 C 0 Definition 1.8 .
Theorem 1.9 is a parabolic analogue of the result [MT20, Theorem 1.1] by the second named author and X. Tolsa, where an A ∞ A_\infty A ∞ L 2 L^2 L 2 n n n R n + 1 \mathbb{R}^{n+1} R n + 1 T 1 T1 T 1 σ ( B ) ω x B ∣ B / ω x B ( B ) \sigma(B)\,\omega^{x_B}|_B/\omega^{x_B}(B) σ ( B ) ω x B ∣ B / ω x B ( B ) Theorem 1.3 . Since the testing measures are non-negative, (2) holds with C 1 = 1 \mathcal{C}_1=1 C 1 = 1 Definition 1.8 with κ ′ ≤ 1 / 200 \kappa'\leq1/200 κ ′ ≤ 1/200 (3) . In [MT20], the analogous condition is assumed with an arbitrary κ ′ ∈ ( 0 , 1 ) \kappa'\in(0,1) κ ′ ∈ ( 0 , 1 ) (4) takes care of the balls on which σ \sigma σ B B B T ∗ T^* T ∗ T T T Lemma 8.9 .
Finally, we remark that Theorem 1.9 is linked to quantitative rectifiability . Indeed, in the elliptic setting, the result in [MT20], combined with the solution of the codimension one David–Semmes problem by F. Nazarov, X. Tolsa, and A. Volberg [69 ] , shows that, if σ \sigma σ A ∞ A_\infty A ∞ supp σ \operatorname{supp}\sigma supp σ [32 ] . A parabolic analogue of this argument would require that, for an ( n + 1 ) (n+1) ( n + 1 ) σ \sigma σ P n \mathbb{P}^n P n L 2 ( σ ) L^2(\sigma) L 2 ( σ ) T σ T_\sigma T σ supp σ \operatorname{supp}\sigma supp σ T σ T_\sigma T σ T σ T_\sigma T σ Theorem 1.9 , would show that, under this condition, the assumptions of Theorem 1.9 imply parabolic uniform rectifiability, see also § 1.4 .
Discussion of the proof, main ideas, and outline of the paper
In the case of a singular integral operator that is not necessarily antisymmetric, the natural assumptions for a local T b Tb T b T T T T ∗ T^* T ∗ Theorem 1.3 , each of the families { v B } B \{v_B\}_B { v B } B { v B ∗ } B \{v_B^*\}_B { v B ∗ } B [70 , 72 ] cannot be applied directly. The approach to non-homogeneous local T b Tb T b [60 ] , which combines a non-homogeneous good-lambda method, the big pieces T b Tb T b [70 ] , and a change of measure argument, was implemented there for antisymmetric kernels, and it was pointed out in [62 ] that the general case involves genuine difficulties. The main achievement of our proof is to make these two a priori separate structures interact. To this end, we develop a framework that brings them into a single tree-wise analysis by means of suppressed singular integral operators and a mixed corona decomposition . Its key ingredient is a tree-wise L 2 L^{2} L 2
We first describe the overall strategy. After regularizing the kernel at small and large scales, which makes all the operators a priori bounded and allows us to absorb small multiples of their norms, see § 3.1 , the proof consists of two steps. In the first one, which is the core of the paper, we prove that, for every ball B 0 ∈ B B_{0}\in\mathcal{B} B 0 ∈ B g ∈ L ∞ ( μ ) g\in L^{\infty}(\mu) g ∈ L ∞ ( μ ) B 0 B_{0} B 0
∫ B 0 ( T μ , ♯ g + ( T ∗ ) μ , ♯ g ) d μ ≲ ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) , \int_{B_{0}}\left(T_{\mu,\sharp}g+(T^{*})_{\mu,\sharp}g\right)\,d\mu\lesssim\lVert g\rVert_{L^{\infty}(\mu)}\mu(B_{0}), ∫ B 0 ( T μ , ♯ g + ( T ∗ ) μ , ♯ g ) d μ ≲ ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) , see (301) . This estimate resembles a local T 1 T1 T 1 L 1 L^{1} L 1 T μ , ♯ χ B 0 T_{\mu,\sharp}\chi_{B_{0}} T μ , ♯ χ B 0 ( T ∗ ) μ , ♯ χ B 0 (T^{*})_{\mu,\sharp}\chi_{B_{0}} ( T ∗ ) μ , ♯ χ B 0 T 1 T1 T 1 L ∞ L^{\infty} L ∞
At this point, the generality of our testing assumptions plays a role in the proof itself. By Chebyshev’s inequality, the measures μ ∣ B \mu|_{B} μ ∣ B L ∞ L^{\infty} L ∞ (6) , and, in the second step, we run our construction once more with these testing measures. In this way, we find, in every doubling David-Mattila cell Q Q Q G Q G_{Q} G Q μ ( G Q ) ≳ μ ( Q ) \mu(G_{Q})\gtrsim\mu(Q) μ ( G Q ) ≳ μ ( Q ) T μ ∣ G Q T_{\mu|_{G_{Q}}} T μ ∣ G Q L 2 ( μ ∣ G Q ) L^{2}(\mu|_{G_{Q}}) L 2 ( μ ∣ G Q ) L 2 ( μ ) L^{2}(\mu) L 2 ( μ ) T T T
Our techniques involve three different geometric objects, and part of the difficulty is to make them compatible with one another.
Metric balls. The testing conditions are formulated on balls, and we work with the testing balls B ~ ( R ) ∈ B \widetilde{B}(R)\in\mathcal{B} B ( R ) ∈ B R R R § 9 .§ 1 . All the exceptional families associated with density, growth, and the operator are defined by conditions on balls, and the weak boundedness property is proved on random balls with small boundary. This allows us to carry out the whole argument in geometrically doubling metric spaces.
Random dyadic systems D ( w ) \mathbb{D}(w) D ( w ) [39 ] , which depend on a parameter w w w § 6 and § 7 .
A fixed David-Mattila lattice D 0 \mathcal{D}_{0} D 0 μ \mu μ T T T T ∗ T^{*} T ∗
The first difficulty concerns the testing measures themselves. We write ν B = b ^ B ∣ ν B ∣ \nu_{B}=\widehat{b}_{B}\lvert\nu_{B}\rvert ν B = b B ∣ ν B ∣ ∣ b ^ B ∣ = 1 \lvert\widehat{b}_{B}\rvert=1 ∣ b B ∣ = 1 polar decomposition of the testing measure. Thus, the testing measure is replaced by the bounded function b ^ B \widehat{b}_{B} b B B B B μ \mu μ ∣ ν B ∣ \lvert\nu_{B}\rvert ∣ ν B ∣ μ \mu μ μ \mu μ L p ( μ ) L^{p}(\mu) L p ( μ ) p > 1 p>1 p > 1
There is a further difficulty in the operator testing. For each η ∈ ( 0 , 1 / 10 ) \eta\in(0,1/10) η ∈ ( 0 , 1/10 ) B B B (5) holds, there is an exceptional set U B = U B ( η ) ⊂ B U_{B}=U_{B}(\eta)\subset B U B = U B ( η ) ⊂ B μ ( U B ) < η μ ( B ) \mu(U_{B})<\eta\mu(B) μ ( U B ) < η μ ( B ) T ♯ ν ^ B ≤ E 2 ( η ) T_{\sharp}\widehat{\nu}_{B}\leq\mathcal{E}_{2}(\eta) T ♯ ν B ≤ E 2 ( η ) ( 1 − τ ) B ∖ U B (1-\tau)B\setminus U_{B} ( 1 − τ ) B ∖ U B η \eta η E 2 ( η ) \mathcal{E}_{2}(\eta) E 2 ( η ) U B U_{B} U B B ∖ ( 1 − τ ) B B\setminus(1-\tau)B B ∖ ( 1 − τ ) B (4) holds instead. Recovering the required growth, comparability, accretivity, and testing properties on a quantitatively large set is one of the main steps of our argument.
In § 3 , we construct, for each testing measure, an exceptional set outside of which ν ^ B \widehat{\nu}_B ν B ∣ ν ^ B ∣ |\widehat{\nu}_B| ∣ ν B ∣ μ \mu μ ∣ ν ^ B ∣ |\widehat{\nu}_B| ∣ ν B ∣ ν ^ B \widehat{\nu}_B ν B (3) and the L 0 L^0 L 0 (6) imply that this set, together with the annulus B ∖ ( 1 − τ ) B B \setminus(1-\tau)B B ∖ ( 1 − τ ) B ν ^ B \widehat{\nu}_B ν B F F F ∣ ν ^ B ∣ ( F ) ≳ ∥ ν ^ B ∥ |\widehat{\nu}_B|(F) \gtrsim\|\widehat{\nu}_B\| ∣ ν B ∣ ( F ) ≳ ∥ ν B ∥
Then, as in [70 ] , we suppress the kernel at the scale Ξ ( x ) : = dist ( x , F ) \Xi(x) := \operatorname{dist}(x,F) Ξ ( x ) := dist ( x , F )
K Ξ ( x , y ) : = K ( x , y ) d ( x , y ) 2 n d ( x , y ) 2 n + Ξ ( x ) n Ξ ( y ) n . K_{\Xi}(x,y) := K(x,y)\frac{d(x,y)^{2n}}{d(x,y)^{2n}+\Xi(x)^n\Xi(y)^n}. K Ξ ( x , y ) := K ( x , y ) d ( x , y ) 2 n + Ξ ( x ) n Ξ ( y ) n d ( x , y ) 2 n . This product-type suppression applies to general Calderón-Zygmund kernels on metric spaces, see Lemma 3.13 and Appendix B, and the suppressed operator satisfies the testing condition on the whole ball, see Lemma 3.15 .
Next, we implement the coronization argument. The exceptional sets depend on the random parameter w w w [70 ] , we replace them by deterministic suppressions which still vanish on a set of measure comparable to the measure of the ball, see Lemmas 4.2 and 4.3. For each testing system, we then construct a corona decomposition on D 0 \mathcal{D}_0 D 0
If the concentration alternative (4) in the testing dichotomy holds on the testing ball of a cell R R R R R R η = η ∗ \eta=\eta_* η = η ∗ (101) gives a packing condition for the densities of the top cells.
Otherwise, we stop at the maximal cells whose size is small compared with the deterministic suppression, which satisfy a Carleson packing condition.
In § 4.3 , we merge the two corona decompositions into a single mixed corona decomposition by intersecting their trees, and we reduce the local estimate above to uniform L 2 L^2 L 2
This is the step in which the two testing systems start to interact. In particular, each testing system is only used on its own good set, and no lower bound on the measure of the set where both systems are accretive is needed.
The second main difficulty arises on the trees of the mixed corona decomposition. Each tree is associated with two nested David-Mattila cells R 1 R_1 R 1 R 2 R_2 R 2 B 1 : = B ~ ( R 1 ) B_1:=\widetilde{B}(R_1) B 1 := B ( R 1 ) B 2 : = B ~ ( R 2 ) B_2:=\widetilde{B}(R_2) B 2 := B ( R 2 ) ν ^ 1 \widehat{\nu}_1 ν 1 T T T ν ^ 2 \widehat{\nu}_2 ν 2 T ∗ T^* T ∗ T b T b T b L 2 ( ∣ ν ^ 1 ∣ ) L^2(|\widehat{\nu}_1|) L 2 ( ∣ ν 1 ∣ ) L 2 ( ∣ ν ^ 2 ∣ ) L^2(|\widehat{\nu}_2|) L 2 ( ∣ ν 2 ∣ ) μ \mu μ
To transfer bounds between these spaces and L 2 ( μ ) L^2(\mu) L 2 ( μ ) § 3.6 a Cotlar inequality for different measures , Lemma 3.16 , which extends [77 Lemmas 5.25–5.27] : if a suppressed operator is bounded from L 2 ( m 1 ) L^2(\mathfrak{m}_1) L 2 ( m 1 ) L 2 ( m 2 ) L^2(\mathfrak{m}_2) L 2 ( m 2 ) L 2 ( m 1 ) L^2(\mathfrak{m}_1) L 2 ( m 1 ) L 2 ( m 3 ) L^2(\mathfrak{m}_3) L 2 ( m 3 ) μ \mu μ ∣ ν 1 ∣ |\nu_1| ∣ ν 1 ∣ ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ ν i : = ν ^ i + μ ∣ B i \nu_i:=\widehat{\nu}_i+\mu|_{B_i} ν i := ν i + μ ∣ B i § 5 , we use this inequality to bound the norm on L 2 ( μ ) L^2(\mu) L 2 ( μ ) B 2 × B 1 B_2\times B_1 B 2 × B 1 L 2 ( ∣ ν ^ 1 ∣ ) L^2(|\widehat{\nu}_1|) L 2 ( ∣ ν 1 ∣ ) L 2 ( ∣ ν ^ 2 ∣ ) L^2(|\widehat{\nu}_2|) L 2 ( ∣ ν 2 ∣ ) Lemma 5.3 . The latter norm is estimated in the Key Lemma 6.7, which is the heart of the paper. Its proof is inspired by the arguments of [70 ] , [77 Chapter 5] , and [63 ] , with two independent random dyadic systems and martingale differences adapted to ∣ ν i ∣ |\nu_i| ∣ ν i ∣ b ^ i \widehat{b}_i b i
The possible imbalance between B 1 B_1 B 1 B 2 B_2 B 2 preliminary randomization which, with large probability, allows us to work with good top cubes containing both balls, see (172) and Lemma 6.11.
Moreover, since K K K § 6.4.1 for balls with small boundary with respect to both ∣ ν 1 ∣ |\nu_1| ∣ ν 1 ∣ ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ ν ^ 1 \widehat{\nu}_1 ν 1 auxiliary parameters that avoids circular arguments, see Table 1 .
Table 1. Main parameters of the proof. The parameters m m m λ \lambda λ υ \upsilon υ a a a (287) is at most 1 / 2 1/2 1/2
Finally, in § 7.1 , we combine the tree-wise estimate with a probabilistic averaging argument over the random suppressions. The norms of the operators with random and deterministic suppressions are controlled by the norm of the averaged operator, again by means of Lemma 3.16, see Lemma 7.2, and the Key Lemma bounds the latter by a constant plus a small multiple of itself, see Lemma 7.3. This gives the tree-wise bounds, uniformly over the trees and the truncations, and hence the local estimate above. The second step and the good-lambda argument of § 7.2 conclude the proof of Theorem 1.3 in § 7.3 . The preliminaries are collected in Section 2, Theorem 1.9 is proved in Section 8, and the appendices contain the constructions, in geometrically doubling metric spaces, of the David-Mattila testing balls, of random and Whitney-type cubes, and of the suppressed kernels.
For the reader’s convenience, we collect the main parameters of the proof in Table 1 .
Related results
While we were at an advanced stage in the preparation of this manuscript, X. Tolsa informed the second named author that M. Fernàndez-Vilaseca simultaneously and independently proved, in [23 ] , a big pieces T b Tb T b
We also announce an upcoming paper by John Hoffman. He proves [HJ25, Conjecture 1.3], which is a parabolic analogue of the codimension one David-Semmes problem , see [21 , 69 ] . More precisely, let μ \mu μ ( n + 1 ) (n+1) ( n + 1 ) R n \mathbb{R}^n R n L 2 ( μ ) L^2(\mu) L 2 ( μ ) μ \mu μ higher spatial dimension condition ), see [HJ25, Section 3]. The latter means that the balls on which μ \mu μ L n ⊗ γ \mathcal{L}^n \otimes\gamma L n ⊗ γ γ \gamma γ supp μ \operatorname{supp}\mu supp μ [34 , 35 ] , that is, the β 2 \beta_2 β 2 μ \mu μ
He will also show that, if σ \sigma σ σ \sigma σ supp σ \operatorname{supp}\sigma supp σ
Acknowledgments
The main ideas and key arguments of this paper were developed during the second-named author’s visit to the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Vienna, for the thematic programme “Free Boundary Problems” in the fall of 2025. He thanks the institute for its hospitality.
AI use
The proof strategy, the mathematical arguments, and an advanced draft of this paper were developed by the authors, with AI used only for editorial assistance. In the final stages of the preparation of the paper, the authors used LLM tools to check the proofs in detail, to proofread the text, to improve the presentation, and to generate Table 1 . The proof checks helped the authors detect and correct some small gaps and inaccuracies. The authors take full responsibility for the content of the manuscript.
Preliminaries and notation
General notation and definitions
Given a , b > 0 a,b>0 a , b > 0 a ≲ b a\lesssim b a ≲ b C > 0 C>0 C > 0 a ≤ C b a\leq Cb a ≤ C b a ≲ t b a\lesssim_t b a ≲ t b C C C t t t a ≈ b a\approx b a ≈ b a ≲ b ≲ a a\lesssim b\lesssim a a ≲ b ≲ a a ≈ t b a\approx_t b a ≈ t b
For A ⊂ X A\subset X A ⊂ X χ A \chi_A χ A M ( X ; C ) M(X;\mathbb{C}) M ( X ; C ) X X X ν ∈ M ( X ; C ) \nu\in M(X;\mathbb{C}) ν ∈ M ( X ; C ) ∥ ν ∥ : = ∣ ν ∣ ( X ) \lVert\nu\rVert:=\lvert\nu\rvert(X) ∥ ν ∥ := ∣ ν ∣ ( X ) M + ( X ) M_+(X) M + ( X ) X X X m ∈ M + ( X ) m\in M_+(X) m ∈ M + ( X ) L l o c 1 ( m ) L^1_{\mathrm{loc}}(m) L loc 1 ( m ) m m m
We recall, see for instance [41 Lemma 2.2] , that, if ( X , d ) (X,d) ( X , d ) D D D B ( x , r ) ⊂ X B(x,r)\subset X B ( x , r ) ⊂ X x i x_i x i D α − log 2 D D\alpha^{-\log_2 D} D α − l o g 2 D B ( x i , α r ) B(x_i,\alpha r) B ( x i , α r ) α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] d : = log 2 D \mathfrak{d}:=\log_2 D d := log 2 D X X X
A ball B ( x , r ) B(x,r) B ( x , r ) m m m ( a , β ) (a,\beta) ( a , β )
m ( B ( x , a r ) ) ≤ β m ( B ( x , r ) ) , (16) m(B(x,ar))\leq\beta m(B(x,r)), \tag*{(16)} m ( B ( x , a r )) ≤ β m ( B ( x , r )) , (16) and we denote by B ( m ; a , β ) \mathcal{B}(m;a,\beta) B ( m ; a , β ) m m m ( a , β ) (a,\beta) ( a , β ) X X X
Remark 2.1. If B ⊂ X B\subset X B ⊂ X (7) for a measure μ \mu μ B B B μ \mu μ ( 5 , β ) (5,\beta) ( 5 , β ) μ \mu μ ( 100 , β ) (100,\beta) ( 100 , β )
μ ( 5 B ) ≤ μ ( 100 B ) ≤ μ ( Λ B ) ≤ β μ ( δ B ) ≤ β μ ( B ) . \mu(5B)\leq\mu(100B)\leq\mu(\Lambda B)\leq\beta\mu(\delta B)\leq\beta\mu(B). μ ( 5 B ) ≤ μ ( 100 B ) ≤ μ ( Λ B ) ≤ β μ ( δ B ) ≤ β μ ( B ) . Let n n n (1) . Given a ball B ⊂ X B\subset X B ⊂ X
Θ m ( B ) : = m ( B ) r ( B ) n . (17) \Theta_m(B):=\frac{m(B)}{r(B)^n}. \tag*{(17)} Θ m ( B ) := r ( B ) n m ( B ) . (17) Given a > 0 a>0 a > 0 m m m X X X B ⊂ X B\subset X B ⊂ X x B x_B x B a a a λ > 0 \lambda>0 λ > 0
m ( { x ∈ 5 B : ∣ dist ( x , x B ) − r ( B ) ∣ ≤ λ r ( B ) } ) ≤ a λ m ( 5 B ) . (18) m\left(\left\{x\in5B:\lvert\operatorname{dist}(x,x_B)-r(B)\rvert\leq\lambda r(B)\right\}\right)\leq a\lambda m(5B). \tag*{(18)} m ( { x ∈ 5 B : ∣ dist ( x , x B ) − r ( B )∣ ≤ λ r ( B ) } ) ≤ aλm ( 5 B ) . (18) We remark that, for 0 < λ ≤ 1 0<\lambda\leq1 0 < λ ≤ 1 B ∖ ( 1 − λ ) B B\setminus(1-\lambda)B B ∖ ( 1 − λ ) B ( 1 + λ ) B ∖ B (1+\lambda)B\setminus B ( 1 + λ ) B ∖ B
{ x ∈ 5 B : ∣ dist ( x , x B ) − r ( B ) ∣ ≤ λ r ( B ) } . \left\{x\in5B:\lvert\operatorname{dist}(x,x_B)-r(B)\rvert\leq\lambda r(B)\right\}. { x ∈ 5 B : ∣ dist ( x , x B ) − r ( B )∣ ≤ λ r ( B ) } . Lemma 2.2. Let m m m X X X x ∈ X x\in X x ∈ X r > 0 r>0 r > 0 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1 ρ ∈ [ r , ( 1 + ε ) r ] \rho\in[r,(1+\varepsilon)r] ρ ∈ [ r , ( 1 + ε ) r ] B ( x , ρ ) B(x,\rho) B ( x , ρ ) C ε C_\varepsilon C ε m m m C ε : = 20 ( 1 + ε ) / ε C_\varepsilon:=20(1+\varepsilon)/\varepsilon C ε := 20 ( 1 + ε ) / ε
Proof. We denote B : = B ( x , r ) B:=B(x,r) B := B ( x , r ) B ~ : = B ( x , 5 r ) \widetilde{B}:=B(x,5r) B := B ( x , 5 r ) ϕ x : X → R \phi_x:X\to\mathbb{R} ϕ x : X → R ϕ x ( y ) : = d ( x , y ) \phi_x(y):=d(x,y) ϕ x ( y ) := d ( x , y ) σ x : = ϕ x ♯ ( m ∣ B ~ ) \sigma_x:=\phi_{x\sharp}(m|_{\widetilde{B}}) σ x := ϕ x ♯ ( m ∣ B ) R \mathbb{R} R
We denote by L 1 L^1 L 1 R \mathbb{R} R M ( R ) → L 1 , ∞ ( L 1 ) M(\mathbb{R})\to L^{1,\infty}(L^1) M ( R ) → L 1 , ∞ ( L 1 )
M σ x ( s ) : = sup t > 0 σ x ( ( s − t , s + t ) ) 2 t , for s ∈ R , M\sigma_x(s):=\sup_{t>0}\frac{\sigma_x((s-t,s+t))}{2t},\qquad\text{for }s\in\mathbb{R}, M σ x ( s ) := t > 0 sup 2 t σ x (( s − t , s + t )) , for s ∈ R , we have
L 1 ( { s ∈ R : M σ x ( s ) > λ } ) ≤ 5 λ ∥ σ x ∥ . (19) L^1\left(\left\{s\in\mathbb{R}:M\sigma_x(s)>\lambda\right\}\right)\leq\frac{5}{\lambda}\lVert\sigma_x\rVert. \tag*{(19)} L 1 ( { s ∈ R : M σ x ( s ) > λ } ) ≤ λ 5 ∥ σ x ∥ . (19) If m ( B ~ ) = 0 \mathfrak{m}(\widetilde{B})=0 m ( B ) = 0 ρ ∈ [ r , ( 1 + ε ) r ] \rho\in[r,(1+\varepsilon)r] ρ ∈ [ r , ( 1 + ε ) r ] m ( B ~ ) > 0 \mathfrak{m}(\widetilde{B})>0 m ( B ) > 0 λ = 10 m ( B ~ ) / ( ε r ) \lambda=10\mathfrak{m}(\widetilde{B})/(\varepsilon r) λ = 10 m ( B ) / ( ε r ) (19) yields
L 1 ( { s ∈ [ r , ( 1 + ε ) r ] : M σ x ( s ) > λ } ) ≤ ε r 2 , \mathcal{L}^{1}\left(\left\{s\in[r,(1+\varepsilon)r]:M\sigma_x(s)>\lambda\right\}\right)\leq\frac{\varepsilon r}{2}, L 1 ( { s ∈ [ r , ( 1 + ε ) r ] : M σ x ( s ) > λ } ) ≤ 2 ε r , so there exists t ∈ [ r , ( 1 + ε ) r ] t\in[r,(1+\varepsilon)r] t ∈ [ r , ( 1 + ε ) r ] M σ x ( t ) ≤ λ M\sigma_x(t)\leq\lambda M σ x ( t ) ≤ λ □ \square □
We remark that the proof of Lemma 2.2 actually implies that the set of radii ρ \rho ρ L 1 \mathcal{L}^{1} L 1 ε r / 2 \varepsilon r/2 ε r /2 ρ \rho ρ ( r , ( 1 + ε ) r ) (r,(1+\varepsilon)r) ( r , ( 1 + ε ) r ) m 1 \mathfrak{m}_1 m 1 m 2 \mathfrak{m}_2 m 2 λ = 20 m i ( B ~ ) / ( ε r ) \lambda=20\mathfrak{m}_i(\widetilde{B})/(\varepsilon r) λ = 20 m i ( B ) / ( ε r ) i = 1 , 2 i=1,2 i = 1 , 2 ρ ∈ ( r , ( 1 + ε ) r ) \rho\in(r,(1+\varepsilon)r) ρ ∈ ( r , ( 1 + ε ) r ) B ( x , ρ ) B(x,\rho) B ( x , ρ ) 2 C ε 2\mathcal{C}_\varepsilon 2 C ε m 1 \mathfrak{m}_1 m 1 m 2 \mathfrak{m}_2 m 2
For G ⊂ X G\subset X G ⊂ X m ∣ G \mathfrak{m}|_G m ∣ G m \mathfrak{m} m G G G p ∈ [ 1 , ∞ ] p\in[1,\infty] p ∈ [ 1 , ∞ ] L p ( G , m ) : = L p ( m ∣ G ) L^p(G,\mathfrak{m}):=L^p(\mathfrak{m}|_G) L p ( G , m ) := L p ( m ∣ G ) m \mathfrak{m} m f , g : X → C f,g:X\to\mathbb{C} f , g : X → C f g ∈ L 1 ( m ) fg\in L^1(\mathfrak{m}) f g ∈ L 1 ( m )
⟨ f , g ⟩ m : = ∫ f g d m (20) \langle f,g\rangle_{\mathfrak{m}}:=\int fg\,\mathrm{d}\mathfrak{m} \tag*{(20)} ⟨ f , g ⟩ m := ∫ f g d m (20) and, if E ⊂ X E\subset X E ⊂ X 0 < m ( E ) < ∞ 0<\mathfrak{m}(E)<\infty 0 < m ( E ) < ∞
\fint_E f\,\mathrm{d}\mathfrak{m}:=\frac{1}{\mathfrak{m}(E)}\int_E f\,\mathrm{d}\mathfrak{m}.
In particular, T ∗ T^* T ∗ T T T
Maximal functions
Let m \mathfrak{m} m X X X n ∈ M ( X ; C ) \mathfrak{n}\in M(X;\mathbb{C}) n ∈ M ( X ; C ) non-centered dilated Hardy–Littlewood maximal function as
M ~ m n ( x ) : = sup B ∋ x ∣ n ∣ ( B ) m ( 5 B ) (21) \widetilde{\mathcal{M}}_{\mathfrak{m}\mathfrak{n}}(x):=\sup_{B\ni x}\frac{|\mathfrak{n}|(B)}{\mathfrak{m}(5B)} \tag*{(21)} M mn ( x ) := B ∋ x sup m ( 5 B ) ∣ n ∣ ( B ) (21) for x ∈ X x\in X x ∈ X
For f ∈ L l o c 1 ( m ) f\in L^1_{\mathrm{loc}}(\mathfrak{m}) f ∈ L loc 1 ( m ) M ~ m f : = M ~ m ( f m ) \widetilde{\mathcal{M}}_{\mathfrak{m}}f:=\widetilde{\mathcal{M}}_{\mathfrak{m}}(f\mathfrak{m}) M m f := M m ( f m ) M ( X ; C ) M(X;\mathbb{C}) M ( X ; C ) L 1 , ∞ ( m ) L^{1,\infty}(\mathfrak{m}) L 1 , ∞ ( m ) L p ( m ) L^p(\mathfrak{m}) L p ( m ) L p ( m ) L^p(\mathfrak{m}) L p ( m ) p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) D D D [37 Proposition 3.5] , or [77 Theorem 9.32] for its Euclidean analogue. In particular, by the 5 r 5r 5 r B ′ B' B ′ ∣ n ∣ ( B ′ ) > λ m ( 5 B ′ ) |\mathfrak{n}|(B')>\lambda\mathfrak{m}(5B') ∣ n ∣ ( B ′ ) > λ m ( 5 B ′ ) λ > 0 \lambda>0 λ > 0
m ( { M ~ m n > λ } ) ≤ ∥ n ∥ λ and m ( { M n r > λ } ) ≤ 5 n C 0 ∥ n ∥ λ , (22) \mathfrak{m}\left(\left\{\widetilde{\mathcal{M}}_{\mathfrak{m}\mathfrak{n}}>\lambda\right\}\right)\leq\frac{\|\mathfrak{n}\|}{\lambda}
\qquad\text{and}\qquad
\mathfrak{m}\left(\left\{M^r_{\mathfrak{n}}>\lambda\right\}\right)\leq\frac{5^n\mathcal{C}_0\|\mathfrak{n}\|}{\lambda}, \tag*{(22)} m ( { M mn > λ } ) ≤ λ ∥ n ∥ and m ( { M n r > λ } ) ≤ λ 5 n C 0 ∥ n ∥ , (22) where, in the second estimate, m \mathfrak{m} m n n n C 0 \mathcal{C}_0 C 0 m ( B ( x , 5 ρ ) ) ≤ 5 n C 0 ρ n \mathfrak{m}(B(x,5\rho))\leq5^n\mathcal{C}_0\rho^n m ( B ( x , 5 ρ )) ≤ 5 n C 0 ρ n
We write
M n r ( x ) : = sup ρ > 0 ∣ n ∣ ( B ( x , ρ ) ) ρ n , M^r_{\mathfrak{n}}(x):=\sup_{\rho>0}\frac{|\mathfrak{n}|(B(x,\rho))}{\rho^n}, M n r ( x ) := ρ > 0 sup ρ n ∣ n ∣ ( B ( x , ρ )) , for x ∈ X x\in X x ∈ X n n n m ∈ M + ( X ) \mathfrak{m}\in M_+(X) m ∈ M + ( X ) n n n C 0 > 0 \mathcal{C}_0>0 C 0 > 0 M n r M^r_{\mathfrak{n}} M n r M ( X ; C ) M(X;\mathbb{C}) M ( X ; C ) L 1 , ∞ ( m ) L^{1,\infty}(\mathfrak{m}) L 1 , ∞ ( m ) C 0 \mathcal{C}_0 C 0 D D D [77 p. 53] for the proof in Euclidean spaces. For f ∈ L l o c 1 ( m ) f\in L^1_{\mathrm{loc}}(\mathfrak{m}) f ∈ L loc 1 ( m ) n n n M m r f : = M n r ( f m ) M^r_{\mathfrak{m}}f:=M^r_{\mathfrak{n}}(f\mathfrak{m}) M m r f := M n r ( f m )
David-Mattila-type cells
The following lemma summarizes the properties of the lattice of dyadic “cubes” adapted to a compactly supported Radon measure σ \sigma σ σ \sigma σ [20 Theorem 3.2] , for measures on the Euclidean space, and further extended to the setting of geometrically doubling spaces by A. Volberg and P. Zorin-Kranich in [81 Lemma 2.7] .2
Lemma 2.3. Let ( X , d ) (X,d) ( X , d ) D D D d : = log 2 ( D ) \mathsf{d}:=\log_2(D) d := log 2 ( D ) σ \sigma σ X X X W : = supp ( σ ) W:=\operatorname{supp}(\sigma) W := supp ( σ ) K 0 > 1 K_0>1 K 0 > 1 A 0 > 5000 K 0 A_0>5000K_0 A 0 > 5000 K 0 ( D σ , k ) k ≥ 0 (\mathcal{D}_{\sigma,k})_{k\geq0} ( D σ , k ) k ≥ 0 W W W Q Q Q σ \sigma σ
For each integer k ≥ 0 k\geq0 k ≥ 0 W W W σ \sigma σ Q Q Q Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k k < ℓ k<\ell k < ℓ Q ∈ D σ , ℓ Q\in\mathcal{D}_{\sigma,\ell} Q ∈ D σ , ℓ R ∈ D σ , k R\in\mathcal{D}_{\sigma,k} R ∈ D σ , k Q ∩ R = ∅ Q\cap R=\varnothing Q ∩ R = ∅ Q ⊂ R Q\subset R Q ⊂ R
For each k ≥ 0 k\geq0 k ≥ 0 σ \sigma σ Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k B ( Q ) = B ( z Q , r ( Q ) ) B(Q)=B(z_Q,r(Q)) B ( Q ) = B ( z Q , r ( Q ))
z Q ∈ W , A 0 − k diam ( W ) ≤ r ( Q ) ≤ K 0 A 0 − k diam ( W ) , z_Q\in W,\qquad A_0^{-k}\operatorname{diam}(W)\leq r(Q)\leq K_0A_0^{-k}\operatorname{diam}(W), z Q ∈ W , A 0 − k diam ( W ) ≤ r ( Q ) ≤ K 0 A 0 − k diam ( W ) , W ∩ B ( Q ) ⊂ Q ⊂ W ∩ 28 B ( Q ) = W ∩ B ( z Q , 28 r ( Q ) ) , W\cap B(Q)\subset Q\subset W\cap28B(Q)=W\cap B(z_Q,28r(Q)), W ∩ B ( Q ) ⊂ Q ⊂ W ∩ 28 B ( Q ) = W ∩ B ( z Q , 28 r ( Q )) , and the balls 5 B ( Q ) 5B(Q) 5 B ( Q ) Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k
The σ \sigma σ Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k ℓ ≥ 0 \ell\geq0 ℓ ≥ 0
N ℓ i n t ( Q ) : = { x ∈ Q : dist ( x , W ∖ Q ) < A 0 − k − ℓ diam ( W ) } , N_\ell^{\mathrm{int}}(Q):=\{x\in Q:\operatorname{dist}(x,W\setminus Q)<A_0^{-k-\ell}\operatorname{diam}(W)\}, N ℓ int ( Q ) := { x ∈ Q : dist ( x , W ∖ Q ) < A 0 − k − ℓ diam ( W )} , N ℓ e x t ( Q ) : = { x ∈ W ∖ Q : dist ( x , Q ) < A 0 − k − ℓ diam ( W ) } , N_\ell^{\mathrm{ext}}(Q):=\{x\in W\setminus Q:\operatorname{dist}(x,Q)<A_0^{-k-\ell}\operatorname{diam}(W)\}, N ℓ ext ( Q ) := { x ∈ W ∖ Q : dist ( x , Q ) < A 0 − k − ℓ diam ( W )} , and N ℓ ( Q ) : = N ℓ i n t ( Q ) ∪ N ℓ e x t ( Q ) N_\ell(Q):=N_\ell^{\mathrm{int}}(Q)\cup N_\ell^{\mathrm{ext}}(Q) N ℓ ( Q ) := N ℓ int ( Q ) ∪ N ℓ ext ( Q )
σ ( N ℓ ( Q ) ) ≤ ϑ D M ℓ σ ( 90 B ( Q ) ) , for ϑ D M : = C D K 0 3 d + 1 A 0 . \sigma(N_\ell(Q))\leq\vartheta_{DM}^{\ell}\sigma(90B(Q)),\qquad\text{for }\vartheta_{DM}:=\frac{C_DK_0^{3\mathsf{d}+1}}{A_0}. σ ( N ℓ ( Q )) ≤ ϑ D M ℓ σ ( 90 B ( Q )) , for ϑ D M := A 0 C D K 0 3 d + 1 . Here C D C_D C D
Let us denote by D σ , k d b \mathcal{D}_{\sigma,k}^{\mathrm{db}} D σ , k db σ \sigma σ Q ∈ D σ , k Q\in\mathcal{D}_{\sigma,k} Q ∈ D σ , k
σ ( 100 B ( Q ) ) ≤ K 0 σ ( B ( Q ) ) . (23) \sigma(100B(Q))\leq K_0\sigma(B(Q)). \tag*{(23)} σ ( 100 B ( Q )) ≤ K 0 σ ( B ( Q )) . (23) For Q ∈ D σ , k ∖ D σ , k d b Q\in\mathcal{D}_{\sigma,k}\setminus\mathcal{D}_{\sigma,k}^{\mathrm{db}} Q ∈ D σ , k ∖ D σ , k db r ( Q ) = A 0 − k diam ( W ) r(Q)=A_0^{-k}\operatorname{diam}(W) r ( Q ) = A 0 − k diam ( W ) ℓ ≥ 1 \ell\geq1 ℓ ≥ 1 100 ℓ ≤ K 0 100^\ell\leq K_0 10 0 ℓ ≤ K 0 Q ∈ D σ , k ∖ D σ , k d b Q\in\mathcal{D}_{\sigma,k}\setminus\mathcal{D}_{\sigma,k}^{\mathrm{db}} Q ∈ D σ , k ∖ D σ , k db
σ ( 100 B ( Q ) ) ≤ K 0 − ℓ σ ( 100 ℓ + 1 B ( Q ) ) . (24) \sigma(100B(Q))\leq K_0^{-\ell}\sigma(100^{\ell+1}B(Q)). \tag*{(24)} σ ( 100 B ( Q )) ≤ K 0 − ℓ σ ( 10 0 ℓ + 1 B ( Q )) . (24) Finally, we denote
D σ : = ⋃ k ≥ 0 D σ , k , D σ d b : = ⋃ k ≥ 0 D σ , k d b , \mathcal{D}_\sigma:=\bigcup_{k\geq0}\mathcal{D}_{\sigma,k},\qquad\mathcal{D}_\sigma^{\mathrm{db}}:=\bigcup_{k\geq0}\mathcal{D}_{\sigma,k}^{\mathrm{db}}, D σ := k ≥ 0 ⋃ D σ , k , D σ db := k ≥ 0 ⋃ D σ , k db , and choose A 0 A_0 A 0
A 0 > 5000 K 0 and ϑ D M < 1. (25) A_0>5000K_0\qquad\text{and}\qquad\vartheta_{DM}<1. \tag*{(25)} A 0 > 5000 K 0 and ϑ D M < 1. (25) Remark 2.4. No small-boundary property is guaranteed for balls B ( Q ) B(Q) B ( Q ) Q Q Q B \mathcal{B} B Theorem 1.3 , so no testing measures are associated with them. This difficulty is overcome in Appendix § 9 .§ 1 by introducing some balls B ~ ( Q ) \widetilde{B}(Q) B ( Q ) Q Q Q
For measures with possibly unbounded support, we need a construction of the David-Mattila lattice with radii comparable to s A 0 − k sA_0^{-k} s A 0 − k k ≥ 0 k \ge0 k ≥ 0 s > 0 s > 0 s > 0
Lemma 2.5 (Auxiliary DM-lattice D σ ( s ) \mathcal{D}^{(s)}_{\sigma} D σ ( s ) ( X , d ) (X,d) ( X , d ) D D D d : = log 2 ( D ) \mathfrak{d} := \log_2(D) d := log 2 ( D ) s > 0 s > 0 s > 0 σ \sigma σ X X X W : = supp ( σ ) W := \operatorname{supp}(\sigma) W := supp ( σ ) K 0 > 1 K_0 > 1 K 0 > 1 A 0 > 5000 K 0 A_0 > 5000K_0 A 0 > 5000 K 0 ( D σ , k ( s ) ) k ≥ 0 (\mathcal{D}^{(s)}_{\sigma,k})_{k \ge0} ( D σ , k ( s ) ) k ≥ 0 W W W Q Q Q
For each integer k ≥ 0 k \ge0 k ≥ 0 W W W Q Q Q Q ∈ D σ , k ( s ) Q \in\mathcal{D}^{(s)}_{\sigma,k} Q ∈ D σ , k ( s ) k < ℓ k < \ell k < ℓ Q ∈ D σ , ℓ ( s ) Q \in\mathcal{D}^{(s)}_{\sigma,\ell} Q ∈ D σ , ℓ ( s ) R ∈ D σ , k ( s ) R \in\mathcal{D}^{(s)}_{\sigma,k} R ∈ D σ , k ( s ) Q ∩ R = ∅ Q \cap R = \varnothing Q ∩ R = ∅ Q ⊂ R Q \subset R Q ⊂ R
For each k ≥ 0 k \ge0 k ≥ 0 Q ∈ D σ , k ( s ) Q \in\mathcal{D}^{(s)}_{\sigma,k} Q ∈ D σ , k ( s ) B ( s ) ( Q ) = B ( z Q , r ( s ) ( Q ) ) B^{(s)}(Q) = B(z_Q,r^{(s)}(Q)) B ( s ) ( Q ) = B ( z Q , r ( s ) ( Q ))
z Q ∈ W , s A 0 − k ≤ r ( s ) ( Q ) ≤ s K 0 A 0 − k , W ∩ B ( s ) ( Q ) ⊂ Q ⊂ W ∩ 28 B ( s ) ( Q ) = W ∩ B ( z Q , 28 r ( s ) ( Q ) ) , (26) \begin{aligned}
z_Q &\in W,\qquad sA_0^{-k} \le r^{(s)}(Q) \le sK_0A_0^{-k},\\
W \cap B^{(s)}(Q) &\subset Q \subset W \cap28B^{(s)}(Q)=W \cap B(z_Q,28r^{(s)}(Q)), \tag*{(26)}
\end{aligned} z Q W ∩ B ( s ) ( Q ) ∈ W , s A 0 − k ≤ r ( s ) ( Q ) ≤ s K 0 A 0 − k , ⊂ Q ⊂ W ∩ 28 B ( s ) ( Q ) = W ∩ B ( z Q , 28 r ( s ) ( Q )) , (26) and the balls 5 B ( s ) ( Q ) 5B^{(s)}(Q) 5 B ( s ) ( Q ) Q ∈ D σ , k ( s ) Q \in\mathcal{D}^{(s)}_{\sigma,k} Q ∈ D σ , k ( s )
For Q ∈ D σ , k ( s ) Q \in\mathcal{D}^{(s)}_{\sigma,k} Q ∈ D σ , k ( s ) ℓ ≥ 0 \ell\ge0 ℓ ≥ 0
N ℓ i n t , ( s ) ( Q ) : = { x ∈ Q : dist ( x , W ∖ Q ) < s A 0 − k − ℓ } , N^{\mathrm{int},(s)}_{\ell}(Q):=\{x \in Q:\operatorname{dist}(x,W\setminus Q)<sA_0^{-k-\ell}\}, N ℓ int , ( s ) ( Q ) := { x ∈ Q : dist ( x , W ∖ Q ) < s A 0 − k − ℓ } , N ℓ e x t , ( s ) ( Q ) : = { x ∈ W ∖ Q : dist ( x , Q ) < s A 0 − k − ℓ } , N^{\mathrm{ext},(s)}_{\ell}(Q):=\{x \in W\setminus Q:\operatorname{dist}(x,Q)<sA_0^{-k-\ell}\}, N ℓ ext , ( s ) ( Q ) := { x ∈ W ∖ Q : dist ( x , Q ) < s A 0 − k − ℓ } , and N ℓ ( s ) ( Q ) : = N ℓ i n t , ( s ) ( Q ) ∪ N ℓ e x t , ( s ) ( Q ) N^{(s)}_{\ell}(Q):=N^{\mathrm{int},(s)}_{\ell}(Q)\cup N^{\mathrm{ext},(s)}_{\ell}(Q) N ℓ ( s ) ( Q ) := N ℓ int , ( s ) ( Q ) ∪ N ℓ ext , ( s ) ( Q )
σ ( N ℓ ( s ) ( Q ) ) ≤ ϑ D M ℓ σ ( 90 B ( s ) ( Q ) ) , for ϑ D M : = C D K 0 3 d + 1 A 0 . \sigma(N^{(s)}_{\ell}(Q))\le\vartheta_{\mathrm{DM}}^\ell\sigma(90B^{(s)}(Q)),\qquad\text{for}\qquad\vartheta_{\mathrm{DM}}:=\frac{C_DK_0^{3\mathfrak{d}+1}}{A_0}. σ ( N ℓ ( s ) ( Q )) ≤ ϑ DM ℓ σ ( 90 B ( s ) ( Q )) , for ϑ DM := A 0 C D K 0 3 d + 1 . Let us denote by D σ , k d b , ( s ) \mathcal{D}^{\mathrm{db},(s)}_{\sigma,k} D σ , k db , ( s ) Q ∈ D σ , k ( s ) Q\in\mathcal{D}^{(s)}_{\sigma,k} Q ∈ D σ , k ( s )
σ ( 100 B ( s ) ( Q ) ) ≤ K 0 σ ( B ( s ) ( Q ) ) . (27) \sigma(100B^{(s)}(Q))\le K_0\sigma(B^{(s)}(Q)). \tag*{(27)} σ ( 100 B ( s ) ( Q )) ≤ K 0 σ ( B ( s ) ( Q )) . (27) For Q ∈ D σ , k ( s ) ∖ D σ , k d b , ( s ) Q\in\mathcal{D}^{(s)}_{\sigma,k}\setminus\mathcal{D}^{\mathrm{db},(s)}_{\sigma,k} Q ∈ D σ , k ( s ) ∖ D σ , k db , ( s ) r ( s ) ( Q ) = s A 0 − k r^{(s)}(Q)=sA_0^{-k} r ( s ) ( Q ) = s A 0 − k ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 100 ℓ ≤ K 0 100^\ell\le K_0 10 0 ℓ ≤ K 0 Q ∈ D σ , k ( s ) ∖ D σ , k d b , ( s ) Q\in\mathcal{D}^{(s)}_{\sigma,k}\setminus\mathcal{D}^{\mathrm{db},(s)}_{\sigma,k} Q ∈ D σ , k ( s ) ∖ D σ , k db , ( s )
σ ( 100 B ( s ) ( Q ) ) ≤ K 0 − ℓ σ ( 100 ℓ + 1 B ( s ) ( Q ) ) . (28) \sigma(100B^{(s)}(Q))\le K_0^{-\ell}\sigma(100^{\ell+1}B^{(s)}(Q)). \tag*{(28)} σ ( 100 B ( s ) ( Q )) ≤ K 0 − ℓ σ ( 10 0 ℓ + 1 B ( s ) ( Q )) . (28) Here C D C_D C D D D D A 0 A_0 A 0 ϑ D M < 1 \vartheta_{\mathrm{DM}}<1 ϑ DM < 1
D σ ( s ) : = ⋃ k ≥ 0 D σ , k ( s ) , D σ d b , ( s ) : = ⋃ k ≥ 0 D σ , k d b , ( s ) . \mathcal{D}^{(s)}_{\sigma}:=\bigcup_{k\ge0}\mathcal{D}^{(s)}_{\sigma,k},\qquad
\mathcal{D}^{\mathrm{db},(s)}_{\sigma}:=\bigcup_{k\ge0}\mathcal{D}^{\mathrm{db},(s)}_{\sigma,k}. D σ ( s ) := k ≥ 0 ⋃ D σ , k ( s ) , D σ db , ( s ) := k ≥ 0 ⋃ D σ , k db , ( s ) . Proof. The existence of a lattice with the properties listed above follows from [81 Lemma 2.7] , applied to the metric space ( X , s − 1 d ) (X,s^{-1}d) ( X , s − 1 d ) D D D σ ( c B ( s ) ( Q ) ) ≤ K 0 − 1 σ ( 100 c B ( s ) ( Q ) ) \sigma(cB^{(s)}(Q))\le K_0^{-1}\sigma(100cB^{(s)}(Q)) σ ( c B ( s ) ( Q )) ≤ K 0 − 1 σ ( 100 c B ( s ) ( Q )) 1 ≤ c ≤ K 0 1\le c\le K_0 1 ≤ c ≤ K 0 (28) . □
Cells are understood together with their generation indices. Distinct generations may give the same underlying subset, and ancestry and strict descent refer to these indexed cells.
Applying the construction in Appendix § 9 .§ 1 to ( X , s − 1 d ) (X,s^{-1}d) ( X , s − 1 d ) B ~ ( s ) ( Q ) \widetilde{B}^{(s)}(Q) B ( s ) ( Q ) Q ∈ D σ d b , ( s ) Q \in\mathcal{D}_{\sigma}^{\mathrm{db},(s)} Q ∈ D σ db , ( s ) Q ∈ D σ , k ( s ) Q \in\mathcal{D}_{\sigma,k}^{(s)} Q ∈ D σ , k ( s )
ℓ ( s ) ( Q ) : = s 56 K 0 A 0 − k . \ell^{(s)}(Q):=s\,56K_0A_0^{-k}. ℓ ( s ) ( Q ) := s 56 K 0 A 0 − k . Auxiliary constants and remarks
We introduce the technical constants used to construct the testing balls in Appendix § 9 .§ 1 . They depend on δ \delta δ Λ \Lambda Λ
We define j ∗ j_* j ∗ j ≥ 0 j \ge0 j ≥ 0 K 0 ϑ D M j ≤ 1 / 2 K_0\vartheta_{DM}^{j} \le1/2 K 0 ϑ D M j ≤ 1/2 c ∗ : = A 0 − j ∗ / K 0 c_* := A_0^{-j_*}/K_0 c ∗ := A 0 − j ∗ / K 0 r Q : = r ( Q ) r_Q := r(Q) r Q := r ( Q )
Lemma 2.6. For every Q ∈ D σ , k d b Q \in\mathcal{D}_{\sigma,k}^{\mathrm{db}} Q ∈ D σ , k db
σ ( Q ∖ N j ∗ i n t ( Q ) ) ≥ 1 2 σ ( Q ) \sigma\left(Q \setminus N_{j_*}^{\mathrm{int}}(Q)\right) \ge\frac{1}{2}\sigma(Q) σ ( Q ∖ N j ∗ int ( Q ) ) ≥ 2 1 σ ( Q ) and
dist ( x , supp ( σ ) ∖ Q ) ≥ c ∗ r Q , for every x ∈ Q ∖ N j ∗ i n t ( Q ) . \operatorname{dist}(x,\operatorname{supp}(\sigma)\setminus Q) \ge c_*r_Q,\qquad\text{for every}\qquad x \in Q \setminus N_{j_*}^{\mathrm{int}}(Q). dist ( x , supp ( σ ) ∖ Q ) ≥ c ∗ r Q , for every x ∈ Q ∖ N j ∗ int ( Q ) . Proof. By Lemma 2.3 and the doubling of Q Q Q
σ ( N j ∗ i n t ( Q ) ) ≤ ϑ D M j ∗ σ ( 90 B ( Q ) ) ≤ K 0 ϑ D M j ∗ σ ( B ( Q ) ) ≤ 1 2 σ ( Q ) . \sigma\left(N_{j_*}^{\mathrm{int}}(Q)\right) \le\vartheta_{DM}^{j_*}\sigma(90B(Q)) \le K_0\vartheta_{DM}^{j_*}\sigma(B(Q)) \le\frac{1}{2}\sigma(Q). σ ( N j ∗ int ( Q ) ) ≤ ϑ D M j ∗ σ ( 90 B ( Q )) ≤ K 0 ϑ D M j ∗ σ ( B ( Q )) ≤ 2 1 σ ( Q ) . For x ∈ Q ∖ N j ∗ i n t ( Q ) x \in Q \setminus N_{j_*}^{\mathrm{int}}(Q) x ∈ Q ∖ N j ∗ int ( Q ) N j ∗ i n t ( Q ) N_{j_*}^{\mathrm{int}}(Q) N j ∗ int ( Q ) r Q ≤ K 0 A 0 − k diam ( W ) r_Q \le K_0A_0^{-k}\operatorname{diam}(W) r Q ≤ K 0 A 0 − k diam ( W )
dist ( x , supp ( σ ) ∖ Q ) ≥ A 0 − k − j ∗ diam ( W ) ≥ c ∗ r Q . \operatorname{dist}(x,\operatorname{supp}(\sigma)\setminus Q) \ge A_0^{-k-j_*}\operatorname{diam}(W) \ge c_*r_Q. dist ( x , supp ( σ ) ∖ Q ) ≥ A 0 − k − j ∗ diam ( W ) ≥ c ∗ r Q . For later use, we also set:
M ∗ : = 224 Λ / c ∗ M_* := 224\Lambda/c_* M ∗ := 224Λ/ c ∗ Q ⊂ M ∗ B ~ ( Q ) Q \subset M_*\widetilde{B}(Q) Q ⊂ M ∗ B ( Q )
A Λ , ∗ : = 512 Λ / c ∗ A_{\Lambda,*} := 512\Lambda/c_* A Λ , ∗ := 512Λ/ c ∗
m ∗ : = 4 + ⌈ log 2 ( M ∗ / δ ) ⌉ = 4 + ⌈ log 2 ( 224 Λ / ( c ∗ δ ) ) ⌉ m_* := 4+\lceil\log_2(M_*/\delta)\rceil= 4+\lceil\log_2(224\Lambda/(c_*\delta))\rceil m ∗ := 4 + ⌈ log 2 ( M ∗ / δ )⌉ = 4 + ⌈ log 2 ( 224Λ/ ( c ∗ δ ))⌉ 28 r Q 28r_Q 28 r Q Q Q Q δ R Q / 2 \delta R_Q/2 δ R Q /2 R Q : = c ∗ r Q / ( 4 Λ ) R_Q := c_*r_Q/(4\Lambda) R Q := c ∗ r Q / ( 4Λ ) B ~ ( Q ) \widetilde{B}(Q) B ( Q )
β ∗ : = 2 D m ∗ \beta_* := 2D^{m_*} β ∗ := 2 D m ∗ Lemma A.1 , and it is defined by a standard covering argument.
Remark 2.7 (Choice of β \beta β Theorem 1.3 , it suffices to assume
β > max { β ∗ , K 0 , ( 1.1 Λ δ ) n , D ⌈ log 2 ( 1.1 Λ / δ ) ⌉ } . \beta> \max\left\{\beta_*,K_0,\left(\frac{1.1\Lambda}{\delta}\right)^n,D^{\lceil\log_2(1.1\Lambda/\delta)\rceil}\right\}. β > max { β ∗ , K 0 , ( δ 1.1Λ ) n , D ⌈ l o g 2 ( 1.1Λ/ δ )⌉ } . The term β ∗ \beta_* β ∗ B \mathcal{B} B Theorem 1.3 . The last two terms on the right-hand side are used to guarantee the existence of balls B B B x x x μ ( 1.1 Λ B ) ≤ β μ ( δ B ) \mu(1.1\Lambda B) \le\beta\mu(\delta B) μ ( 1.1Λ B ) ≤ β μ ( δ B ) [37 Lemma 3.3] , where the factor 1.1 1.1 1.1
Here, we list some quantities and properties associated with Q ∈ D σ Q \in\mathcal{D}_{\sigma} Q ∈ D σ
diam ( Q ) ≤ diam ( 28 B ( Q ) ) ≤ 56 r ( Q ) ≤ ℓ ( Q ) , (29) \operatorname{diam}(Q) \le\operatorname{diam}(28B(Q)) \le56r(Q) \le\ell(Q), \tag*{(29)} diam ( Q ) ≤ diam ( 28 B ( Q )) ≤ 56 r ( Q ) ≤ ℓ ( Q ) , (29) and that r ( Q ) ≈ ℓ ( Q ) r(Q) \approx\ell(Q) r ( Q ) ≈ ℓ ( Q ) K 0 K_0 K 0
1 1000 K 0 − 1 ℓ ( Q ) ≤ r ( Q ) ≤ ℓ ( Q ) . (30) \frac{1}{1000}K_0^{-1}\ell(Q) \le r(Q) \le\ell(Q). \tag*{(30)} 1000 1 K 0 − 1 ℓ ( Q ) ≤ r ( Q ) ≤ ℓ ( Q ) . (30) Denoting by z Q z_Q z Q center of Q Q Q B Q : = 28 B ( Q ) = B ( z Q , 28 r ( Q ) ) B_Q := 28B(Q) = B(z_Q,28r(Q)) B Q := 28 B ( Q ) = B ( z Q , 28 r ( Q ))
W ∩ 1 28 B Q ⊂ Q ⊂ B Q . W \cap\frac{1}{28}B_Q \subset Q \subset B_Q. W ∩ 28 1 B Q ⊂ Q ⊂ B Q . For Q ∈ D σ Q \in\mathcal{D}_\sigma Q ∈ D σ D σ ( Q ) : = { P ∈ D σ : P ⊂ Q } \mathcal{D}_\sigma(Q) := \{P \in\mathcal{D}_\sigma: P \subset Q\} D σ ( Q ) := { P ∈ D σ : P ⊂ Q }
If Q ⊂ Q ′ Q \subset Q' Q ⊂ Q ′ D σ \mathcal{D}_\sigma D σ A 0 A_0 A 0 30 B ( Q ) ⊂ 30 B ( Q ′ ) 30B(Q) \subset30B(Q') 30 B ( Q ) ⊂ 30 B ( Q ′ ) [81 Section 3] .
Every Q ∈ D σ Q \in\mathcal{D}_\sigma Q ∈ D σ σ \sigma σ D σ d b \mathcal{D}_\sigma^{\mathrm{db}} D σ db [20 Lemma 5.28] .
For a measure with n n n
Θ μ ( 100 B ( Q ) ) : = μ ( 100 B ( Q ) ) ( 100 r ( Q ) ) n . \Theta_\mu(100B(Q)) := \frac{\mu(100B(Q))}{(100r(Q))^n}. Θ μ ( 100 B ( Q )) := ( 100 r ( Q ) ) n μ ( 100 B ( Q )) . More generally, in the upper-doubling setting of Remark 1.7, the denominator may be replaced by λ ( z Q , 100 r ( Q ) ) \lambda(z_Q,100r(Q)) λ ( z Q , 100 r ( Q ))
λ ( x , 2 r ) ≤ C λ λ ( x , r ) , λ ( x , r ) ≤ C λ λ ( y , r ) if d ( x , y ) ≤ r . \lambda(x,2r) \le C_\lambda\lambda(x,r), \qquad\lambda(x,r) \le C_\lambda\lambda(y,r) \quad\text{if } d(x,y) \le r. λ ( x , 2 r ) ≤ C λ λ ( x , r ) , λ ( x , r ) ≤ C λ λ ( y , r ) if d ( x , y ) ≤ r . Such a dominating function is supplied by Lemma B.2 . For the n n n λ ( x , r ) = r n \lambda(x,r) = r^n λ ( x , r ) = r n C λ = 2 n C_\lambda= 2^n C λ = 2 n
Lemma 2.8. Let K 0 ≥ 10 4 K_0 \ge10^4 K 0 ≥ 1 0 4 k 0 k_0 k 0 100 k 0 + 1 ≤ K 0 100^{k_0+1} \le K_0 10 0 k 0 + 1 ≤ K 0 K 0 K_0 K 0 A 0 A_0 A 0
K 0 k 0 / 2 > C λ ⌈ log 2 A 0 ⌉ + 1 . K_0^{k_0/2} > C_\lambda^{\lceil\log_2 A_0\rceil+1}. K 0 k 0 /2 > C λ ⌈ l o g 2 A 0 ⌉ + 1 . Let Q ⊂ R Q \subset R Q ⊂ R μ \mu μ S S S Q ⊊ S ⊊ R Q \subsetneq S \subsetneq R Q ⊊ S ⊊ R Q Q Q R R R
Θ μ ( 100 B ( Q ) ) ≲ K 0 − ( J ( Q ) − J ( R ) − 1 ) + k 0 / 2 Θ μ ( 100 B ( R ) ) \Theta_\mu(100B(Q)) \lesssim K_0^{-(J(Q)-J(R)-1)+k_0/2}\Theta_\mu(100B(R)) Θ μ ( 100 B ( Q )) ≲ K 0 − ( J ( Q ) − J ( R ) − 1 ) + k 0 /2 Θ μ ( 100 B ( R )) and
∑ Q ⊂ S ⊂ R Θ μ ( 100 B ( S ) ) ≲ Θ μ ( 100 B ( R ) ) . \sum_{Q\subset S\subset R}\Theta_\mu(100B(S)) \lesssim\Theta_\mu(100B(R)). Q ⊂ S ⊂ R ∑ Θ μ ( 100 B ( S )) ≲ Θ μ ( 100 B ( R )) . The implicit constants depend only on the parameters of the lattice and of the dominating function.
Proof. For the proof we refer, for instance, to [81 Lemma 3.6] and [78 Lemma 4.4] . □
These conditions hold if A 0 = C D ′ K 0 3 ∂ + 2 A_0 = C'_D K_0^{3\partial+2} A 0 = C D ′ K 0 3 ∂ + 2 C D ′ C'_D C D ′ (25) , and K 0 K_0 K 0 D D D C λ C_\lambda C λ
k 0 ≥ log K 0 log 100 − 2 , ⌈ log 2 A 0 ⌉ + 1 ≤ log 2 C D ′ + ( 3 ∂ + 2 ) log 2 K 0 + 2 , k_0 \ge\frac{\log K_0}{\log100}-2, \qquad\lceil\log_2 A_0\rceil+1 \le\log_2 C'_D+(3\partial+2)\log_2 K_0+2, k 0 ≥ log 100 log K 0 − 2 , ⌈ log 2 A 0 ⌉ + 1 ≤ log 2 C D ′ + ( 3 ∂ + 2 ) log 2 K 0 + 2 , which imply K 0 k 0 / 2 > C λ ⌈ log 2 A 0 ⌉ + 1 K_0^{k_0/2} > C_\lambda^{\lceil\log_2 A_0\rceil+1} K 0 k 0 /2 > C λ ⌈ l o g 2 A 0 ⌉ + 1 K 0 K_0 K 0
Finally, given two non-negative Radon measures m , n m,n m , n X X X dyadic maximal Hardy-Littlewood operator with respect to D σ \mathcal{D}_{\sigma} D σ
M m , D n ( x ) : = sup Q ∈ D σ , Q ∋ x m ( Q ) > 0 n ( Q ) m ( Q ) (31) M_{m,\mathcal{D}n}(x):=\sup_{\substack{Q\in\mathcal{D}_{\sigma},\,Q\ni x\\m(Q)>0}}\frac{n(Q)}{m(Q)} \tag*{(31)} M m , D n ( x ) := Q ∈ D σ , Q ∋ x m ( Q ) > 0 sup m ( Q ) n ( Q ) (31) which defines a bounded operator from the finite measures in M + ( X ) \mathcal{M}_{+}(X) M + ( X ) L 1 , ∞ ( m ) L^{1,\infty}(m) L 1 , ∞ ( m ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ L p ( m ) L^{p}(m) L p ( m ) L p ( m ) L^{p}(m) L p ( m ) p p p [77 Theorem 5.7] for a proof using a dyadic grid in X X X (31) .
Dyadic cubes in geometrically doubling metric spaces
Let ℓ ∈ ( 0 , 1 ) \ell\in(0,1) ℓ ∈ ( 0 , 1 ) 144 ℓ ≤ 1 144\ell\leq1 144 ℓ ≤ 1 [39 ] , which we use as the suitable analogue of standard dyadic cubes on the whole space, and that in the original work was formulated for geometrically doubling quasi-metric spaces.
Theorem 2.9 ([39 ] , Theorem 2.2). Let ( X , d ) (X,d) ( X , d ) D D D 0 < c 0 ≤ C 0 < ∞ 0<c_{0}\leq C_{0}<\infty 0 < c 0 ≤ C 0 < ∞
12 C 0 ℓ ≤ c 0 . 12C_{0}\ell\leq c_{0}. 12 C 0 ℓ ≤ c 0 . Assume that, for every k ∈ Z k\in\mathbb{Z} k ∈ Z { z α k } α ∈ A k \{z_{\alpha}^{k}\}_{\alpha\in A_{k}} { z α k } α ∈ A k
d ( z α k , z β k ) ≥ c 0 ℓ k ( α ≠ β ) , min α ∈ A k d ( x , z α k ) < C 0 ℓ k , for every x ∈ X . d(z_{\alpha}^{k},z_{\beta}^{k})\geq c_{0}\ell^{k}\quad(\alpha\ne\beta),\qquad\min_{\alpha\in A_{k}}d(x,z_{\alpha}^{k})<C_{0}\ell^{k},\qquad\text{for every }x\in X. d ( z α k , z β k ) ≥ c 0 ℓ k ( α = β ) , α ∈ A k min d ( x , z α k ) < C 0 ℓ k , for every x ∈ X . Then, there exist families of sets
Q ~ α k ⊂ Q α k ⊂ Q ‾ α k , k ∈ Z , \widetilde{Q}_{\alpha}^{k}\subset Q_{\alpha}^{k}\subset\overline{Q}_{\alpha}^{k},\qquad k\in\mathbb{Z}, Q α k ⊂ Q α k ⊂ Q α k , k ∈ Z , called open, half-open and closed dyadic cubes, such that Q ~ α k \widetilde{Q}_{\alpha}^{k} Q α k Q ‾ α k \overline{Q}_{\alpha}^{k} Q α k Q α k Q_{\alpha}^{k} Q α k
if j ≥ k , then either Q β j ⊂ Q α k or Q α k ∩ Q β j = ∅ , \text{if }j\geq k,\text{ then either }Q_{\beta}^{j}\subset Q_{\alpha}^{k}\text{ or }Q_{\alpha}^{k}\cap Q_{\beta}^{j}=\varnothing, if j ≥ k , then either Q β j ⊂ Q α k or Q α k ∩ Q β j = ∅ , X = ⋃ α ∈ A k Q α k disjointly, for every k ∈ Z , X=\bigcup_{\alpha\in A_{k}}Q_{\alpha}^{k}\quad\text{disjointly, for every }k\in\mathbb{Z}, X = α ∈ A k ⋃ Q α k disjointly, for every k ∈ Z , and
B ( z α k , c 1 ℓ k ) ⊂ Q α k ⊂ B ( z α k , C 1 ℓ k ) = : B ( Q α k ) , c 1 : = c 0 3 , C 1 : = 2 C 0 . B(z_{\alpha}^{k},c_{1}\ell^{k})\subset Q_{\alpha}^{k}\subset B(z_{\alpha}^{k},C_{1}\ell^{k})=:B(Q_{\alpha}^{k}),\qquad c_{1}:=\frac{c_{0}}{3},\quad C_{1}:=2C_{0}. B ( z α k , c 1 ℓ k ) ⊂ Q α k ⊂ B ( z α k , C 1 ℓ k ) =: B ( Q α k ) , c 1 := 3 c 0 , C 1 := 2 C 0 . Moreover, if k ≤ j k\leq j k ≤ j Q β j ⊂ Q α k Q_{\beta}^{j}\subset Q_{\alpha}^{k} Q β j ⊂ Q α k B ( Q β j ) ⊂ B ( Q α k ) B(Q_{\beta}^{j})\subset B(Q_{\alpha}^{k}) B ( Q β j ) ⊂ B ( Q α k )
diam Q α k ≤ 2 C 1 ℓ k . (32) \operatorname{diam}Q_{\alpha}^{k}\leq2C_{1}\ell^{k}. \tag*{(32)} diam Q α k ≤ 2 C 1 ℓ k . (32) For Q = Q α k Q=Q_{\alpha}^{k} Q = Q α k ℓ ( Q ) : = ℓ k \ell(Q):=\ell^{k} ℓ ( Q ) := ℓ k (32) reads
diam ( Q ) ≤ 2 C 1 ℓ ( Q ) . (33) \operatorname{diam}(Q)\leq2C_{1}\ell(Q). \tag*{(33)} diam ( Q ) ≤ 2 C 1 ℓ ( Q ) . (33) Remark 2.10. For a geometrically doubling metric space ( X , d ) (X,d) ( X , d ) D D D B ( x , r ) B(x,r) B ( x , r ) D t − log 2 D Dt^{-\log_{2}D} D t − l o g 2 D t r tr t r t ∈ ( 0 , 1 ] t\in(0,1] t ∈ ( 0 , 1 ] B ( x , r ) B(x,r) B ( x , r ) D t − log 2 D Dt^{-\log_{2}D} D t − l o g 2 D x i x_{i} x i B ( x i , t r ) B(x_{i},tr) B ( x i , t r ) [37 ] .
We now define a randomization of the dyadic system of cubes. We say that a collection { x α k } k , α ⊂ X \{x_{\alpha}^{k}\}_{k,\alpha}\subset X { x α k } k , α ⊂ X family of reference points if for every k ∈ Z k\in\mathbb{Z} k ∈ Z
d ( x α k , x β k ) ≥ ℓ k ( α ≠ β ) , min α ∈ A k d ( x , x α k ) < ℓ k for every x ∈ X . d(x_{\alpha}^{k},x_{\beta}^{k})\geq\ell^{k}\quad(\alpha\ne\beta),\qquad\min_{\alpha\in A_{k}}d(x,x_{\alpha}^{k})<\ell^{k}\quad\text{for every }x\in X. d ( x α k , x β k ) ≥ ℓ k ( α = β ) , α ∈ A k min d ( x , x α k ) < ℓ k for every x ∈ X . Theorem 2.11 ([39 ] , Theorem 5.1). Given reference points { x α k } k , α ⊂ X \{x_{\alpha}^{k}\}_{k,\alpha} \subset X { x α k } k , α ⊂ X 96 ℓ ≤ 1 96\ell\le1 96 ℓ ≤ 1 ( Ω , P ) (\Omega,\mathbb{P}) ( Ω , P ) w ∈ Ω w \in\Omega w ∈ Ω
D ( w ) = { Q α k ( w ) } k , α , \mathbb{D}(w)=\{Q_{\alpha}^{k}(w)\}_{k,\alpha}, D ( w ) = { Q α k ( w ) } k , α , related to new dyadic points { z α k ( w ) } k , α \{z_{\alpha}^{k}(w)\}_{k,\alpha} { z α k ( w ) } k , α Theorem 2.9 . Moreover,
Ω = ∏ k ∈ Z Ω k w = ( w k ) k ∈ Z , \Omega=\prod_{k\in\mathbb{Z}}\Omega_k \qquad w=(w_k)_{k\in\mathbb{Z}}, Ω = k ∈ Z ∏ Ω k w = ( w k ) k ∈ Z , where the coordinates w k w_k w k
z α k ( w ) = z α k ( w k ) , z_{\alpha}^{k}(w)=z_{\alpha}^{k}(w_k), z α k ( w ) = z α k ( w k ) , and there exists τ 0 > 0 \tau_0>0 τ 0 > 0 ( k + 1 , β ) (k+1,\beta) ( k + 1 , β ) ( k , α ) (k,\alpha) ( k , α )
P ( { w ∈ Ω : z α k ( w ) = x β k + 1 } ) ≥ τ 0 . \mathbb{P}\left(\{w\in\Omega:z_{\alpha}^{k}(w)=x_{\beta}^{k+1}\}\right)\ge\tau_0. P ( { w ∈ Ω : z α k ( w ) = x β k + 1 } ) ≥ τ 0 . For further properties of dyadic “random” cubes used in our manuscript, we refer to Appendix § 9 . We also remark that, in the Euclidean setting, random dyadic grids are obtained by means of random translations of the standard dyadic grid, see for instance [77 Section 5] . For w ∈ Ω w\in\Omega w ∈ Ω
D k ( w ) : = { Q ∈ D ( w ) : ℓ ( Q ) = ℓ k } , k ∈ Z , \mathbb{D}_k(w):=\{Q\in\mathbb{D}(w):\ell(Q)=\ell^k\},\qquad k\in\mathbb{Z}, D k ( w ) := { Q ∈ D ( w ) : ℓ ( Q ) = ℓ k } , k ∈ Z , so that D ( w ) = ⋃ k ∈ Z D k ( w ) \mathbb{D}(w)=\bigcup_{k\in\mathbb{Z}}\mathbb{D}_k(w) D ( w ) = ⋃ k ∈ Z D k ( w )
The Exceptional Sets and the Induced Suppressed Operator
Normalization of measure testing
The proof of Theorem 1.3 can be reduced to the following normalized statement, which is proved in the remainder of the paper.
Proposition 3.1. Let ( X , d ) (X,d) ( X , d ) μ \mu μ T T T B \mathcal{B} B Theorem 1.3 , let C 1 ≥ 1 \mathcal{C}_1\ge1 C 1 ≥ 1 ε 0 ∈ ( 0 , 1 ) \varepsilon_0\in(0,1) ε 0 ∈ ( 0 , 1 ) 0 < τ < τ 0 0<\tau<\tau_0 0 < τ < τ 0 0 < ζ < ζ 0 0<\zeta<\zeta_0 0 < ζ < ζ 0 τ 0 \tau_0 τ 0 ζ 0 \zeta_0 ζ 0 { ν ^ B } B ∈ B \{\widehat{\nu}_B\}_{B\in\mathcal{B}} { ν B } B ∈ B { ν ^ B ∗ } B ∈ B \{\widehat{\nu}_B^*\}_{B\in\mathcal{B}} { ν B ∗ } B ∈ B η ∗ \eta_* η ∗ B ∈ B B\in\mathcal{B} B ∈ B U B ⊂ B U_B\subset B U B ⊂ B η = η ∗ \eta=\eta_* η = η ∗ V B ⊂ U B V_B\subset U_B V B ⊂ U B
μ ( U B ) < ζ μ ( B ) and T ν ^ B ≤ 1 on ( 1 − τ ) B ∖ U B . (34) \mu(U_B)<\zeta\mu(B)\qquad\text{and}\qquad T\widehat{\nu}_B\le1\quad\text{on }(1-\tau)B\setminus U_B. \tag*{(34)} μ ( U B ) < ζ μ ( B ) and T ν B ≤ 1 on ( 1 − τ ) B ∖ U B . (34) and that the analogous condition holds for T ∗ T^* T ∗ ν ^ B ∗ \widehat{\nu}_B^* ν B ∗
sup ε > 0 ∥ T μ , ε ∥ L 2 ( μ ) → L 2 ( μ ) ≤ C n o r m , (35) \sup_{\varepsilon>0}\|T_{\mu,\varepsilon}\|_{L^2(\mu)\to L^2(\mu)}\le C_{\mathrm{norm}}, \tag*{(35)} ε > 0 sup ∥ T μ , ε ∥ L 2 ( μ ) → L 2 ( μ ) ≤ C norm , (35) where C n o r m C_{\mathrm{norm}} C norm D D D n n n C 0 C_0 C 0 C C C γ \gamma γ δ \delta δ Λ \Lambda Λ β \beta β C 3 \mathcal{C}_3 C 3 C 1 \mathcal{C}_1 C 1 ε 0 \varepsilon_0 ε 0 ζ \zeta ζ
We first show that it suffices to prove Proposition 3.1 for auxiliary kernels with cut-offs. The associated operators are a priori bounded on L 2 ( μ ) L^2(\mu) L 2 ( μ ) (287) .
To this end, we fix smooth functions φ , ψ : [ 0 , ∞ ) → [ 0 , 1 ] \varphi,\psi:[0,\infty)\to[0,1] φ , ψ : [ 0 , ∞ ) → [ 0 , 1 ] φ \varphi φ ψ \psi ψ φ = 0 \varphi=0 φ = 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] φ = 1 \varphi=1 φ = 1 [ 2 , ∞ ) [2,\infty) [ 2 , ∞ ) ψ = 1 \psi=1 ψ = 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] ψ = 0 \psi=0 ψ = 0 [ 2 , ∞ ) [2,\infty) [ 2 , ∞ ) 0 < h < L 0<h<L 0 < h < L T h , L T^{h,L} T h , L
K h , L ( x , y ) : = K ( x , y ) φ ( d ( x , y ) h ) ψ ( d ( x , y ) L ) . (36) K^{h,L}(x,y):=K(x,y)\varphi\left(\frac{d(x,y)}{h}\right)\psi\left(\frac{d(x,y)}{L}\right). \tag*{(36)} K h , L ( x , y ) := K ( x , y ) φ ( h d ( x , y ) ) ψ ( L d ( x , y ) ) . (36) With η ∗ \eta_* η ∗ Theorem 1.3 (so, as in (101) ), we also set
A : = 1 + 3 C 2 ( η ∗ ) , T : = A − 1 T h , L , (37) A := 1 + 3\mathcal{C}_2(\eta_*), \qquad\mathcal{T} := A^{-1}T^{h,L}, \tag*{(37)} A := 1 + 3 C 2 ( η ∗ ) , T := A − 1 T h , L , (37) and we prove the lemma that reduces the main theorem to the proposition above.
Lemma 3.2. For every 0 < h < L 0<h<L 0 < h < L T \mathcal{T} T Theorem 1.3 satisfy the hypotheses of Proposition 3.1 , with Calderón–Zygmund constants independent of h h h L L L Proposition 3.1 gives
∥ T μ h , L ∥ L 2 ( μ ) → L 2 ( μ ) ≤ C n o r m A , for 0 < h < L , (38) \left\|T^{h,L}_{\mu}\right\|_{L^2(\mu)\to L^2(\mu)} \le C_{\mathrm{norm}} A, \qquad\text{for }0<h<L, \tag*{(38)} T μ h , L L 2 ( μ ) → L 2 ( μ ) ≤ C norm A , for 0 < h < L , (38) and Theorem 1.3 holds with
sup ε > 0 ∥ T μ , ε ∥ L 2 ( μ ) → L 2 ( μ ) ≤ C n o r m A + C 2 n C 0 . (39) \sup_{\varepsilon>0}\left\|T_{\mu,\varepsilon}\right\|_{L^2(\mu)\to L^2(\mu)} \le C_{\mathrm{norm}} A+C2^n\mathcal{C}_0. \tag*{(39)} ε > 0 sup ∥ T μ , ε ∥ L 2 ( μ ) → L 2 ( μ ) ≤ C norm A + C 2 n C 0 . (39) Proof. For t ≥ 0 t\ge0 t ≥ 0 a ( t ) : = φ ( t / h ) ψ ( t / L ) a(t):=\varphi(t/h)\psi(t/L) a ( t ) := φ ( t / h ) ψ ( t / L ) ∣ a ′ ( t ) ∣ ≲ 1 / t \lvert a'(t)\rvert\lesssim1/t ∣ a ′ ( t )∣ ≲ 1/ t 1 1 1 K h , L K^{h,L} K h , L h h h L L L σ ∈ M ( X ; C ) \sigma\in M(X;\mathbb{C}) σ ∈ M ( X ; C ) x ∈ X x\in X x ∈ X ρ > 0 \rho>0 ρ > 0
T ρ h , L σ ( x ) = a ( ρ ) T ρ σ ( x ) + ∫ ρ ∞ a ′ ( t ) T t σ ( x ) d t , T^{h,L}_{\rho}\sigma(x)=a(\rho)T_{\rho}\sigma(x)+\int_{\rho}^{\infty}a'(t)T_t\sigma(x)\,dt, T ρ h , L σ ( x ) = a ( ρ ) T ρ σ ( x ) + ∫ ρ ∞ a ′ ( t ) T t σ ( x ) d t , which, since 0 ≤ a ≤ 1 0\le a\le1 0 ≤ a ≤ 1 ∫ 0 ∞ ∣ a ′ ∣ ≤ 2 \int_0^\infty\lvert a'\rvert\le2 ∫ 0 ∞ ∣ a ′ ∣ ≤ 2 ( T h , L ) ♯ σ ≤ 3 T ♯ σ (T^{h,L})^\sharp\sigma\le3T^\sharp\sigma ( T h , L ) ♯ σ ≤ 3 T ♯ σ
Hence, if (5) holds with η = η ∗ \eta=\eta_* η = η ∗ T ♯ ν ^ B ≤ 3 C 2 ( η ∗ ) / A < 1 T^\sharp\widehat{\nu}_B\le3\mathcal{C}_2(\eta_*)/A<1 T ♯ ν B ≤ 3 C 2 ( η ∗ ) / A < 1 ( 1 − τ ) B ∖ U B (1-\tau)B\setminus U_B ( 1 − τ ) B ∖ U B (34) holds for T \mathcal{T} T T ∗ \mathcal{T}^* T ∗ ν ^ B ∗ \widehat{\nu}_B^* ν B ∗ Proposition 3.1 do not involve the operator. Since K h , L ( x , y ) = 0 K^{h,L}(x,y)=0 K h , L ( x , y ) = 0 d ( x , y ) ≤ h d(x,y)\le h d ( x , y ) ≤ h T μ T_\mu T μ h h h T μ T_\mu T μ (35) for T \mathcal{T} T (38) .
Let f f f ε > 0 \varepsilon>0 ε > 0 L → ∞ L\to\infty L → ∞ (38) with h = ε h=\varepsilon h = ε ∥ T μ ε , ∞ f ∥ L 2 ( μ ) ≤ C n o r m A ∥ f ∥ L 2 ( μ ) \left\|T^{\varepsilon,\infty}_\mu f\right\|_{L^2(\mu)}\le C_{\mathrm{norm}}A\lVert f\rVert_{L^2(\mu)} T μ ε , ∞ f L 2 ( μ ) ≤ C norm A ∥ f ∥ L 2 ( μ ) T ε , ∞ T^{\varepsilon,\infty} T ε , ∞ K ( x , y ) φ ( d ( x , y ) / ε ) K(x,y)\varphi(d(x,y)/\varepsilon) K ( x , y ) φ ( d ( x , y ) / ε ) T μ , ε − T μ ε , ∞ T_{\mu,\varepsilon}-T^{\varepsilon,\infty}_\mu T μ , ε − T μ ε , ∞ { ε < d ( x , y ) < 2 ε } \{\varepsilon<d(x,y)<2\varepsilon\} { ε < d ( x , y ) < 2 ε } C ε − n C\varepsilon^{-n} C ε − n n n n μ \mu μ C 2 n C 0 C2^n\mathcal{C}_0 C 2 n C 0 (39) follows by density. □ \square □
By Lemma 3.2 , it suffices to prove Proposition 3.1 for T = T \mathcal{T}=\mathcal{T} T = T h h h L L L T \mathcal{T} T T \mathcal{T} T K h , L K^{h,L} K h , L { h < d ( x , y ) < 2 L } \{h<d(x,y)<2L\} { h < d ( x , y ) < 2 L } n n n μ \mu μ T μ T_\mu T μ L 2 ( μ ) L^2(\mu) L 2 ( μ )
Throughout this section, we consider a geometrically doubling metric space ( X , d ) (X,d) ( X , d ) D D D
The construction of the exceptional sets
Let B ⊂ X B\subset X B ⊂ X μ \mu μ ν ^ \widehat{\nu} ν X X X C 0 > 0 \mathcal{C}_0>0 C 0 > 0 C 1 ≥ 1 \mathcal{C}_1\ge1 C 1 ≥ 1 C 2 > 0 \mathcal{C}_2>0 C 2 > 0 C 3 > 1 \mathcal{C}_3>1 C 3 > 1 β > 5 log 2 D \beta>5^{\log_2 D} β > 5 l o g 2 D ε 0 ∈ ( 0 , 1 ) \varepsilon_0\in(0,1) ε 0 ∈ ( 0 , 1 )
(A0) The ball B B B C 3 \mathcal{C}_3 C 3 μ \mu μ
μ ( 5 B ) ≤ β μ ( B ) . (40) \mu(5B)\le\beta\mu(B). \tag*{(40)} μ ( 5 B ) ≤ β μ ( B ) . (40) (A1) The measure μ \mu μ n n n C 0 \mathcal{C}_0 C 0
(A2) We have ∣ ν ^ ∣ ( X ∖ B ) = 0 \lvert\widehat{\nu}\rvert(X\setminus B)=0 ∣ ν ∣ ( X ∖ B ) = 0
(A3) We have ∣ ν ^ ( B ) ∣ = μ ( B ) > 0 \lvert\widehat{\nu}(B)\rvert=\mu(B)>0 ∣ ν ( B )∣ = μ ( B ) > 0 (A4) We have
∥ ν ^ ∥ ≤ C 1 μ ( B ) . (41) \lVert\widehat{\nu}\rVert\le\mathfrak{C}_1\mu(B). \tag*{(41)} ∥ ν ∥ ≤ C 1 μ ( B ) . (41) (A5) For every Borel set A ⊂ B A\subset B A ⊂ B μ ( A ) ≤ ε 0 μ ( B ) \mu(A)\le\varepsilon_0\mu(B) μ ( A ) ≤ ε 0 μ ( B )
∣ ν ^ ∣ ( A ) ≤ ∥ ν ^ ∥ 200 C 1 . (42) |\widehat{\nu}|(A)\le\frac{\lVert\widehat{\nu}\rVert}{200\mathfrak{C}_1}. \tag*{(42)} ∣ ν ∣ ( A ) ≤ 200 C 1 ∥ ν ∥ . (42) (A6) There is a Borel set U B ⊂ B U_B\subset B U B ⊂ B
μ ( U B ) ≤ ε 0 μ ( B ) and T # ν ^ ≤ C 2 on B ∖ U B . (43) \mu(U_B)\le\varepsilon_0\mu(B)\qquad\text{and}\qquad T^\#\widehat{\nu}\le\mathfrak{C}_2\quad\text{on }B\setminus U_B. \tag*{(43)} μ ( U B ) ≤ ε 0 μ ( B ) and T # ν ≤ C 2 on B ∖ U B . (43) Possibly by normalizing the measure ν ^ \widehat{\nu} ν ν ^ ( B ) = μ ( B ) \widehat{\nu}(B)=\mu(B) ν ( B ) = μ ( B )
ν : = ν ^ + μ ∣ X ∖ B , so that ∣ ν ∣ = ∣ ν ^ ∣ + μ ∣ X ∖ B . \nu:=\widehat{\nu}+\mu|_{X\setminus B},\qquad\text{so that}\qquad|\nu|=|\widehat{\nu}|+\mu|_{X\setminus B}. ν := ν + μ ∣ X ∖ B , so that ∣ ν ∣ = ∣ ν ∣ + μ ∣ X ∖ B . Write ν ^ = b ^ ∣ ν ^ ∣ \widehat{\nu}=\widehat{b}|\widehat{\nu}| ν = b ∣ ν ∣ B B B b ^ \widehat{b} b 1 1 1 X ∖ B X\setminus B X ∖ B
ν = b ^ ∣ ν ∣ , \nu=\widehat{b}|\nu|, ν = b ∣ ν ∣ , where ∣ b ^ ∣ = 1 |\widehat{b}|=1 ∣ b ∣ = 1 ∣ ν ∣ |\nu| ∣ ν ∣
The assumptions concern ν ^ \widehat{\nu} ν ν \nu ν ∣ ν ∣ |\nu| ∣ ν ∣
Assumption (A6) has the same form as the normalized testing condition (34) . In the proof of Proposition 3.1, it holds with C 2 = 1 \mathfrak{C}_2=1 C 2 = 1 B ∖ ( 1 − τ ) B B\setminus(1-\tau)B B ∖ ( 1 − τ ) B (34) , see Section 4.
The set of non-accretive cubes. Next, fix η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 ) ( Ω , P ) (\Omega,\mathbb{P}) ( Ω , P ) w ∈ Ω w\in\Omega w ∈ Ω D ( w ) \mathbb{D}(w) D ( w ) A ( w ) \mathcal{A}(w) A ( w ) S ∈ D ( w ) S\in\mathbb{D}(w) S ∈ D ( w ) S ∩ B ≠ ∅ S\cap B\ne\varnothing S ∩ B = ∅
∣ ν ( S ) ∣ < η ∣ ν ∣ ( S ) . |\nu(S)|<\eta|\nu|(S). ∣ ν ( S ) ∣ < η ∣ ν ∣ ( S ) . We then set
T ( w ) : = ⋃ S ∈ A ( w ) S . \mathcal{T}(w):=\bigcup_{S\in\mathcal{A}(w)}S. T ( w ) := S ∈ A ( w ) ⋃ S . Lemma 3.3. For η = ( 2 C 1 ) − 1 \eta=(2\mathfrak{C}_1)^{-1} η = ( 2 C 1 ) − 1
∣ ν ^ ∣ ( T ( w ) ) ≤ ( 1 − η ) ∣ ν ^ ∣ ( B ) , for all w ∈ Ω . |\widehat{\nu}|(\mathcal{T}(w))\le(1-\eta)|\widehat{\nu}|(B),\qquad\text{for all }w\in\Omega. ∣ ν ∣ ( T ( w )) ≤ ( 1 − η ) ∣ ν ∣ ( B ) , for all w ∈ Ω. Proof. Let w ∈ Ω w\in\Omega w ∈ Ω N ∈ N N\in\mathbb{N} N ∈ N A N ( w ) \mathcal{A}_N(w) A N ( w ) A ( w ) \mathcal{A}(w) A ( w ) − N -N − N T N ( w ) : = ⋃ S ∈ A N ( w ) S \mathcal{T}_N(w):=\bigcup_{S\in\mathcal{A}_N(w)}S T N ( w ) := ⋃ S ∈ A N ( w ) S S ∈ A N ( w ) S\in\mathcal{A}_N(w) S ∈ A N ( w )
Re ν ^ ( S ) + μ ( S ∖ B ) ≤ ∣ ν ^ ( S ) + μ ( S ∖ B ) ∣ < η ( ∣ ν ^ ∣ ( S ) + μ ( S ∖ B ) ) , \operatorname{Re}\widehat{\nu}(S)+\mu(S\setminus B)\le|\widehat{\nu}(S)+\mu(S\setminus B)|<\eta\bigl(|\widehat{\nu}|(S)+\mu(S\setminus B)\bigr), Re ν ( S ) + μ ( S ∖ B ) ≤ ∣ ν ( S ) + μ ( S ∖ B ) ∣ < η ( ∣ ν ∣ ( S ) + μ ( S ∖ B ) ) , and hence Re ν ^ ( S ) ≤ η ∣ ν ^ ∣ ( S ) \operatorname{Re}\widehat{\nu}(S)\le\eta|\widehat{\nu}|(S) Re ν ( S ) ≤ η ∣ ν ∣ ( S )
μ ( B ) = Re ν ^ ( B ) ≤ ∣ ν ^ ∣ ( B ∖ T N ( w ) ) + ∑ S ∈ A N ( w ) Re ν ^ ( S ) ≤ ∣ ν ^ ∣ ( B ∖ T N ( w ) ) + η ∣ ν ^ ∣ ( B ) . \mu(B)=\operatorname{Re}\widehat{\nu}(B)\le|\widehat{\nu}|(B\setminus\mathcal{T}_N(w))+\sum_{S\in\mathcal{A}_N(w)}\operatorname{Re}\widehat{\nu}(S)\le|\widehat{\nu}|(B\setminus\mathcal{T}_N(w))+\eta|\widehat{\nu}|(B). μ ( B ) = Re ν ( B ) ≤ ∣ ν ∣ ( B ∖ T N ( w )) + S ∈ A N ( w ) ∑ Re ν ( S ) ≤ ∣ ν ∣ ( B ∖ T N ( w )) + η ∣ ν ∣ ( B ) . Since T N ( w ) ↑ T ( w ) \mathcal{T}_N(w)\uparrow\mathcal{T}(w) T N ( w ) ↑ T ( w ) ∣ ν ^ ∣ ( B ) < ∞ |\widehat{\nu}|(B)<\infty ∣ ν ∣ ( B ) < ∞ N → ∞ N\to\infty N → ∞
μ ( B ) ≤ ∣ ν ^ ∣ ( B ∖ T ( w ) ) + η ∣ ν ^ ∣ ( B ) . \mu(B)\le|\widehat{\nu}|(B\setminus\mathcal{T}(w))+\eta|\widehat{\nu}|(B). μ ( B ) ≤ ∣ ν ∣ ( B ∖ T ( w )) + η ∣ ν ∣ ( B ) . Combining the above inequalities, (41) , and using the definition of η \eta η
∣ ν ^ ∣ ( B ∖ T ( w ) ) ≥ C 1 − 1 ∣ ν ^ ∣ ( B ) − η ∣ ν ^ ∣ ( B ) = η ∣ ν ^ ∣ ( B ) , |\widehat{\nu}|(B\setminus\mathcal{T}(w))\ge\mathfrak{C}_1^{-1}|\widehat{\nu}|(B)-\eta|\widehat{\nu}|(B)=\eta|\widehat{\nu}|(B), ∣ ν ∣ ( B ∖ T ( w )) ≥ C 1 − 1 ∣ ν ∣ ( B ) − η ∣ ν ∣ ( B ) = η ∣ ν ∣ ( B ) , which immediately implies
∣ ν ^ ∣ ( T ( w ) ) ≤ ( 1 − η ) ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathcal{T}(w))\le(1-\eta)|\widehat{\nu}|(B). ∣ ν ∣ ( T ( w )) ≤ ( 1 − η ) ∣ ν ∣ ( B ) . The sets of high and low density. For n ∈ M ( X ; C ) \mathbf{n} \in M(X;\mathbb{C}) n ∈ M ( X ; C ) m ∈ M + ( X ) \mathbf{m} \in M_{+}(X) m ∈ M + ( X ) λ > 0 \lambda> 0 λ > 0
E λ ( m , n ) : = { x ∈ B : M ~ m n ( x ) > λ } . (44) E_{\lambda}(\mathbf{m},\mathbf{n}) := \{x \in B : \widetilde{\mathcal{M}}_{\mathbf{m}\mathbf{n}}(x) > \lambda\}. \tag*{(44)} E λ ( m , n ) := { x ∈ B : M mn ( x ) > λ } . (44) Let μ \mu μ ν ^ \widehat{\nu} ν x ∈ B x \in B x ∈ B m 1 > C 1 m_1 > \mathfrak{C}_1 m 1 > C 1 m 2 > 1 m_2 > 1 m 2 > 1
r 1 ( x ) : = { sup { r > 0 : ∣ ν ^ ∣ ( B ( x , r ) ) μ ( B ( x , 5 r ) ) > m 1 } , if M ~ μ ν ^ ( x ) > m 1 , 0 , otherwise , r_1(x) :=
\begin{cases}
\sup\left\{r>0:\dfrac{|\widehat{\nu}|(B(x,r))}{\mu(B(x,5r))}>m_1\right\}, & \text{if }\widetilde{\mathcal{M}}_{\mu}\widehat{\nu}(x)>m_1,\\
0, & \text{otherwise},
\end{cases} r 1 ( x ) := ⎩ ⎨ ⎧ sup { r > 0 : μ ( B ( x , 5 r )) ∣ ν ∣ ( B ( x , r )) > m 1 } , 0 , if M μ ν ( x ) > m 1 , otherwise , and
r 2 ( x ) : = { sup { r > 0 : μ ( B ( x , r ) ∩ B ) ∣ ν ^ ∣ ( B ( x , 5 r ) ) > m 2 } , if M ~ ∣ ν ^ ∣ ( μ ∣ B ) ( x ) > m 2 , 0 , otherwise . r_2(x) :=
\begin{cases}
\sup\left\{r>0:\dfrac{\mu(B(x,r)\cap B)}{|\widehat{\nu}|(B(x,5r))}>m_2\right\}, & \text{if }\widetilde{\mathcal{M}}_{|\widehat{\nu}|}(\mu|B)(x)>m_2,\\
0, & \text{otherwise}.
\end{cases} r 2 ( x ) := ⎩ ⎨ ⎧ sup { r > 0 : ∣ ν ∣ ( B ( x , 5 r )) μ ( B ( x , r ) ∩ B ) > m 2 } , 0 , if M ∣ ν ∣ ( μ ∣ B ) ( x ) > m 2 , otherwise . Finally, setting
S ^ 1 : = E m 1 ( μ , ν ^ ) and S ^ 2 : = E m 2 ( ∣ ν ^ ∣ , μ ∣ B ) , \widehat{\mathcal{S}}_1:=E_{m_1}(\mu,\widehat{\nu})\qquad\text{and}\qquad
\widehat{\mathcal{S}}_2:=E_{m_2}(|\widehat{\nu}|,\mu|B), S 1 := E m 1 ( μ , ν ) and S 2 := E m 2 ( ∣ ν ∣ , μ ∣ B ) , we define the exceptional sets
S 1 : = ⋃ x ∈ S ^ 1 B ( x , r 1 ( x ) ) and S 2 : = ⋃ x ∈ S ^ 2 B ( x , r 2 ( x ) ) . \mathcal{S}_1:=\bigcup_{x\in\widehat{\mathcal{S}}_1}B(x,r_1(x))
\qquad\text{and}\qquad
\mathcal{S}_2:=\bigcup_{x\in\widehat{\mathcal{S}}_2}B(x,r_2(x)). S 1 := x ∈ S 1 ⋃ B ( x , r 1 ( x )) and S 2 := x ∈ S 2 ⋃ B ( x , r 2 ( x )) . Here the supremum of an empty set of admissible radii is defined to be zero, and B ( x , 0 ) = ∅ B(x,0)=\varnothing B ( x , 0 ) = ∅ B B B m 1 > C 1 m_1>\mathfrak{C}_1 m 1 > C 1 m 2 > 1 m_2>1 m 2 > 1 S : = S 1 ∪ S 2 \mathcal{S}:=\mathcal{S}_1\cup\mathcal{S}_2 S := S 1 ∪ S 2
Lemma 3.4. For m 1 ≥ C 1 / ε 0 m_1\geq\mathfrak{C}_1/\varepsilon_0 m 1 ≥ C 1 / ε 0 m 2 ≥ 200 C 1 m_2\geq200\mathfrak{C}_1 m 2 ≥ 200 C 1
∣ ν ^ ∣ ( S 1 ) ≤ η 100 ∣ ν ^ ∣ ( B ) and ∣ ν ^ ∣ ( S 2 ) ≤ η 100 ∣ ν ^ ∣ ( B ) . (45) |\widehat{\nu}|(\mathcal{S}_1)\leq\frac{\eta}{100}|\widehat{\nu}|(B)
\qquad\text{and}\qquad
|\widehat{\nu}|(\mathcal{S}_2)\leq\frac{\eta}{100}|\widehat{\nu}|(B). \tag*{(45)} ∣ ν ∣ ( S 1 ) ≤ 100 η ∣ ν ∣ ( B ) and ∣ ν ∣ ( S 2 ) ≤ 100 η ∣ ν ∣ ( B ) . (45) Proof. We denote
E λ 1 : = E λ ( μ , ν ^ ) = { x ∈ B : M ~ μ ν ^ ( x ) > λ } , E^1_\lambda:=E_\lambda(\mu,\widehat{\nu})=\{x\in B:\widetilde{\mathcal{M}}_\mu\widehat{\nu}(x)>\lambda\}, E λ 1 := E λ ( μ , ν ) = { x ∈ B : M μ ν ( x ) > λ } , where M ~ μ \widetilde{\mathcal{M}}_\mu M μ (21) . By (2.7) and (A4), we have
μ ( E λ 1 ) ≤ ∥ ν ^ ∥ λ ≤ C 1 μ ( B ) λ . (46) \mu(E^1_\lambda)\leq\frac{\|\widehat{\nu}\|}{\lambda}\leq\frac{\mathfrak{C}_1\mu(B)}{\lambda}. \tag*{(46)} μ ( E λ 1 ) ≤ λ ∥ ν ∥ ≤ λ C 1 μ ( B ) . (46) Let y ∈ S 1 ∩ B y\in\mathcal{S}_1\cap B y ∈ S 1 ∩ B x ∈ B x\in B x ∈ B d ( x , y ) < r 1 ( x ) d(x,y)<r_1(x) d ( x , y ) < r 1 ( x ) r 1 ( x ) r_1(x) r 1 ( x ) r > d ( x , y ) r>d(x,y) r > d ( x , y )
∣ ν ^ ∣ ( B ( x , r ) ) > m 1 μ ( B ( x , 5 r ) ) . (47) |\widehat{\nu}|(B(x,r))>m_1\mu(B(x,5r)). \tag*{(47)} ∣ ν ∣ ( B ( x , r )) > m 1 μ ( B ( x , 5 r )) . (47) Since y ∈ B ( x , r ) y\in B(x,r) y ∈ B ( x , r ) (21) gives M ~ μ ν ^ ( y ) > m 1 \widetilde{\mathcal{M}}_\mu\widehat{\nu}(y)>m_1 M μ ν ( y ) > m 1 S 1 ∩ B ⊂ E m 1 1 \mathcal{S}_1\cap B\subset E^1_{m_1} S 1 ∩ B ⊂ E m 1 1 m 1 ≥ C 1 / ε 0 m_1\geq\mathfrak{C}_1/\varepsilon_0 m 1 ≥ C 1 / ε 0 (46) gives μ ( E m 1 1 ) ≤ ε 0 μ ( B ) \mu(E^1_{m_1})\leq\varepsilon_0\mu(B) μ ( E m 1 1 ) ≤ ε 0 μ ( B ) ν ^ \widehat{\nu} ν B B B (42) implies
∣ ν ^ ∣ ( S 1 ) = ∣ ν ^ ∣ ( S 1 ∩ B ) ≤ ∣ ν ^ ∣ ( E m 1 1 ) ≤ η 100 ∣ ν ^ ∣ ( B ) . (48) |\widehat{\nu}|(\mathcal{S}_1)=|\widehat{\nu}|(\mathcal{S}_1\cap B)\leq|\widehat{\nu}|(E^1_{m_1})\leq\frac{\eta}{100}|\widehat{\nu}|(B). \tag*{(48)} ∣ ν ∣ ( S 1 ) = ∣ ν ∣ ( S 1 ∩ B ) ≤ ∣ ν ∣ ( E m 1 1 ) ≤ 100 η ∣ ν ∣ ( B ) . (48) To prove the second estimate in (45) , define
E λ 2 : = E λ ( ∣ ν ^ ∣ , μ ∣ B ) = { x ∈ B : M ~ ∣ ν ^ ∣ ( μ ∣ B ) ( x ) > λ } . E^2_\lambda:=E_\lambda(|\widehat{\nu}|,\mu|B)=\{x\in B:\widetilde{\mathcal{M}}_{|\widehat{\nu}|}(\mu|B)(x)>\lambda\}. E λ 2 := E λ ( ∣ ν ∣ , μ ∣ B ) = { x ∈ B : M ∣ ν ∣ ( μ ∣ B ) ( x ) > λ } . By (22) , we have
∣ ν ^ ∣ ( E λ 2 ) ≤ μ ( B ) λ ≤ ∣ ν ^ ∣ ( B ) λ , |\widehat{\nu}|(E^2_\lambda)\leq\frac{\mu(B)}{\lambda}\leq\frac{|\widehat{\nu}|(B)}{\lambda}, ∣ ν ∣ ( E λ 2 ) ≤ λ μ ( B ) ≤ λ ∣ ν ∣ ( B ) , where the last inequality follows from (A3). If y ∈ S 2 ∩ B y\in\mathcal{S}_2\cap B y ∈ S 2 ∩ B x ∈ B x\in B x ∈ B d ( x , y ) < r 2 ( x ) d(x,y)<r_2(x) d ( x , y ) < r 2 ( x ) r > d ( x , y ) r>d(x,y) r > d ( x , y )
μ ( B ( x , r ) ∩ B ) > m 2 ∣ ν ^ ∣ ( B ( x , 5 r ) ) . \mu(B(x,r)\cap B)>m_2|\widehat{\nu}|(B(x,5r)). μ ( B ( x , r ) ∩ B ) > m 2 ∣ ν ∣ ( B ( x , 5 r )) . Since y ∈ B ( x , r ) y \in B(x,r) y ∈ B ( x , r ) M ∣ ν ^ ∣ ( μ ∣ B ) ( y ) > m 2 \mathcal{M}_{|\widehat{\nu}|}(\mu|_B)(y)>m_2 M ∣ ν ∣ ( μ ∣ B ) ( y ) > m 2 S 2 ∩ B ⊂ E m 2 2 \mathcal{S}_2\cap B\subset E^2_{m_2} S 2 ∩ B ⊂ E m 2 2 ν ^ \widehat{\nu} ν B B B
∣ ν ^ ∣ ( S 2 ) = ∣ ν ^ ∣ ( S 2 ∩ B ) ≤ ∣ ν ^ ∣ ( E m 2 2 ) ≤ ∣ ν ^ ∣ ( B ) m 2 . |\widehat{\nu}|(\mathcal{S}_2)=|\widehat{\nu}|(\mathcal{S}_2\cap B)\leq|\widehat{\nu}|(E^2_{m_2})\leq\frac{|\widehat{\nu}|(B)}{m_2}. ∣ ν ∣ ( S 2 ) = ∣ ν ∣ ( S 2 ∩ B ) ≤ ∣ ν ∣ ( E m 2 2 ) ≤ m 2 ∣ ν ∣ ( B ) . Taking m 2 ≥ 200 ϵ 1 = 100 / η m_2\geq200\epsilon_1=100/\eta m 2 ≥ 200 ϵ 1 = 100/ η
Lemma 3.5. In the following formula, the Radon-Nikodym derivative is that of the restriction of ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ B ∖ S B\setminus\mathcal{S} B ∖ S μ \mu μ β > 5 log 2 D \beta>5^{\log_2 D} β > 5 l o g 2 D m 1 , m 2 m_1,m_2 m 1 , m 2 ∣ ν ^ ∣ B ∖ S |\widehat{\nu}|_{B\setminus\mathcal{S}} ∣ ν ∣ B ∖ S μ ∣ B ∖ S \mu|_{B\setminus\mathcal{S}} μ ∣ B ∖ S
1 β m 2 ≤ d ∣ ν ^ ∣ d μ ( x ) ≤ β m 1 , for ∣ ν ^ ∣ -a.e. x ∈ B ∖ S . (49) \frac{1}{\beta m_2}\leq\frac{d|\widehat{\nu}|}{d\mu}(x)\leq\beta m_1,\qquad\text{for }|\widehat{\nu}|\text{-a.e. }x\in B\setminus\mathcal{S}. \tag*{(49)} β m 2 1 ≤ d μ d ∣ ν ∣ ( x ) ≤ β m 1 , for ∣ ν ∣ -a.e. x ∈ B ∖ S . (49) Proof. Put G = B ∖ S G=B\setminus\mathcal{S} G = B ∖ S x ∈ G x\in G x ∈ G r > 0 r>0 r > 0
∣ ν ^ ∣ ( B ( x , r ) ) ≤ m 1 μ ( B ( x , 5 r ) ) , μ ( B ( x , r ) ) ≤ m 2 ∣ ν ^ ∣ ( B ( x , 5 r ) ) . |\widehat{\nu}|(B(x,r))\leq m_1\mu(B(x,5r)),\qquad\mu(B(x,r))\leq m_2|\widehat{\nu}|(B(x,5r)). ∣ ν ∣ ( B ( x , r )) ≤ m 1 μ ( B ( x , 5 r )) , μ ( B ( x , r )) ≤ m 2 ∣ ν ∣ ( B ( x , 5 r )) . We first justify absolute continuity on G G G ρ , λ \rho,\lambda ρ , λ ρ ( B ( x , r ) ) ≤ M λ ( B ( x , 5 r ) ) \rho(B(x,r))\leq M\lambda(B(x,5r)) ρ ( B ( x , r )) ≤ M λ ( B ( x , 5 r )) r r r x ∈ G x\in G x ∈ G ρ ∣ G ≪ λ \rho|_G\ll\lambda ρ ∣ G ≪ λ A ⊂ G A\subset G A ⊂ G λ ( A ) = 0 \lambda(A)=0 λ ( A ) = 0 O ⊃ A O\supset A O ⊃ A λ ( O ) < ε \lambda(O)<\varepsilon λ ( O ) < ε β ′ > 25 log 2 D \beta'>25^{\log_2 D} β ′ > 2 5 l o g 2 D [37 Lemma 3.3] , for ρ \rho ρ x ∈ A x\in A x ∈ A ρ \rho ρ ( 25 , β ′ ) (25,\beta') ( 25 , β ′ ) B x B_x B x x x x 25 B x ⊂ O 25B_x\subset O 25 B x ⊂ O 5 r 5r 5 r 5 B x 5B_x 5 B x 5 B i 5B_i 5 B i A A A ρ \rho ρ
ρ ( A ) ≤ ∑ i ρ ( 25 B i ) ≤ β ′ ∑ i ρ ( B i ) ≤ β ′ M ∑ i λ ( 5 B i ) ≤ β ′ M ε . \rho(A)\leq\sum_i\rho(25B_i)\leq\beta'\sum_i\rho(B_i)\leq\beta'M\sum_i\lambda(5B_i)\leq\beta'M\varepsilon. ρ ( A ) ≤ i ∑ ρ ( 25 B i ) ≤ β ′ i ∑ ρ ( B i ) ≤ β ′ M i ∑ λ ( 5 B i ) ≤ β ′ M ε . Letting ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ( ρ , λ , M ) = ( ∣ ν ^ ∣ , μ , m 1 ) (\rho,\lambda,M)=(|\widehat{\nu}|,\mu,m_1) ( ρ , λ , M ) = ( ∣ ν ∣ , μ , m 1 ) ( ρ , λ , M ) = ( μ , ∣ ν ^ ∣ , m 2 ) (\rho,\lambda,M)=(\mu,|\widehat{\nu}|,m_2) ( ρ , λ , M ) = ( μ , ∣ ν ∣ , m 2 ) ∣ ν ^ ∣ G ≪ μ |\widehat{\nu}|_G\ll\mu ∣ ν ∣ G ≪ μ μ ∣ G ≪ ∣ ν ^ ∣ \mu|_G\ll|\widehat{\nu}| μ ∣ G ≪ ∣ ν ∣
Let ϕ = d ( ∣ ν ^ ∣ G ) / d μ \phi=d(|\widehat{\nu}|_G)/d\mu ϕ = d ( ∣ ν ∣ G ) / d μ G G G [37 Lemma 3.3 and Corollary 3.6] , at μ \mu μ x ∈ G x\in G x ∈ G ( 5 , β ) (5,\beta) ( 5 , β ) B j = B ( x , r j ) B_j=B(x,r_j) B j = B ( x , r j ) μ \mu μ r j → 0 r_j\to0 r j → 0 5 B j ⊂ B 5B_j\subset B 5 B j ⊂ B
ϕ ( x ) = lim j → ∞ ∣ ν ^ ∣ ( B j ∩ G ) μ ( B j ) ≤ lim sup j → ∞ ∣ ν ^ ∣ ( B j ) μ ( B j ) ≤ m 1 lim sup j → ∞ μ ( 5 B j ) μ ( B j ) ≤ β m 1 . \phi(x)=\lim_{j\to\infty}\frac{|\widehat{\nu}|(B_j\cap G)}{\mu(B_j)}
\leq\limsup_{j\to\infty}\frac{|\widehat{\nu}|(B_j)}{\mu(B_j)}
\leq m_1\limsup_{j\to\infty}\frac{\mu(5B_j)}{\mu(B_j)}
\leq\beta m_1. ϕ ( x ) = j → ∞ lim μ ( B j ) ∣ ν ∣ ( B j ∩ G ) ≤ j → ∞ lim sup μ ( B j ) ∣ ν ∣ ( B j ) ≤ m 1 j → ∞ lim sup μ ( B j ) μ ( 5 B j ) ≤ β m 1 . This inequality also holds ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ G G G
Finally, let ψ = d ( μ ∣ G ) / d ∣ ν ^ ∣ \psi=d(\mu|_G)/d|\widehat{\nu}| ψ = d ( μ ∣ G ) / d ∣ ν ∣ ψ \psi ψ ( 5 , β ) (5,\beta) ( 5 , β ) B j = B ( x , r j ) B_j=B(x,r_j) B j = B ( x , r j ) ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ 5 B j ⊂ B 5B_j\subset B 5 B j ⊂ B ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ x ∈ G x\in G x ∈ G
ψ ( x ) = lim j → ∞ μ ( B j ∩ G ) ∣ ν ^ ∣ ( B j ) ≤ lim sup j → ∞ μ ( B j ) ∣ ν ^ ∣ ( B j ) ≤ β m 2 . \psi(x)=\lim_{j\to\infty}\frac{\mu(B_j\cap G)}{|\widehat{\nu}|(B_j)}
\leq\limsup_{j\to\infty}\frac{\mu(B_j)}{|\widehat{\nu}|(B_j)}
\leq\beta m_2. ψ ( x ) = j → ∞ lim ∣ ν ∣ ( B j ) μ ( B j ∩ G ) ≤ j → ∞ lim sup ∣ ν ∣ ( B j ) μ ( B j ) ≤ β m 2 . Since the restricted measures are mutually absolutely continuous, ψ = 1 / ϕ \psi=1/\phi ψ = 1/ ϕ G G G (49) . □
We next define the high- and low-density sets for the random grid D ( w ) \mathbb{D}(w) D ( w ) m ^ > 1 \widehat{m}>1 m > 1 S m ^ , + ( w ) \mathcal{S}^{\widehat{m},+}(w) S m , + ( w ) Q ∈ D ( w ) Q\in\mathbb{D}(w) Q ∈ D ( w )
∣ ν ∣ ( Q ) > m ^ μ ( Q ) . (50) |\nu|(Q)>\widehat{m}\mu(Q). \tag*{(50)} ∣ ν ∣ ( Q ) > m μ ( Q ) . (50) Moreover, we define S m ^ , − ( w ) \mathscr{S}^{\widehat{\mathfrak m},-}(w) S m , − ( w ) Q ∈ D ( w ) Q \in\mathbb{D}(w) Q ∈ D ( w )
∣ ν ∣ ( Q ) < m ^ − 1 μ ( Q ) . (51) |\nu|(Q) < \widehat{\mathfrak m}^{-1}\mu(Q). \tag*{(51)} ∣ ν ∣ ( Q ) < m − 1 μ ( Q ) . (51) Finally, we define
S m ^ , ± ( w ) : = ⋃ Q ∈ S m ^ , ± ( w ) Q and S m ^ ( w ) : = S m ^ , + ( w ) ∪ S m ^ , − ( w ) . \mathscr{S}^{\widehat{\mathfrak m},\pm}(w) := \bigcup_{Q\in\mathscr{S}^{\widehat{\mathfrak m},\pm}(w)} Q
\quad\text{and}\quad
\mathscr{S}^{\widehat{\mathfrak m}}(w):=\mathscr{S}^{\widehat{\mathfrak m},+}(w)\cup\mathscr{S}^{\widehat{\mathfrak m},-}(w). S m , ± ( w ) := Q ∈ S m , ± ( w ) ⋃ Q and S m ( w ) := S m , + ( w ) ∪ S m , − ( w ) . We have the following estimate.
Lemma 3.6. If m ^ \widehat{\mathfrak m} m C 1 \mathfrak C_1 C 1 ε 0 \varepsilon_0 ε 0
∣ ν ^ ∣ ( S m ^ ( w ) ) ≤ η 50 ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathscr{S}^{\widehat{\mathfrak m}}(w)) \le\frac{\eta}{50}|\widehat{\nu}|(B). ∣ ν ∣ ( S m ( w )) ≤ 50 η ∣ ν ∣ ( B ) . Proof. For N ∈ N N\in\mathbb{N} N ∈ N S N m ^ , ± ( w ) \mathscr{S}^{\widehat{\mathfrak m},\pm}_N(w) S N m , ± ( w ) S m ^ , ± ( w ) \mathscr{S}^{\widehat{\mathfrak m},\pm}(w) S m , ± ( w ) − N -N − N S N m ^ , ± ( w ) \mathscr{S}^{\widehat{\mathfrak m},\pm}_N(w) S N m , ± ( w ) Q ∈ S N m ^ , + ( w ) Q\in\mathscr{S}^{\widehat{\mathfrak m},+}_N(w) Q ∈ S N m , + ( w ) (50) gives
∣ ν ^ ∣ ( Q ) > m ^ μ ( Q ∩ B ) + ( m ^ − 1 ) μ ( Q ∖ B ) ≥ m ^ μ ( Q ∩ B ) . |\widehat{\nu}|(Q)>\widehat{\mathfrak m}\mu(Q\cap B)+(\widehat{\mathfrak m}-1)\mu(Q\setminus B)\ge\widehat{\mathfrak m}\mu(Q\cap B). ∣ ν ∣ ( Q ) > m μ ( Q ∩ B ) + ( m − 1 ) μ ( Q ∖ B ) ≥ m μ ( Q ∩ B ) . Consequently,
μ ( S N m ^ , + ( w ) ∩ B ) ≤ m ^ − 1 ∑ Q ∈ S N m ^ , + ( w ) ∣ ν ^ ∣ ( Q ) ≤ m ^ − 1 ∣ ν ^ ∣ ( B ) ≤ m ^ − 1 C 1 μ ( B ) . \mu(\mathscr{S}^{\widehat{\mathfrak m},+}_N(w)\cap B)
\le\widehat{\mathfrak m}^{-1}\sum_{Q\in\mathscr{S}^{\widehat{\mathfrak m},+}_N(w)}|\widehat{\nu}|(Q)
\le\widehat{\mathfrak m}^{-1}|\widehat{\nu}|(B)
\le\widehat{\mathfrak m}^{-1}\mathfrak C_1\mu(B). μ ( S N m , + ( w ) ∩ B ) ≤ m − 1 Q ∈ S N m , + ( w ) ∑ ∣ ν ∣ ( Q ) ≤ m − 1 ∣ ν ∣ ( B ) ≤ m − 1 C 1 μ ( B ) . Thus, for m ^ ≥ C 1 ε 0 − 1 \widehat{\mathfrak m}\ge\mathfrak C_1\varepsilon_0^{-1} m ≥ C 1 ε 0 − 1
∣ ν ^ ∣ ( S N m ^ , + ( w ) ) = ∣ ν ^ ∣ ( S N m ^ , + ( w ) ∩ B ) ≤ ∣ ν ^ ∣ ( B ) 200 C 1 ≤ η 100 ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathscr{S}^{\widehat{\mathfrak m},+}_N(w))
=|\widehat{\nu}|(\mathscr{S}^{\widehat{\mathfrak m},+}_N(w)\cap B)
\le\frac{|\widehat{\nu}|(B)}{200\mathfrak C_1}
\le\frac{\eta}{100}|\widehat{\nu}|(B). ∣ ν ∣ ( S N m , + ( w )) = ∣ ν ∣ ( S N m , + ( w ) ∩ B ) ≤ 200 C 1 ∣ ν ∣ ( B ) ≤ 100 η ∣ ν ∣ ( B ) . For Q ∈ S N m ^ , − ( w ) Q\in\mathscr{S}^{\widehat{\mathfrak m},-}_N(w) Q ∈ S N m , − ( w ) (51) implies
∣ ν ^ ∣ ( Q ) < m ^ − 1 μ ( Q ∩ B ) − ( 1 − m ^ − 1 ) μ ( Q ∖ B ) ≤ m ^ − 1 μ ( Q ∩ B ) . |\widehat{\nu}|(Q)<\widehat{\mathfrak m}^{-1}\mu(Q\cap B)-(1-\widehat{\mathfrak m}^{-1})\mu(Q\setminus B)\le\widehat{\mathfrak m}^{-1}\mu(Q\cap B). ∣ ν ∣ ( Q ) < m − 1 μ ( Q ∩ B ) − ( 1 − m − 1 ) μ ( Q ∖ B ) ≤ m − 1 μ ( Q ∩ B ) . Therefore,
∣ ν ^ ∣ ( S N m ^ , − ( w ) ) ≤ m ^ − 1 ∑ Q ∈ S N m ^ , − ( w ) μ ( Q ∩ B ) ≤ m ^ − 1 μ ( B ) ≤ η 100 ∣ ν ^ ∣ ( B ) |\widehat{\nu}|(\mathscr{S}^{\widehat{\mathfrak m},-}_N(w))
\le\widehat{\mathfrak m}^{-1}\sum_{Q\in\mathscr{S}^{\widehat{\mathfrak m},-}_N(w)}\mu(Q\cap B)
\le\widehat{\mathfrak m}^{-1}\mu(B)
\le\frac{\eta}{100}|\widehat{\nu}|(B) ∣ ν ∣ ( S N m , − ( w )) ≤ m − 1 Q ∈ S N m , − ( w ) ∑ μ ( Q ∩ B ) ≤ m − 1 μ ( B ) ≤ 100 η ∣ ν ∣ ( B ) whenever m ^ ≥ 100 / η \widehat{\mathfrak m}\ge100/\eta m ≥ 100/ η S N m ^ , ± ( w ) \mathscr{S}^{\widehat{\mathfrak m},\pm}_N(w) S N m , ± ( w ) S m ^ , ± ( w ) \mathscr{S}^{\widehat{\mathfrak m},\pm}(w) S m , ± ( w ) N → ∞ N\to\infty N → ∞ m ^ ≥ max { C 1 ε 0 − 1 , 100 / η } \widehat{\mathfrak m}\ge\max\{\mathfrak C_1\varepsilon_0^{-1},100/\eta\} m ≥ max { C 1 ε 0 − 1 , 100/ η } □ \square □
The set of non-Ahlfors regular balls
Fix m > 0 m>0 m > 0 x ∈ X x\in X x ∈ X
R ( x ) : = { sup { r > 0 : ∣ ν ^ ∣ ( B ( x , r ) ) > m r η } , if M n r ν ^ ( x ) > m , 0 , otherwise . (52) \mathcal{R}(x):=
\begin{cases}
\sup\{r>0:|\widehat{\nu}|(B(x,r))>mr^\eta\}, & \text{if }M_n^r\widehat{\nu}(x)>m,\\
0, & \text{otherwise}.
\end{cases} \tag*{(52)} R ( x ) := { sup { r > 0 : ∣ ν ∣ ( B ( x , r )) > m r η } , 0 , if M n r ν ( x ) > m , otherwise . (52) We remark that the radii R ( x ) \mathcal{R}(x) R ( x ) x ∈ X x\in X x ∈ X ν ^ \widehat{\nu} ν
H ^ : = { x ∈ X : M n r ν ^ ( x ) > m } and H : = ⋃ x ∈ H ^ B ( x , R ( x ) ) . (53) \widehat{\mathcal{H}}:=\{x\in X:M_n^r\widehat{\nu}(x)>m\}
\quad\text{and}\quad
\mathcal{H}:=\bigcup_{x\in\widehat{\mathcal{H}}}B(x,\mathcal{R}(x)). \tag*{(53)} H := { x ∈ X : M n r ν ( x ) > m } and H := x ∈ H ⋃ B ( x , R ( x )) . (53) Thus, if B ( x , r ) B(x,r) B ( x , r ) ∣ ν ^ ∣ ( B ( x , r ) ) > m r n |\widehat{\nu}|(B(x,r))>\mathbf{m}\,r^n ∣ ν ∣ ( B ( x , r )) > m r n B ( x , r ) ⊂ H B(x,r)\subset\mathcal{H} B ( x , r ) ⊂ H x ∈ X x\in X x ∈ X r > R ( x ) r>\mathcal{R}(x) r > R ( x ) n n n μ \mu μ
∣ ν ∣ ( B ( x , r ) ) ≤ ∣ ν ^ ∣ ( B ( x , r ) ) + μ ( B ( x , r ) ) ≤ ( m + C 0 ) r n . |\nu|(B(x,r))\leq|\widehat{\nu}|(B(x,r))+\mu(B(x,r))\leq(\mathbf{m}+\mathcal{C}_0)r^n. ∣ ν ∣ ( B ( x , r )) ≤ ∣ ν ∣ ( B ( x , r )) + μ ( B ( x , r )) ≤ ( m + C 0 ) r n . We also have the following estimate for the ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ H \mathcal{H} H
Lemma 3.7. For m \mathbf{m} m n n n ε 0 \varepsilon_0 ε 0 C 0 \mathcal{C}_0 C 0 C 1 \mathcal{C}_1 C 1
∣ ν ^ ∣ ( H ) ≤ η 100 ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathcal{H})\leq\frac{\eta}{100}|\widehat{\nu}|(B). ∣ ν ∣ ( H ) ≤ 100 η ∣ ν ∣ ( B ) . Proof. For λ > 0 \lambda>0 λ > 0 E λ = { M n r ν ^ > λ } E_\lambda=\{M_n^r\widehat{\nu}>\lambda\} E λ = { M n r ν > λ } (22) and (A4),
μ ( E λ ) ≤ 5 n C 0 λ ∥ ν ^ ∥ ≤ 5 n C 0 C 1 λ μ ( B ) . \mu(E_\lambda)\leq\frac{5^n\mathcal{C}_0}{\lambda}\|\widehat{\nu}\|\leq\frac{5^n\mathcal{C}_0\mathcal{C}_1}{\lambda}\mu(B). μ ( E λ ) ≤ λ 5 n C 0 ∥ ν ∥ ≤ λ 5 n C 0 C 1 μ ( B ) . If y ∈ H y\in\mathcal{H} y ∈ H x x x d ( x , y ) < R ( x ) d(x,y)<\mathcal{R}(x) d ( x , y ) < R ( x ) (52) , there exists r > d ( x , y ) r>d(x,y) r > d ( x , y ) ∣ ν ^ ∣ ( B ( x , r ) ) > m r n |\widehat{\nu}|(B(x,r))>\mathbf{m}\,r^n ∣ ν ∣ ( B ( x , r )) > m r n
∣ ν ^ ∣ ( B ( y , 2 r ) ) ≥ ∣ ν ^ ∣ ( B ( x , r ) ) > m r n = 2 − n m ( 2 r ) n . (54) |\widehat{\nu}|(B(y,2r))\geq|\widehat{\nu}|(B(x,r))>\mathbf{m}\,r^n=2^{-n}\mathbf{m}\,(2r)^n. \tag*{(54)} ∣ ν ∣ ( B ( y , 2 r )) ≥ ∣ ν ∣ ( B ( x , r )) > m r n = 2 − n m ( 2 r ) n . (54) Thus H ⊂ E m / 2 n \mathcal{H}\subset E_{\mathbf{m}/2^n} H ⊂ E m / 2 n m ≥ 10 n C 0 C 1 / ε 0 \mathbf{m}\geq10^n\mathcal{C}_0\mathcal{C}_1/\varepsilon_0 m ≥ 1 0 n C 0 C 1 / ε 0 μ ( B ∩ E m / 2 n ) ≤ ε 0 μ ( B ) \mu(B\cap E_{\mathbf{m}/2^n})\leq\varepsilon_0\mu(B) μ ( B ∩ E m / 2 n ) ≤ ε 0 μ ( B ) ν ^ \widehat{\nu} ν B B B
∣ ν ^ ∣ ( H ) ≤ ∣ ν ^ ∣ ( B ∩ E m / 2 n ) ≤ ∥ ν ^ ∥ 200 C 1 ≤ η 100 ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathcal{H})\leq|\widehat{\nu}|(B\cap E_{\mathbf{m}/2^n})\leq\frac{\|\widehat{\nu}\|}{200\mathcal{C}_1}\leq\frac{\eta}{100}|\widehat{\nu}|(B). ∣ ν ∣ ( H ) ≤ ∣ ν ∣ ( B ∩ E m / 2 n ) ≤ 200 C 1 ∥ ν ∥ ≤ 100 η ∣ ν ∣ ( B ) . The set where T ν ^ T\widehat{\nu} T ν
Fix α > 1 \alpha>1 α > 1 x ∈ B x\in B x ∈ B
e ( x ) : = { sup { ε > 0 : ∣ T ε ν ^ ( x ) ∣ > α } , if T ♯ ν ^ ( x ) > α , 0 , otherwise. (55) e(x):=
\begin{cases}
\sup\{\varepsilon>0:|T_\varepsilon\widehat{\nu}(x)|>\alpha\},&\text{if }T_\sharp\widehat{\nu}(x)>\alpha,\\
0,&\text{otherwise.}
\end{cases} \tag*{(55)} e ( x ) := { sup { ε > 0 : ∣ T ε ν ( x ) ∣ > α } , 0 , if T ♯ ν ( x ) > α , otherwise. (55) Note that these radii are finite, since ∣ T ε ν ^ ( x ) ∣ ≤ C ε − n ∥ ν ^ ∥ |T_\varepsilon\widehat{\nu}(x)|\leq C\varepsilon^{-n}\|\widehat{\nu}\| ∣ T ε ν ( x ) ∣ ≤ C ε − n ∥ ν ∥
E ^ : = { x ∈ B : T ♯ ν ^ ( x ) > α } and E : = ⋃ x ∈ E ^ B ( x , e ( x ) ) . (56) \widehat{\mathcal{E}}:=\{x\in B:T_\sharp\widehat{\nu}(x)>\alpha\}
\qquad\text{and}\qquad
\mathcal{E}:=\bigcup_{x\in\widehat{\mathcal{E}}}B(x,e(x)). \tag*{(56)} E := { x ∈ B : T ♯ ν ( x ) > α } and E := x ∈ E ⋃ B ( x , e ( x )) . (56) The proof of the following lemma follows the scheme of [77 Lemma 5.2] .
Lemma 3.8. If y ∈ E ∖ H y\in\mathcal{E}\setminus\mathcal{H} y ∈ E ∖ H
T ♯ ν ^ ( y ) > α − α E ′ , for α E ′ : = C m 2 n 3 ⋅ 2 γ − 2 2 γ − 1 . T_\sharp\widehat{\nu}(y)>\alpha-\alpha'_{\mathcal{E}},
\qquad\text{for }\alpha'_{\mathcal{E}}:=C\mathbf{m}\,2^{n}\frac{3\cdot2^\gamma-2}{2^\gamma-1}. T ♯ ν ( y ) > α − α E ′ , for α E ′ := C m 2 n 2 γ − 1 3 ⋅ 2 γ − 2 . Moreover, if α ≥ 2 max { α E ′ , C 2 } \alpha\geq2\max\{\alpha'_{\mathcal{E}},\mathcal{C}_2\} α ≥ 2 max { α E ′ , C 2 } B ∩ E ⊂ H ∪ U B B\cap\mathcal{E}\subset\mathcal{H}\cup U_B B ∩ E ⊂ H ∪ U B
Proof. Let y ∈ E ∖ H y\in\mathcal{E}\setminus\mathcal{H} y ∈ E ∖ H x ∈ E ^ x\in\widehat{\mathcal{E}} x ∈ E d ( x , y ) < e ( x ) d(x,y)<e(x) d ( x , y ) < e ( x ) (55) , there exists ε \varepsilon ε
d ( x , y ) < ε ≤ e ( x ) and ∣ T ε ν ^ ( x ) ∣ > α . d(x,y)<\varepsilon\leq e(x)
\qquad\text{and}\qquad
|T_\varepsilon\widehat{\nu}(x)|>\alpha. d ( x , y ) < ε ≤ e ( x ) and ∣ T ε ν ( x ) ∣ > α . The triangle inequality gives
∣ T ε ν ^ ( x ) − T ε ν ^ ( y ) ∣ ≤ ∣ T ε ( ν ^ ∣ B ( y , 2 ε ) ) ( x ) ∣ + ∣ T ε ( ν ^ ∣ B ( y , 2 ε ) ) ( y ) ∣ + ∣ T ε ( ν ^ ∣ X ∖ B ( y , 2 ε ) ) ( x ) − T ε ( ν ^ ∣ X ∖ B ( y , 2 ε ) ) ( y ) ∣ = : I 1 + I 2 + I 3 . (57) \begin{aligned}
|T_\varepsilon\widehat{\nu}(x)-T_\varepsilon\widehat{\nu}(y)|
&\leq|T_\varepsilon(\widehat{\nu}|_{B(y,2\varepsilon)})(x)|
+|T_\varepsilon(\widehat{\nu}|_{B(y,2\varepsilon)})(y)|\\
&\quad+|T_\varepsilon(\widehat{\nu}|_{X\setminus B(y,2\varepsilon)})(x)
-T_\varepsilon(\widehat{\nu}|_{X\setminus B(y,2\varepsilon)})(y)|\\
&=:I_1+I_2+I_3. \tag*{(57)}
\end{aligned} ∣ T ε ν ( x ) − T ε ν ( y ) ∣ ≤ ∣ T ε ( ν ∣ B ( y , 2 ε ) ) ( x ) ∣ + ∣ T ε ( ν ∣ B ( y , 2 ε ) ) ( y ) ∣ + ∣ T ε ( ν ∣ X ∖ B ( y , 2 ε ) ) ( x ) − T ε ( ν ∣ X ∖ B ( y , 2 ε ) ) ( y ) ∣ =: I 1 + I 2 + I 3 . (57) Since y ∉ H y\notin\mathcal{H} y ∈ / H y y y n n n m \mathbf{m} m
I 1 + I 2 ≤ 2 C ε − n ∣ ν ^ ∣ ( B ( y , 2 ε ) ) ≤ 2 C m 2 n . I_1+I_2\leq2C\varepsilon^{-n}|\widehat{\nu}|(B(y,2\varepsilon))\leq2C\mathbf{m}\,2^n. I 1 + I 2 ≤ 2 C ε − n ∣ ν ∣ ( B ( y , 2 ε )) ≤ 2 C m 2 n . For z ∉ B ( y , 2 ε ) z \notin B(y,2\varepsilon) z ∈ / B ( y , 2 ε ) d ( x , z ) ≥ d ( y , z ) − d ( x , y ) > ε d(x,z) \ge d(y,z)-d(x,y) > \varepsilon d ( x , z ) ≥ d ( y , z ) − d ( x , y ) > ε d ( y , z ) > 2 d ( x , y ) d(y,z) > 2d(x,y) d ( y , z ) > 2 d ( x , y ) K K K
I 3 ≤ C ε γ ∫ X ∖ B ( y , 2 ε ) d ∣ ν ∣ ( z ) d ( y , z ) n + γ ≤ C m 2 n + γ 2 γ − 1 . \mathcal{I}_3 \le C\varepsilon^\gamma\int_{X\setminus B(y,2\varepsilon)} \frac{d|\nu|(z)}{d(y,z)^{n+\gamma}} \le C\mathbf{m}\frac{2^{n+\gamma}}{2^\gamma-1}. I 3 ≤ C ε γ ∫ X ∖ B ( y , 2 ε ) d ( y , z ) n + γ d ∣ ν ∣ ( z ) ≤ C m 2 γ − 1 2 n + γ . So, we have
∣ T ε ν ^ ( y ) ∣ ≥ ∣ T ε ν ^ ( x ) ∣ − α E ′ > α − α E ′ , |T_\varepsilon\widehat{\nu}(y)| \ge|T_\varepsilon\widehat{\nu}(x)|-\alpha'_E > \alpha-\alpha'_E, ∣ T ε ν ( y ) ∣ ≥ ∣ T ε ν ( x ) ∣ − α E ′ > α − α E ′ , which proves the first assertion. If α ≥ 2 max { α E ′ , C 2 } \alpha\ge2\max\{\alpha'_E,\mathcal{C}_2\} α ≥ 2 max { α E ′ , C 2 } y ∈ B ∩ E ∖ H y \in B\cap\mathcal{E}\setminus\mathcal{H} y ∈ B ∩ E ∖ H T ♯ ν ^ ( y ) > α − α E ′ ≥ α / 2 ≥ C 2 T^\sharp\widehat{\nu}(y)>\alpha-\alpha'_E\ge\alpha/2\ge\mathcal{C}_2 T ♯ ν ( y ) > α − α E ′ ≥ α /2 ≥ C 2 y ∈ U B y\in U_B y ∈ U B
We remark that the thresholds m 1 m_1 m 1 m 2 m_2 m 2 m ^ \widehat{m} m m \mathbf{m} m α \alpha α n n n C C C γ \gamma γ C 0 \mathcal{C}_0 C 0 C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2 ε 0 \varepsilon_0 ε 0 h h h L L L c ♯ c_{\sharp} c ♯
The exceptional set W 0 ( w ) \mathcal{W}^0(w) W 0 ( w ) We now collect all the exceptional sets constructed so far, and define
W 0 ( w ) : = T ( w ) ∪ S ∪ S m ^ ( w ) ∪ H ∪ E ∪ U B . (58) \mathcal{W}^0(w):=\mathcal{T}(w)\cup\mathcal{S}\cup\mathcal{S}^{\widehat{m}}(w)\cup\mathcal{H}\cup\mathcal{E}\cup U_B. \tag*{(58)} W 0 ( w ) := T ( w ) ∪ S ∪ S m ( w ) ∪ H ∪ E ∪ U B . (58) Lemma 3.9. Let us assume that the constants in the constructions of S \mathcal{S} S S m ^ ( w ) \mathcal{S}^{\widehat{m}}(w) S m ( w ) H \mathcal{H} H E \mathcal{E} E δ 1 : = 1 − 9 10 η = 1 − 9 10 ( 2 C 1 ) − 1 \delta_1:=1-\frac{9}{10}\eta=1-\frac{9}{10}(2\mathcal{C}_1)^{-1} δ 1 := 1 − 10 9 η = 1 − 10 9 ( 2 C 1 ) − 1
∣ ν ^ ∣ ( B ∖ W 0 ( w ) ) ≥ ( 1 − δ 1 ) ∣ ν ^ ∣ ( B ) , for all w ∈ Ω . (59) |\widehat{\nu}|(B\setminus\mathcal{W}^0(w))\ge(1-\delta_1)|\widehat{\nu}|(B),\qquad\text{for all }w\in\Omega. \tag*{(59)} ∣ ν ∣ ( B ∖ W 0 ( w )) ≥ ( 1 − δ 1 ) ∣ ν ∣ ( B ) , for all w ∈ Ω. (59) Moreover,
μ ( B ∖ W 0 ( w ) ) ≥ ( β m 1 ) − 1 ( 1 − δ 1 ) μ ( B ) , for all w ∈ Ω . (60) \mu(B\setminus\mathcal{W}^0(w))\ge(\beta m_1)^{-1}(1-\delta_1)\mu(B),\qquad\text{for all }w\in\Omega. \tag*{(60)} μ ( B ∖ W 0 ( w )) ≥ ( β m 1 ) − 1 ( 1 − δ 1 ) μ ( B ) , for all w ∈ Ω. (60) Proof. Since ν ^ \widehat{\nu} ν B B B ∣ ν ^ ∣ ( U B ) ≤ ∥ ν ^ ∥ / ( 200 C 1 ) = η ∣ ν ^ ∣ ( B ) / 100 |\widehat{\nu}|(U_B)\le\|\widehat{\nu}\|/(200\mathcal{C}_1)=\eta|\widehat{\nu}|(B)/100 ∣ ν ∣ ( U B ) ≤ ∥ ν ∥/ ( 200 C 1 ) = η ∣ ν ∣ ( B ) /100 B ∩ E ⊂ H ∪ U B B\cap\mathcal{E}\subset\mathcal{H}\cup U_B B ∩ E ⊂ H ∪ U B E \mathcal{E} E ∣ ν ^ ∣ |\widehat{\nu}| ∣ ν ∣ W 0 ( w ) \mathcal{W}^0(w) W 0 ( w )
∣ ν ^ ∣ ( W 0 ( w ) ) ≤ ( 1 − η + 6 100 η ) ∣ ν ^ ∣ ( B ) . |\widehat{\nu}|(\mathcal{W}^0(w))\le\left(1-\eta+\frac{6}{100}\eta\right)|\widehat{\nu}|(B). ∣ ν ∣ ( W 0 ( w )) ≤ ( 1 − η + 100 6 η ) ∣ ν ∣ ( B ) . Thus, the choice δ 1 = 1 − 9 10 η \delta_1=1-\frac{9}{10}\eta δ 1 = 1 − 10 9 η (59) , which, in turn, by (49) and (A3), implies (60) . □
The joint stopping family. Suppose that w ∈ Ω w\in\Omega w ∈ Ω Q 0 ∈ D ( w ) Q_0\in\mathcal{D}(w) Q 0 ∈ D ( w ) B ⊂ Q 0 B\subset Q_0 B ⊂ Q 0 m ^ \widehat{m} m St ( Q 0 , w ) \operatorname{St}(Q_0,w) St ( Q 0 , w )
{ S ∈ A ( w ) ∪ S m ^ , + ( w ) ∪ S m ^ , − ( w ) : S ⊊ Q 0 } . \left\{S\in\mathcal{A}(w)\cup\mathcal{S}^{\widehat{m},+}(w)\cup\mathcal{S}^{\widehat{m},-}(w):S\subsetneq Q_0\right\}. { S ∈ A ( w ) ∪ S m , + ( w ) ∪ S m , − ( w ) : S ⊊ Q 0 } . Thus a cube belongs to one of the stopping families if it satisfies the non-accretivity condition or one of (50) and (51) .
Lemma 3.10. The cubes in St ( Q 0 , w ) \operatorname{St}(Q_0,w) St ( Q 0 , w )
⋃ S ∈ St ( Q 0 , w ) S = T ( w ) ∪ S m ^ ( w ) . \bigcup_{S\in\operatorname{St}(Q_0,w)}S=\mathcal{T}(w)\cup\mathcal{S}^{\widehat{m}}(w). S ∈ St ( Q 0 , w ) ⋃ S = T ( w ) ∪ S m ( w ) . In particular, this union does not depend on the choice of Q 0 Q_0 Q 0 If S ∈ St ( Q 0 , w ) S\in\operatorname{St}(Q_0,w) S ∈ St ( Q 0 , w ) S ∗ S^* S ∗
∣ ν ( S ∗ ) ∣ ≥ η ∣ ν ∣ ( S ∗ ) and m ^ − 1 μ ( S ∗ ) ≤ ∣ ν ∣ ( S ∗ ) ≤ m ^ μ ( S ∗ ) . |\nu(S^*)|\ge\eta|\nu|(S^*)\qquad\text{and}\qquad\widehat{m}^{-1}\mu(S^*)\le|\nu|(S^*)\le\widehat{m}\mu(S^*). ∣ ν ( S ∗ ) ∣ ≥ η ∣ ν ∣ ( S ∗ ) and m − 1 μ ( S ∗ ) ≤ ∣ ν ∣ ( S ∗ ) ≤ m μ ( S ∗ ) . The same inequalities hold for every cube Q ⊂ Q 0 Q \subset Q_0 Q ⊂ Q 0 St ( Q 0 , w ) \operatorname{St}(Q_0,w) St ( Q 0 , w )
Proof. Let F : = A ( w ) ∪ S m ^ , + ( w ) ∪ S m ^ , − ( w ) \mathcal{F}:=\mathcal{A}(w)\cup\mathcal{S}^{\widehat{m},+}(w)\cup\mathcal{S}^{\widehat{m},-}(w) F := A ( w ) ∪ S m , + ( w ) ∪ S m , − ( w ) Q ∈ D ( w ) Q\in\mathbb{D}(w) Q ∈ D ( w ) B B B ν = ∣ ν ∣ = μ \nu=|\nu|=\mu ν = ∣ ν ∣ = μ Q Q Q Q ∉ F Q\notin\mathcal{F} Q ∈ / F
On the other hand, if B ⊂ Q B\subset Q B ⊂ Q ν ^ ( B ) = μ ( B ) \widehat{\nu}(B)=\mu(B) ν ( B ) = μ ( B ) (41) yield
ν ( Q ) = μ ( Q ) ≤ ∣ ν ∣ ( Q ) ≤ C 1 μ ( Q ) . \nu(Q)=\mu(Q)\leq|\nu|(Q)\leq\mathfrak{C}_1\mu(Q). ν ( Q ) = μ ( Q ) ≤ ∣ ν ∣ ( Q ) ≤ C 1 μ ( Q ) . so Q ∉ F Q\notin\mathcal{F} Q ∈ / F η = ( 2 C 1 ) − 1 \eta=(2\mathfrak{C}_1)^{-1} η = ( 2 C 1 ) − 1 m ^ ≥ C 1 \widehat{m}\geq\mathfrak{C}_1 m ≥ C 1 S ∈ F S\in\mathcal{F} S ∈ F B ⊂ Q 0 B\subset Q_0 B ⊂ Q 0 Q 0 Q_0 Q 0 S ⊊ Q 0 S\subsetneq Q_0 S ⊊ Q 0 S S S Q 0 Q_0 Q 0 S S S F \mathcal{F} F F \mathcal{F} F St ( Q 0 , w ) \operatorname{St}(Q_0,w) St ( Q 0 , w )
If S ∈ St ( Q 0 , w ) S\in\operatorname{St}(Q_0,w) S ∈ St ( Q 0 , w ) S ∗ ⊂ Q 0 S^*\subset Q_0 S ∗ ⊂ Q 0 F \mathcal{F} F S ∗ = Q 0 S^*=Q_0 S ∗ = Q 0 Q ⊂ Q 0 Q\subset Q_0 Q ⊂ Q 0 St ( Q 0 , w ) \operatorname{St}(Q_0,w) St ( Q 0 , w ) F \mathcal{F} F B B B ν = ∣ ν ∣ = μ \nu=|\nu|=\mu ν = ∣ ν ∣ = μ
By Lemma 3.10 and (58) , we have the set inclusion
⋃ S ∈ St ( Q 0 , w ) S ⊂ W 0 ( w ) . \bigcup_{S\in\operatorname{St}(Q_0,w)}S\subset\mathcal{W}^0(w). S ∈ St ( Q 0 , w ) ⋃ S ⊂ W 0 ( w ) . The total exceptional set. Let R ∈ D 0 d b R\in\mathbb{D}_0^{\mathrm{db}} R ∈ D 0 db D 0 d b \mathbb{D}_0^{\mathrm{db}} D 0 db D 0 \mathbb{D}_0 D 0 § 4 , and let B : = B ~ ( R ) B:=\widetilde{B}(R) B := B ( R ) W 0 ( w ) \mathcal{W}^0(w) W 0 ( w ) (58) , let θ : = min { 1 / 4 , ε 0 / ( 2 β C 3 ) } \theta:=\min\{1/4,\varepsilon_0/(2\beta\mathcal{C}_3)\} θ := min { 1/4 , ε 0 / ( 2 β C 3 )} B θ : = ( 1 − θ ) B B_\theta:=(1-\theta)B B θ := ( 1 − θ ) B
W R ( w ) : = W 0 ( w ) , (61) \mathcal{W}_R(w):=\mathcal{W}^0(w), \tag*{(61)} W R ( w ) := W 0 ( w ) , (61) and we denote G R ( w ) : = B θ ∖ W R ( w ) G_R(w):=B_\theta\setminus\mathcal{W}_R(w) G R ( w ) := B θ ∖ W R ( w )
Lemma 3.11. We put θ ′ : = C 3 β θ \theta':=\mathcal{C}_3\beta\theta θ ′ := C 3 β θ θ ′ ≤ ε 0 / 2 \theta'\leq\varepsilon_0/2 θ ′ ≤ ε 0 /2 B θ B_\theta B θ μ \mu μ ( 5 , β / ( 1 − θ ′ ) ) (5,\beta/(1-\theta')) ( 5 , β / ( 1 − θ ′ ))
∣ ν ^ ∣ ( G R ( w ) ) ≥ 1 − δ 1 2 ∣ ν ^ ∣ ( B ~ ( R ) ) , w ∈ Ω , (62) |\widehat{\nu}|(G_R(w))\geq\frac{1-\delta_1}{2}|\widehat{\nu}|(\widetilde{B}(R)),\qquad w\in\Omega, \tag*{(62)} ∣ ν ∣ ( G R ( w )) ≥ 2 1 − δ 1 ∣ ν ∣ ( B ( R )) , w ∈ Ω , (62) μ ( G R ( w ) ) ≥ ( β m 1 ) − 1 1 − δ 1 2 μ ( B ~ ( R ) ) , w ∈ Ω . (63) \mu(G_R(w))\geq(\beta m_1)^{-1}\frac{1-\delta_1}{2}\mu(\widetilde{B}(R)),\qquad w\in\Omega. \tag*{(63)} μ ( G R ( w )) ≥ ( β m 1 ) − 1 2 1 − δ 1 μ ( B ( R )) , w ∈ Ω. (63) Proof. The small-boundary and doubling properties of B B B
μ ( B ∖ B θ ) ≤ C 3 θ μ ( 5 B ) ≤ θ ′ μ ( B ) . (64) \mu(B\setminus B_\theta)\leq\mathcal{C}_3\theta\mu(5B)\leq\theta'\mu(B). \tag*{(64)} μ ( B ∖ B θ ) ≤ C 3 θ μ ( 5 B ) ≤ θ ′ μ ( B ) . (64) Consequently,
μ ( B θ ) ≥ ( 1 − θ ′ ) μ ( B ) ≥ 1 − θ ′ β μ ( 5 B ) ≥ 1 − θ ′ β μ ( 5 B θ ) , \mu(B_\theta)\geq(1-\theta')\mu(B)\geq\frac{1-\theta'}{\beta}\mu(5B)\geq\frac{1-\theta'}{\beta}\mu(5B_\theta), μ ( B θ ) ≥ ( 1 − θ ′ ) μ ( B ) ≥ β 1 − θ ′ μ ( 5 B ) ≥ β 1 − θ ′ μ ( 5 B θ ) , which proves the doubling assertion. Since θ ′ ≤ ε 0 \theta'\leq\varepsilon_0 θ ′ ≤ ε 0
∣ ν ^ ∣ ( B ∖ B θ ) ≤ ∥ ν ^ ∥ 200 C 1 ≤ η 100 ∣ ν ^ ∣ ( B ) . (65) |\widehat{\nu}|(B\setminus B_\theta)\leq\frac{\|\widehat{\nu}\|}{200\mathfrak{C}_1}\leq\frac{\eta}{100}|\widehat{\nu}|(B). \tag*{(65)} ∣ ν ∣ ( B ∖ B θ ) ≤ 200 C 1 ∥ ν ∥ ≤ 100 η ∣ ν ∣ ( B ) . (65) In particular,
∣ ν ^ ∣ ( B θ ) ≥ ( 1 − η 100 ) ∣ ν ^ ∣ ( B ) . (66) |\widehat{\nu}|(B_\theta)\geq\left(1-\frac{\eta}{100}\right)|\widehat{\nu}|(B). \tag*{(66)} ∣ ν ∣ ( B θ ) ≥ ( 1 − 100 η ) ∣ ν ∣ ( B ) . (66) Note that W R ( w ) = W 0 ( w ) \mathcal{W}_{R}(w)=\mathcal{W}^{0}(w) W R ( w ) = W 0 ( w )
∣ ν ^ ∣ ( G R ( w ) ) ≥ ∣ ν ^ ∣ ( B ∖ W 0 ( w ) ) − ∣ ν ^ ∣ ( B ∖ B θ ) ≥ ( 1 − δ 1 − η 100 ) ∣ ν ^ ∣ ( B ) ≥ 1 − δ 1 2 ∣ ν ^ ∣ ( B ) , \begin{aligned}
|\hat{\nu}|(G_{R}(w))\geq|\hat{\nu}|(B\setminus\mathcal{W}^{0}(w))-|\hat{\nu}|(B\setminus B_{\theta}) \\
&\geq\left(1-\delta_{1}-\frac{\eta}{100}\right)|\hat{\nu}|(B)\geq\frac{1-\delta_{1}}{2}|\hat{\nu}|(B),
\end{aligned} ∣ ν ^ ∣ ( G R ( w )) ≥ ∣ ν ^ ∣ ( B ∖ W 0 ( w )) − ∣ ν ^ ∣ ( B ∖ B θ ) ≥ ( 1 − δ 1 − 100 η ) ∣ ν ^ ∣ ( B ) ≥ 2 1 − δ 1 ∣ ν ^ ∣ ( B ) , since 1 − δ 1 = 9 η / 10 1-\delta_{1}=9\eta/10 1 − δ 1 = 9 η /10 (63) follows from (49) and (A3).
Suppressed maximal functions and singular integral operators
Let Ξ : X → [ 0 , ∞ ) \Xi:X\to[0,\infty) Ξ : X → [ 0 , ∞ )
∣ Ξ ( x ) − Ξ ( y ) ∣ ≤ d ( x , y ) , for x , y ∈ X . |\Xi(x)-\Xi(y)|\leq d(x,y),\qquad\text{for }x,y\in X. ∣Ξ ( x ) − Ξ ( y ) ∣ ≤ d ( x , y ) , for x , y ∈ X . For a complex Radon measure ν \nu ν X X X
M Ξ r ( ν ) ( x ) : = sup ρ > 0 , ρ ≥ Ξ ( x ) ∣ ν ∣ ( B ( x , ρ ) ) ρ n . M_{\Xi}^{r}(\nu)(x):=\sup_{\rho>0,\rho\geq\Xi(x)}\frac{|\nu|(B(x,\rho))}{\rho^{n}}. M Ξ r ( ν ) ( x ) := ρ > 0 , ρ ≥ Ξ ( x ) sup ρ n ∣ ν ∣ ( B ( x , ρ )) . For f ∈ L l o c 1 ( m ) f\in L_{\mathrm{loc}}^{1}(\mathfrak{m}) f ∈ L loc 1 ( m ) M Ξ , m r f : = M Ξ r ( f m ) M_{\Xi,\mathfrak{m}}^{r}f:=M_{\Xi}^{r}(f\mathfrak{m}) M Ξ , m r f := M Ξ r ( f m )
Lemma 3.12. Let m 1 , m 2 \mathfrak{m}_{1},\mathfrak{m}_{2} m 1 , m 2 X X X R 1 , R 2 R_{1},R_{2} R 1 , R 2 c 1 , c 2 c_{1},c_{2} c 1 , c 2
m j ( B ( x , r ) ) ≤ c j r n if r > R j ( x ) , j = 1 , 2. \mathfrak{m}_{j}(B(x,r))\leq c_{j}r^{n}\qquad\text{if }r>R_{j}(x),\quad j=1,2. m j ( B ( x , r )) ≤ c j r n if r > R j ( x ) , j = 1 , 2. If Ξ ≥ max { R 1 , R 2 } \Xi\geq\max\{R_{1},R_{2}\} Ξ ≥ max { R 1 , R 2 } M Ξ , m 2 r M_{\Xi,\mathfrak{m}_{2}}^{r} M Ξ , m 2 r L 1 ( m 2 ) L^{1}(\mathfrak{m}_{2}) L 1 ( m 2 ) L 1 , ∞ ( m 1 ) L^{1,\infty}(\mathfrak{m}_{1}) L 1 , ∞ ( m 1 ) L p ( m 2 ) L^{p}(\mathfrak{m}_{2}) L p ( m 2 ) L p ( m 1 ) L^{p}(\mathfrak{m}_{1}) L p ( m 1 ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ D D D n n n c 1 c_{1} c 1 c 2 c_{2} c 2 p p p
Proof. The growth estimates hold also at a positive radius r = R j ( x ) r=R_{j}(x) r = R j ( x ) r r r M Ξ , m 2 r f M_{\Xi,\mathfrak{m}_{2}}^{r}f M Ξ , m 2 r f 5 r 5r 5 r B i : = B ( x i , r i ) B_{i}:=B(x_{i},r_{i}) B i := B ( x i , r i ) r i ≥ Ξ ( x i ) r_{i}\geq\Xi(x_{i}) r i ≥ Ξ ( x i )
λ r i n < ∫ B i ∣ f ∣ d m 2 . \lambda r_{i}^{n}<\int_{B_{i}}|f|\,d\mathfrak{m}_{2}. λ r i n < ∫ B i ∣ f ∣ d m 2 . The corresponding dilated balls form a covering of the level set so, applying the argument first to bounded collections of radii gives
m 1 ( { M Ξ , m 2 r f > λ } ) ≤ ∑ i m 1 ( 5 B i ) ≤ 5 n c 1 ∑ i r i n ≤ 5 n c 1 λ ∥ f ∥ L 1 ( m 2 ) . (67) \mathfrak{m}_{1}\left(\{M_{\Xi,\mathfrak{m}_{2}}^{r}f>\lambda\}\right)\leq\sum_{i}\mathfrak{m}_{1}(5B_{i})\leq5^{n}c_{1}\sum_{i}r_{i}^{n}\leq\frac{5^{n}c_{1}}{\lambda}\|f\|_{L^{1}(\mathfrak{m}_{2})}. \tag*{(67)} m 1 ( { M Ξ , m 2 r f > λ } ) ≤ i ∑ m 1 ( 5 B i ) ≤ 5 n c 1 i ∑ r i n ≤ λ 5 n c 1 ∥ f ∥ L 1 ( m 2 ) . (67) The L p ( m 2 ) → L p ( m 1 ) L^{p}(\mathfrak{m}_{2})\to L^{p}(\mathfrak{m}_{1}) L p ( m 2 ) → L p ( m 1 ) (67) and the trivial L ∞ ( m 2 ) → L ∞ ( m 1 ) L^{\infty}(\mathfrak{m}_{2})\to L^{\infty}(\mathfrak{m}_{1}) L ∞ ( m 2 ) → L ∞ ( m 1 )
Let K K K n n n C C C γ \gamma γ
K Ξ ( x , y ) : = K ( x , y ) d ( x , y ) 2 n d ( x , y ) 2 n + Ξ ( x ) n Ξ ( y ) n (68) K_{\Xi}(x,y):=K(x,y)\frac{d(x,y)^{2n}}{d(x,y)^{2n}+\Xi(x)^{n}\Xi(y)^{n}} \tag*{(68)} K Ξ ( x , y ) := K ( x , y ) d ( x , y ) 2 n + Ξ ( x ) n Ξ ( y ) n d ( x , y ) 2 n (68) and
T Ξ , m , ε f ( x ) : = ∫ d ( x , y ) > ε K Ξ ( x , y ) f ( y ) d m ( y ) . (69) T_{\Xi,\mathfrak{m},\varepsilon}f(x):=\int_{d(x,y)>\varepsilon}K_{\Xi}(x,y)f(y)\,d\mathfrak{m}(y). \tag*{(69)} T Ξ , m , ε f ( x ) := ∫ d ( x , y ) > ε K Ξ ( x , y ) f ( y ) d m ( y ) . (69) Note that, if K ( x , y ) = − K ( y , x ) K(x,y)=-K(y,x) K ( x , y ) = − K ( y , x ) (68) gives K Ξ ( x , y ) = − K Ξ ( y , x ) K_{\Xi}(x,y)=-K_{\Xi}(y,x) K Ξ ( x , y ) = − K Ξ ( y , x )
T Ξ , m , # f ( x ) : = sup ε > 0 ∣ T Ξ , m , ε f ( x ) ∣ , and T Ξ , m , # , δ f ( x ) : = sup ε > δ ∣ T Ξ , m , ε f ( x ) ∣ . T_{\Xi,\mathfrak{m},\#}f(x):=\sup_{\varepsilon>0}|T_{\Xi,\mathfrak{m},\varepsilon}f(x)|,\qquad\text{and}\qquad T_{\Xi,\mathfrak{m},\#, \delta}f(x):=\sup_{\varepsilon>\delta}|T_{\Xi,\mathfrak{m},\varepsilon}f(x)|. T Ξ , m , # f ( x ) := ε > 0 sup ∣ T Ξ , m , ε f ( x ) ∣ , and T Ξ , m , # , δ f ( x ) := ε > δ sup ∣ T Ξ , m , ε f ( x ) ∣. For positive Radon measures m m m n n n T Ξ , m T_{\Xi,m} T Ξ , m L 2 ( m ) L^2(m) L 2 ( m ) L 2 ( n ) L^2(n) L 2 ( n )
∥ T Ξ , m ∥ L 2 ( m ) → L 2 ( n ) : = sup ε > 0 ∥ T Ξ , m , ε ∥ L 2 ( m ) → L 2 ( n ) . \|T_{\Xi,m}\|_{L^2(m)\to L^2(n)}:=\sup_{\varepsilon>0}\|T_{\Xi,m,\varepsilon}\|_{L^2(m)\to L^2(n)}. ∥ T Ξ , m ∥ L 2 ( m ) → L 2 ( n ) := ε > 0 sup ∥ T Ξ , m , ε ∥ L 2 ( m ) → L 2 ( n ) . In the case of a complex measure n n n X X X T Ξ , ε n : = T Ξ , n , ε 1 T_{\Xi,\varepsilon}n:=T_{\Xi,n,\varepsilon}1 T Ξ , ε n := T Ξ , n , ε 1 T Ξ , ♯ n : = T Ξ , n , ♯ 1 T_{\Xi,\sharp}n:=T_{\Xi,n,\sharp}1 T Ξ , ♯ n := T Ξ , n , ♯ 1
Lemma 3.13. The kernel K Ξ K_{\Xi} K Ξ (68) has the following properties, with constants depending only on n n n γ \gamma γ K K K
(1) K Ξ K_{\Xi} K Ξ n n n γ \gamma γ n n n γ \gamma γ C C C
(2) If max { Ξ ( x ) , Ξ ( y ) } > 0 \max\{\Xi(x),\Xi(y)\}>0 max { Ξ ( x ) , Ξ ( y )} > 0
∣ K Ξ ( x , y ) ∣ ≤ C max { Ξ ( x ) n , Ξ ( y ) n } . (70) |K_{\Xi}(x,y)|\le\frac{C}{\max\{\Xi(x)^n,\Xi(y)^n\}}. \tag*{(70)} ∣ K Ξ ( x , y ) ∣ ≤ max { Ξ ( x ) n , Ξ ( y ) n } C . (70) (3) If Ξ ( x ) = 0 \Xi(x)=0 Ξ ( x ) = 0 Ξ ( y ) = 0 \Xi(y)=0 Ξ ( y ) = 0 K Ξ ( x , y ) = K ( x , y ) K_{\Xi}(x,y)=K(x,y) K Ξ ( x , y ) = K ( x , y ) n ∈ M ( X ; C ) n\in M(X;\mathbb{C}) n ∈ M ( X ; C ) x ∈ X x\in X x ∈ X ε ≥ Ξ ( x ) > 0 \varepsilon\ge\Xi(x)>0 ε ≥ Ξ ( x ) > 0
∣ T Ξ , ε n ( x ) − T ε n ( x ) ∣ ≤ C sup r > ε ∣ n ∣ ( B ( x , r ) ) r n . (71) |T_{\Xi,\varepsilon}n(x)-T_{\varepsilon}n(x)|\le C\sup_{r>\varepsilon}\frac{|n|(B(x,r))}{r^n}. \tag*{(71)} ∣ T Ξ , ε n ( x ) − T ε n ( x ) ∣ ≤ C r > ε sup r n ∣ n ∣ ( B ( x , r )) . (71) Proof. Since μ \mu μ n n n C 0 C_0 C 0 ( X , d , C 0 − 1 μ ) (X,d,C_0^{-1}\mu) ( X , d , C 0 − 1 μ ) λ ( x , r ) : = r n \lambda(x,r):=r^n λ ( x , r ) := r n K K K Λ \Lambda Λ Λ ( x , r ) = c n r n \Lambda(x,r)=c_n r^n Λ ( x , r ) = c n r n c n > 0 c_n>0 c n > 0 d λ = n d_\lambda=n d λ = n n ≥ 1 n\ge1 n ≥ 1 M = n M=n M = n K Ξ K_{\Xi} K Ξ t = ε t=\varepsilon t = ε
We next estimate T Ξ , ε n T_{\Xi,\varepsilon}n T Ξ , ε n T ε n T_{\varepsilon}n T ε n
Lemma 3.14. Let Ξ : X → [ 0 , ∞ ) \Xi:X\to[0,\infty) Ξ : X → [ 0 , ∞ ) n ∈ M ( X ; C ) n\in M(X;\mathbb{C}) n ∈ M ( X ; C ) α , m > 0 \alpha,m>0 α , m > 0 x ∈ X x\in X x ∈ X R 0 ≥ 0 R_0\ge0 R 0 ≥ 0 ε > R 0 \varepsilon>R_0 ε > R 0 r > R 0 r>R_0 r > R 0
∣ n ∣ ( B ( x , r ) ) ≤ m r n and ∣ T ε n ( x ) ∣ ≤ α . |n|(B(x,r))\le mr^n
\qquad\text{and}\qquad
|T_{\varepsilon}n(x)|\le\alpha. ∣ n ∣ ( B ( x , r )) ≤ m r n and ∣ T ε n ( x ) ∣ ≤ α . If Ξ ( x ) ≥ R 0 \Xi(x)\ge R_0 Ξ ( x ) ≥ R 0
∣ T Ξ , ε n ( x ) ∣ ≤ α + C m , for ε > 0. (72) |T_{\Xi,\varepsilon}n(x)|\le\alpha+Cm,\qquad\text{for }\varepsilon>0. \tag*{(72)} ∣ T Ξ , ε n ( x ) ∣ ≤ α + C m , for ε > 0. (72) Proof. Let h : = Ξ ( x ) h:=\Xi(x) h := Ξ ( x ) h = 0 h=0 h = 0 R 0 = 0 R_0=0 R 0 = 0 T Ξ , ε n ( x ) = T ε n ( x ) T_{\Xi,\varepsilon}n(x)=T_{\varepsilon}n(x) T Ξ , ε n ( x ) = T ε n ( x ) ε > h > 0 \varepsilon>h>0 ε > h > 0
∣ T Ξ , ε n ( x ) ∣ ≤ ∣ T ε n ( x ) ∣ + C sup r > ε ∣ n ∣ ( B ( x , r ) ) r n ≤ α + C m . (73) |T_{\Xi,\varepsilon}n(x)|\le|T_{\varepsilon}n(x)|+C\sup_{r>\varepsilon}\frac{|n|(B(x,r))}{r^n}\le\alpha+Cm. \tag*{(73)} ∣ T Ξ , ε n ( x ) ∣ ≤ ∣ T ε n ( x ) ∣ + C r > ε sup r n ∣ n ∣ ( B ( x , r )) ≤ α + C m . (73) If 0 < ε ≤ h 0<\varepsilon\le h 0 < ε ≤ h
∣ T Ξ , ε n ( x ) ∣ ≤ ∫ ε < d ( x , y ) ≤ 2 h ∣ K Ξ ( x , y ) ∣ d ∣ n ∣ ( y ) + ∣ T Ξ , 2 h n ( x ) ∣ |T_{\Xi,\varepsilon}n(x)|\le\int_{\varepsilon<d(x,y)\le2h}|K_{\Xi}(x,y)|\,d|n|(y)+|T_{\Xi,2h}n(x)| ∣ T Ξ , ε n ( x ) ∣ ≤ ∫ ε < d ( x , y ) ≤ 2 h ∣ K Ξ ( x , y ) ∣ d ∣ n ∣ ( y ) + ∣ T Ξ , 2 h n ( x ) ∣ ≤ C h − n ∣ n ∣ ( B ( x , 3 h ) ) + α + C m ≤ α + C m . \le Ch^{-n}|n|(B(x,3h))+\alpha+Cm\le\alpha+Cm. ≤ C h − n ∣ n ∣ ( B ( x , 3 h )) + α + C m ≤ α + C m . Here we used (3.37), ∣ n ∣ ( B ( x , 3 h ) ) ≤ m ( 3 h ) n |n|(B(x,3h))\le m(3h)^n ∣ n ∣ ( B ( x , 3 h )) ≤ m ( 3 h ) n ε = 2 h \varepsilon=2h ε = 2 h
Let μ \mu μ ν ^ \widehat{\nu} ν R \mathcal{R} R e e e (52) and (55) , respectively. If the suppression is such that
Ξ ( x ) ≥ dist ( x , B ∖ ( H ∪ E ) ) (74) \Xi(x) \ge\operatorname{dist}(x,B\setminus(\mathcal{H}\cup\mathcal{E})) \tag*{(74)} Ξ ( x ) ≥ dist ( x , B ∖ ( H ∪ E )) (74) for every x ∈ X x\in X x ∈ X B ( x , R ( x ) ) ⊂ H B(x,\mathcal{R}(x))\subset\mathcal{H} B ( x , R ( x )) ⊂ H B ( x , e ( x ) ) ⊂ E B(x,e(x))\subset\mathcal{E} B ( x , e ( x )) ⊂ E
Ξ ( x ) ≥ max { R ( x ) , e ( x ) } , for x ∈ B . \Xi(x)\ge\max\{\mathcal{R}(x),e(x)\},\qquad\text{for }x\in B. Ξ ( x ) ≥ max { R ( x ) , e ( x )} , for x ∈ B . By (52) and (55) , Lemma 3.14 applies with R 0 = max { R ( x ) , e ( x ) } R_0=\max\{\mathcal{R}(x),e(x)\} R 0 = max { R ( x ) , e ( x )} m = m m=m m = m α \alpha α Lemma 3.8 .
Lemma 3.15. Suppose that Ξ : X → [ 0 , ∞ ) \Xi:X\to[0,\infty) Ξ : X → [ 0 , ∞ ) (74) . Then there is c ♯ > 0 c_{\sharp}>0 c ♯ > 0 n , D , γ , ε 0 , C , C 0 , C 1 , C 2 n,D,\gamma,\varepsilon_0,C,C_0,C_1,C_2 n , D , γ , ε 0 , C , C 0 , C 1 , C 2
T Ξ , ν ^ ♯ ( x ) ≤ c ♯ , x ∈ B . T_{\Xi,\widehat{\nu}}^{\sharp}(x)\le c_{\sharp},\qquad x\in B. T Ξ , ν ♯ ( x ) ≤ c ♯ , x ∈ B . A Cotlar inequality for different measures
The following SIO boundedness result is a variant of To14, Lemma 5.25 for different measures, formulated in geometrically doubling metric measure spaces, and it will be applied to the measures μ \mu μ ∣ ν i ∣ = ∣ ν ^ i ∣ + μ ∣ X ∖ B i |\nu_i|=|\widehat{\nu}_i|+\mu|_{X\setminus B_i} ∣ ν i ∣ = ∣ ν i ∣ + μ ∣ X ∖ B i i = 1 , 2 i=1,2 i = 1 , 2
Lemma 3.16. Let n > 0 n>0 n > 0 0 < γ ≤ 1 0<\gamma\le1 0 < γ ≤ 1 ( X , d ) (X,d) ( X , d ) D D D Ξ : X → [ 0 , ∞ ) \Xi:X\to[0,\infty) Ξ : X → [ 0 , ∞ ) m 1 m_1 m 1 m 2 m_2 m 2 m 3 m_3 m 3 X X X C , A ≥ 1 C,A\ge1 C , A ≥ 1 x ∈ X x\in X x ∈ X r > 0 r>0 r > 0 r ≥ Ξ ( x ) r\ge\Xi(x) r ≥ Ξ ( x ) i , j ∈ { 1 , 2 , 3 } i,j\in\{1,2,3\} i , j ∈ { 1 , 2 , 3 }
m i ( B ( x , r ) ) ≤ C r n and m i ( B ( x , r ) ) ≤ C m j ( B ( x , A r ) ) . (75) m_i(B(x,r))\le Cr^n\qquad\text{and}\qquad m_i(B(x,r))\le C\,m_j(B(x,Ar)). \tag*{(75)} m i ( B ( x , r )) ≤ C r n and m i ( B ( x , r )) ≤ C m j ( B ( x , A r )) . (75) Let k k k n n n C C C γ \gamma γ
∣ k ( x , y ) ∣ ≤ C min { d ( x , y ) − n , Ξ ( x ) − n , Ξ ( y ) − n } . (76) |k(x,y)|\le C\min\{d(x,y)^{-n},\Xi(x)^{-n},\Xi(y)^{-n}\}. \tag*{(76)} ∣ k ( x , y ) ∣ ≤ C min { d ( x , y ) − n , Ξ ( x ) − n , Ξ ( y ) − n } . (76) with the formal convention 0 − n = + ∞ 0^{-n}=+\infty 0 − n = + ∞ δ 0 > 0 \delta_0>0 δ 0 > 0 k ( x , y ) = 0 k(x,y)=0 k ( x , y ) = 0 d ( x , y ) ≤ δ 0 d(x,y)\le\delta_0 d ( x , y ) ≤ δ 0 f f f
S m 1 f ( x ) : = ∫ X k ( x , y ) f ( y ) d m 1 ( y ) , S_{m_1}f(x):=\int_X k(x,y)f(y)\,dm_1(y), S m 1 f ( x ) := ∫ X k ( x , y ) f ( y ) d m 1 ( y ) , S m 1 , δ f ( x ) : = ∫ d ( x , y ) > δ k ( x , y ) f ( y ) d m 1 ( y ) , for δ > 0 , S_{m_1,\delta}f(x):=\int_{d(x,y)>\delta} k(x,y)f(y)\,dm_1(y),\qquad\text{for }\delta>0, S m 1 , δ f ( x ) := ∫ d ( x , y ) > δ k ( x , y ) f ( y ) d m 1 ( y ) , for δ > 0 , and S m 1 , ♯ f ( x ) : = sup δ > 0 ∣ S m 1 , δ f ( x ) ∣ S_{m_1,\sharp}f(x):=\sup_{\delta>0}|S_{m_1,\delta}f(x)| S m 1 , ♯ f ( x ) := sup δ > 0 ∣ S m 1 , δ f ( x ) ∣ S m 1 S_{m_1} S m 1 L 2 ( m 1 ) L^2(m_1) L 2 ( m 1 ) L 2 ( m 2 ) L^2(m_2) L 2 ( m 2 ) ∥ S m 1 ∥ o p : = ∥ S m 1 ∥ L 2 ( m 1 ) → L 2 ( m 2 ) \|S_{m_1}\|_{\mathrm{op}}:=\|S_{m_1}\|_{L^2(m_1)\to L^2(m_2)} ∥ S m 1 ∥ op := ∥ S m 1 ∥ L 2 ( m 1 ) → L 2 ( m 2 )
∥ S m 1 ∥ L 1 ( m 1 ) → L 1 , ∞ ( m 2 ) ≲ ( 1 + ∥ S m 1 ∥ o p ) 2 (77) \|S_{m_1}\|_{L^1(m_1)\to L^{1,\infty}(m_2)}\lesssim(1+\|S_{m_1}\|_{\mathrm{op}})^2 \tag*{(77)} ∥ S m 1 ∥ L 1 ( m 1 ) → L 1 , ∞ ( m 2 ) ≲ ( 1 + ∥ S m 1 ∥ op ) 2 (77) and
∥ S m 1 , ♯ f ∥ L 2 ( m 3 ) ≲ ( 1 + ∥ S m 1 ∥ o p ) ∥ f ∥ L 2 ( m 1 ) , for f ∈ L 2 ( m 1 ) . (78) \|S_{m_1,\sharp}f\|_{L^2(m_3)}\lesssim(1+\|S_{m_1}\|_{\mathrm{op}})\|f\|_{L^2(m_1)},\qquad\text{for }f\in L^2(m_1). \tag*{(78)} ∥ S m 1 , ♯ f ∥ L 2 ( m 3 ) ≲ ( 1 + ∥ S m 1 ∥ op ) ∥ f ∥ L 2 ( m 1 ) , for f ∈ L 2 ( m 1 ) . (78) The implicit constants depend only on n n n D D D γ \gamma γ C C C A A A δ 0 \delta_0 δ 0
The proof of Lemma 3.16 follows the ideas in [77 Lemmas 5.25-5.27] , and it combines the nonhomogeneous Calderón-Zygmund decomposition and Cotlar inequality (see [77 Chapter 2] ) with the comparisons in (75) . We give the details only where the suppression Ξ \Xi Ξ
Throughout the rest of this subsection, we assume that the hypotheses of Lemma 3.16 hold, and, to simplify the notation, we write S : = S m 1 S := S_{\mathfrak m_1} S := S m 1 S δ : = S m 1 , δ S_\delta:= S_{\mathfrak m_1,\delta} S δ := S m 1 , δ S ♯ : = S m 1 , ♯ S_\sharp:= S_{\mathfrak m_1,\sharp} S ♯ := S m 1 , ♯ ∥ S ∥ o p : = ∥ S m 1 ∥ o p \|S\|_{\mathrm{op}} := \|S_{\mathfrak m_1}\|_{\mathrm{op}} ∥ S ∥ op := ∥ S m 1 ∥ op c ≥ 1 c \ge1 c ≥ 1 [39 Theorem 4.1 and Remark 4.13] , so that every ball B B B Q Q Q
Q ⊂ c B and c − 1 r ( B ) ≤ ℓ ( Q ) ≤ c r ( B ) . (79) Q \subset cB \qquad\text{and}\qquad c^{-1}r(B) \le\ell(Q) \le cr(B). \tag*{(79)} Q ⊂ c B and c − 1 r ( B ) ≤ ℓ ( Q ) ≤ cr ( B ) . (79) We also say that such a cell Q Q Q B B B A A A A > 200 c A > 200c A > 200 c α > 100 A 2 c 3 \alpha> 100A^{2}c^{3} α > 100 A 2 c 3 β > max { α n + 1 , α log 2 D } \beta> \max\{\alpha^{n+1},\alpha^{\log_{2}D}\} β > max { α n + 1 , α l o g 2 D } B B B m \mathfrak m m ( α , β ) (\alpha,\beta) ( α , β ) m ( α B ) ≤ β m ( B ) \mathfrak m(\alpha B) \le\beta\mathfrak m(B) m ( α B ) ≤ β m ( B )
Following the notation in (21) , for a locally finite complex measure n \mathfrak n n
M ~ Ξ , m 2 n ( x ) : = sup r > 0 , r ≥ Ξ ( x ) ∣ n ∣ ( B ( x , r ) ) m 2 ( B ( x , 10 A r ) ) , for x ∈ X , \widetilde{\mathcal M}_{\Xi,\mathfrak m_2}\mathfrak n(x) := \sup_{r>0,\ r\ge\Xi(x)} \frac{|\mathfrak n|(B(x,r))}{\mathfrak m_2(B(x,10Ar))}, \qquad\text{for }x\in X, M Ξ , m 2 n ( x ) := r > 0 , r ≥ Ξ ( x ) sup m 2 ( B ( x , 10 A r )) ∣ n ∣ ( B ( x , r )) , for x ∈ X , with the convention that the quantity on the right-hand side equals 0 0 0 M Ξ r M^\mathrm r_{\Xi} M Ξ r
M Ξ r ( h m i ) ( x ) : = sup r > 0 , r ≥ Ξ ( x ) 1 r n ∫ B ( x , r ) ∣ h ∣ d m i . M^\mathrm r_{\Xi}(h\mathfrak m_i)(x) := \sup_{r>0,\ r\ge\Xi(x)} \frac{1}{r^n}\int_{B(x,r)} |h|\,d\mathfrak m_i. M Ξ r ( h m i ) ( x ) := r > 0 , r ≥ Ξ ( x ) sup r n 1 ∫ B ( x , r ) ∣ h ∣ d m i . These two maximal operators satisfy the following bounds.
Lemma 3.17. Let i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } 1 < s < ∞ 1<s<\infty 1 < s < ∞ h ∈ L s ( m i ) h\in L^s(\mathfrak m_i) h ∈ L s ( m i )
∥ M ~ Ξ , m 2 ( h m i ) ∥ L s ( m 3 ) + ∥ M Ξ r ( h m i ) ∥ L s ( m 3 ) ≲ s ∥ h ∥ L s ( m i ) . (80) \left\|\widetilde{\mathcal M}_{\Xi,\mathfrak m_2}(h\mathfrak m_i)\right\|_{L^s(\mathfrak m_3)}
+
\left\|M^\mathrm r_{\Xi}(h\mathfrak m_i)\right\|_{L^s(\mathfrak m_3)}
\lesssim_s
\|h\|_{L^s(\mathfrak m_i)}. \tag*{(80)} M Ξ , m 2 ( h m i ) L s ( m 3 ) + ∥ M Ξ r ( h m i ) ∥ L s ( m 3 ) ≲ s ∥ h ∥ L s ( m i ) . (80) Proof. For M Ξ r M^\mathrm r_{\Xi} M Ξ r m 3 \mathfrak m_3 m 3 m i \mathfrak m_i m i m 1 \mathfrak m_1 m 1 m 2 \mathfrak m_2 m 2 Ξ \Xi Ξ R 1 \mathcal R_1 R 1 R 2 \mathcal R_2 R 2 (75) provides the required growth. For M ~ Ξ , m 2 \widetilde{\mathcal M}_{\Xi,\mathfrak m_2} M Ξ , m 2 (75) , every ball B = B ( x , r ) B=B(x,r) B = B ( x , r ) r ≥ Ξ ( x ) r\ge\Xi(x) r ≥ Ξ ( x )
m 3 ( 5 B ) ≤ C m 2 ( 5 A B ) ≤ C m 2 ( 10 A B ) , \mathfrak m_3(5B)\le C\mathfrak m_2(5AB)\le C\mathfrak m_2(10AB), m 3 ( 5 B ) ≤ C m 2 ( 5 A B ) ≤ C m 2 ( 10 A B ) , so the 5 r 5r 5 r L 1 ( m i ) L^1(\mathfrak m_i) L 1 ( m i ) L 1 , ∞ ( m 3 ) L^{1,\infty}(\mathfrak m_3) L 1 , ∞ ( m 3 ) (75) also gives M ~ Ξ , m 2 ( h m i ) ≤ C ∥ h ∥ L ∞ ( m i ) \widetilde{\mathcal M}_{\Xi,\mathfrak m_2}(h\mathfrak m_i)\le C\|h\|_{L^\infty(\mathfrak m_i)} M Ξ , m 2 ( h m i ) ≤ C ∥ h ∥ L ∞ ( m i ) (80) follows by interpolation. □
The proof of (78) relies on the following pointwise Cotlar-type inequality.
Lemma 3.18. Suppose that there exists C w > 0 C_w>0 C w > 0
m 2 ( { ∣ S h ∣ > λ } ) ≤ C w λ ∥ h ∥ L 1 ( m 1 ) , (81) \mathfrak m_2\bigl(\{|Sh|>\lambda\}\bigr)\le\frac{C_w}{\lambda}\|h\|_{L^1(\mathfrak m_1)}, \tag*{(81)} m 2 ( { ∣ S h ∣ > λ } ) ≤ λ C w ∥ h ∥ L 1 ( m 1 ) , (81) for every λ > 0 \lambda>0 λ > 0 h h h f f f x ∈ supp m 3 x\in\operatorname{supp}\mathfrak m_3 x ∈ supp m 3
S ♯ f ( x ) ≲ M ~ Ξ , m 2 ( S f m 2 ) ( x ) + M ~ Ξ , m 2 ( f m 1 ) ( x ) + M Ξ r ( f m 1 ) ( x ) , (82) S_\sharp f(x)\lesssim\widetilde{\mathcal M}_{\Xi,\mathfrak m_2}(Sf\mathfrak m_2)(x)+\widetilde{\mathcal M}_{\Xi,\mathfrak m_2}(f\mathfrak m_1)(x)+M^\mathrm r_{\Xi}(f\mathfrak m_1)(x), \tag*{(82)} S ♯ f ( x ) ≲ M Ξ , m 2 ( S f m 2 ) ( x ) + M Ξ , m 2 ( f m 1 ) ( x ) + M Ξ r ( f m 1 ) ( x ) , (82) where the implicit constant depends only on n n n D D D γ \gamma γ C C C A A A C w C_w C w
Proof. We follow the standard proof of Cotlar’s inequality for non-doubling measures, see [77 Chapter 2] . Let x ∈ supp m 3 x \in\operatorname{supp} m_3 x ∈ supp m 3 δ > 0 \delta> 0 δ > 0 δ < Ξ ( x ) \delta< \Xi(x) δ < Ξ ( x ) (76) gives
∣ S δ f ( x ) − S Ξ ( x ) f ( x ) ∣ ≲ M Ξ r ( f m 1 ) ( x ) , \lvert S_\delta f(x)-S_{\Xi(x)}f(x)\rvert\lesssim M^\mathrm{r}_{\Xi}(f m_1)(x), ∣ S δ f ( x ) − S Ξ ( x ) f ( x )∣ ≲ M Ξ r ( f m 1 ) ( x ) , so we can assume without loss of generality that δ ≥ Ξ ( x ) \delta\geq\Xi(x) δ ≥ Ξ ( x )
We set s 0 : = A δ s_0 := A\delta s 0 := A δ B k : = B ( x , α k s 0 ) B_k := B(x,\alpha^k s_0) B k := B ( x , α k s 0 ) k ≥ 0 k \geq0 k ≥ 0 (75) , we have
m 2 ( B 0 ) ≥ C − 1 m 3 ( B ( x , δ ) ) > 0 m_2(B_0) \geq C^{-1}m_3(B(x,\delta)) > 0 m 2 ( B 0 ) ≥ C − 1 m 3 ( B ( x , δ )) > 0 and m 2 ( B k ) ≤ C α k n s 0 n m_2(B_k) \leq C\alpha^{kn}s_0^n m 2 ( B k ) ≤ C α k n s 0 n β > α n \beta> \alpha^n β > α n B k B_k B k m 2 m_2 m 2 ( α , β ) (\alpha,\beta) ( α , β ) m m m k k k R : = α m s 0 R := \alpha^m s_0 R := α m s 0 B : = B m B := B_m B := B m m 2 ( B k ) ≤ β k − m m 2 ( B ) m_2(B_k) \leq\beta^{k-m}m_2(B) m 2 ( B k ) ≤ β k − m m 2 ( B ) 0 ≤ k ≤ m 0 \leq k \leq m 0 ≤ k ≤ m
Since S f ( y ) = S ( f χ 4 B ) ( y ) + S ( f χ X ∖ 4 B ) ( y ) Sf(y)=S(f\chi_{4B})(y)+S(f\chi_{X\setminus4B})(y) S f ( y ) = S ( f χ 4 B ) ( y ) + S ( f χ X ∖ 4 B ) ( y ) y ∈ B y \in B y ∈ B
∣ S δ f ( x ) ∣ ≤ ∣ S δ f ( x ) − S 4 R f ( x ) ∣ + ∣ S 4 R f ( x ) − S ( f χ X ∖ 4 B ) ( y ) ∣ + ∣ S ( f χ 4 B ) ( y ) ∣ + ∣ S f ( y ) ∣ = : I 1 + I 2 ( y ) + I 3 ( y ) + I 4 ( y ) . (83) \begin{aligned}
\lvert S_\delta f(x)\rvert\leq\lvert S_\delta f(x)-S_{4R}f(x)\rvert+\lvert S_{4R}f(x)-S(f\chi_{X\setminus4B})(y)\rvert+\lvert S(f\chi_{4B})(y)\rvert+\lvert Sf(y)\rvert\\
&=: \mathcal{I}_1+\mathcal{I}_2(y)+\mathcal{I}_3(y)+\mathcal{I}_4(y). \tag*{(83)}
\end{aligned} ∣ S δ f ( x )∣ ≤ ∣ S δ f ( x ) − S 4 R f ( x )∣ + ∣ S 4 R f ( x ) − S ( f χ X ∖ 4 B ) ( y )∣ + ∣ S ( f χ 4 B ) ( y )∣ + ∣ S f ( y )∣ =: I 1 + I 2 ( y ) + I 3 ( y ) + I 4 ( y ) . (83) To bound I 1 \mathcal{I}_1 I 1 { z : δ < d ( x , z ) ≤ 4 R } \{z:\delta<d(x,z)\leq4R\} { z : δ < d ( x , z ) ≤ 4 R } { z : δ < d ( x , z ) ≤ s 0 } \{z:\delta<d(x,z)\leq s_0\} { z : δ < d ( x , z ) ≤ s 0 } B k + 1 ∖ B k B_{k+1}\setminus B_k B k + 1 ∖ B k 0 ≤ k ≤ m − 2 0\leq k\leq m-2 0 ≤ k ≤ m − 2 { z : α m ′ s 0 ≤ d ( x , z ) ≤ 4 R } \{z:\alpha^{m'}s_0\leq d(x,z)\leq4R\} { z : α m ′ s 0 ≤ d ( x , z ) ≤ 4 R } m ′ : = max { m − 1 , 0 } m':=\max\{m-1,0\} m ′ := max { m − 1 , 0 } α k + 1 s 0 ≥ Ξ ( x ) \alpha^{k+1}s_0\geq\Xi(x) α k + 1 s 0 ≥ Ξ ( x ) 10 A α k + 1 s 0 ≤ α k + 2 s 0 10A\alpha^{k+1}s_0\leq\alpha^{k+2}s_0 10 A α k + 1 s 0 ≤ α k + 2 s 0
I 1 ≲ 1 δ n ∫ B ( x , 2 s 0 ) ∣ f ∣ d m 1 + ∑ k = 0 m − 2 1 ( α k s 0 ) n ∫ B k + 1 ∣ f ∣ d m 1 + 1 ( α m ′ s 0 ) n ∫ 8 B ∣ f ∣ d m 1 \mathcal{I}_1\lesssim\frac{1}{\delta^n}\int_{B(x,2s_0)}\lvert f\rvert\,\mathrm{d}m_1+\sum_{k=0}^{m-2}\frac{1}{(\alpha^k s_0)^n}\int_{B_{k+1}}\lvert f\rvert\,\mathrm{d}m_1+\frac{1}{(\alpha^{m'}s_0)^n}\int_{8B}\lvert f\rvert\,\mathrm{d}m_1 I 1 ≲ δ n 1 ∫ B ( x , 2 s 0 ) ∣ f ∣ d m 1 + k = 0 ∑ m − 2 ( α k s 0 ) n 1 ∫ B k + 1 ∣ f ∣ d m 1 + ( α m ′ s 0 ) n 1 ∫ 8 B ∣ f ∣ d m 1 ≲ M Ξ r ( f m 1 ) ( x ) + M ~ Ξ , m 2 ( f m 1 ) ( x ) ∑ k = 0 m − 2 m 2 ( B k + 2 ) ( α k s 0 ) n ≲ M Ξ r ( f m 1 ) ( x ) + M ~ Ξ , m 2 ( f m 1 ) ( x ) , (84) \lesssim M^\mathrm{r}_{\Xi}(f m_1)(x)+\widetilde{M}_{\Xi,m_2}(f m_1)(x)\sum_{k=0}^{m-2}\frac{m_2(B_{k+2})}{(\alpha^k s_0)^n}\lesssim M^\mathrm{r}_{\Xi}(f m_1)(x)+\widetilde{M}_{\Xi,m_2}(f m_1)(x), \tag*{(84)} ≲ M Ξ r ( f m 1 ) ( x ) + M Ξ , m 2 ( f m 1 ) ( x ) k = 0 ∑ m − 2 ( α k s 0 ) n m 2 ( B k + 2 ) ≲ M Ξ r ( f m 1 ) ( x ) + M Ξ , m 2 ( f m 1 ) ( x ) , (84) where, in the last inequality, we used that β > α n \beta>\alpha^n β > α n
m 2 ( B k + 2 ) ≤ β k + 2 − m m 2 ( B ) ≤ C β k + 2 − m R n . m_2(B_{k+2})\leq\beta^{k+2-m}m_2(B)\leq C\beta^{k+2-m}R^n. m 2 ( B k + 2 ) ≤ β k + 2 − m m 2 ( B ) ≤ C β k + 2 − m R n . To bound I 2 ( y ) \mathcal{I}_2(y) I 2 ( y ) y ∈ B y\in B y ∈ B k k k 2 ℓ + 2 R ≤ d ( x , z ) < 2 ℓ + 3 R 2^{\ell+2}R\leq d(x,z)<2^{\ell+3}R 2 ℓ + 2 R ≤ d ( x , z ) < 2 ℓ + 3 R ℓ ≥ 0 \ell\geq0 ℓ ≥ 0 d ( x , z ) = 4 R d(x,z)=4R d ( x , z ) = 4 R
I 2 ( y ) ≲ 1 R n ∫ 8 B ∣ f ∣ d m 1 + ∑ ℓ ≥ 0 R γ ( 2 ℓ + 2 R ) n + γ ∫ B ( x , 2 ℓ + 3 R ) ∣ f ∣ d m 1 ≲ M Ξ r ( f m 1 ) ( x ) . (85) \mathcal{I}_2(y)\lesssim\frac{1}{R^n}\int_{8B}\lvert f\rvert\,\mathrm{d}m_1+\sum_{\ell\geq0}\frac{R^\gamma}{(2^{\ell+2}R)^{n+\gamma}}\int_{B(x,2^{\ell+3}R)}\lvert f\rvert\,\mathrm{d}m_1\lesssim M^\mathrm{r}_{\Xi}(f m_1)(x). \tag*{(85)} I 2 ( y ) ≲ R n 1 ∫ 8 B ∣ f ∣ d m 1 + ℓ ≥ 0 ∑ ( 2 ℓ + 2 R ) n + γ R γ ∫ B ( x , 2 ℓ + 3 R ) ∣ f ∣ d m 1 ≲ M Ξ r ( f m 1 ) ( x ) . (85) To bound I 3 ( y ) \mathcal{I}_3(y) I 3 ( y ) I 4 ( y ) \mathcal{I}_4(y) I 4 ( y ) y y y (81) and by Chebyshev’s inequality, respectively, each of the inequalities
∣ S ( f χ 4 B ) ( y ) ∣ > 4 C w m 2 ( B ) ∫ 4 B ∣ f ∣ d m 1 and ∣ S f ( y ) ∣ > 4 m 2 ( B ) ∫ B ∣ S f ∣ d m 2 \lvert S(f\chi_{4B})(y)\rvert>\frac{4C_w}{m_2(B)}\int_{4B}\lvert f\rvert\,\mathrm{d}m_1
\qquad\text{and}\qquad
\lvert Sf(y)\rvert>\frac{4}{m_2(B)}\int_B\lvert Sf\rvert\,\mathrm{d}m_2 ∣ S ( f χ 4 B ) ( y )∣ > m 2 ( B ) 4 C w ∫ 4 B ∣ f ∣ d m 1 and ∣ S f ( y )∣ > m 2 ( B ) 4 ∫ B ∣ S f ∣ d m 2 holds on a subset of B B B m 2 m_2 m 2 m 2 ( B ) / 4 m_2(B)/4 m 2 ( B ) /4 m 2 ( B ) ≥ m 2 ( B 0 ) > 0 m_2(B)\geq m_2(B_0)>0 m 2 ( B ) ≥ m 2 ( B 0 ) > 0 y ∈ B y\in B y ∈ B α > 40 A \alpha>40A α > 40 A B B B m 2 m_2 m 2 ( α , β ) (\alpha,\beta) ( α , β ) m 2 ( 40 A B ) ≤ β m 2 ( B ) m_2(40AB)\leq\beta m_2(B) m 2 ( 40 A B ) ≤ β m 2 ( B )
I 3 ( y ) ≤ 4 C w β M ~ Ξ , m 2 ( f m 1 ) ( x ) and I 4 ( y ) ≤ 4 β M ~ Ξ , m 2 ( S f m 2 ) ( x ) , \mathcal{I}_3(y)\leq4C_w\beta\widetilde{M}_{\Xi,m_2}(f m_1)(x)
\qquad\text{and}\qquad
\mathcal{I}_4(y)\leq4\beta\widetilde{M}_{\Xi,m_2}(Sf m_2)(x), I 3 ( y ) ≤ 4 C w β M Ξ , m 2 ( f m 1 ) ( x ) and I 4 ( y ) ≤ 4 β M Ξ , m 2 ( S f m 2 ) ( x ) , which, together with (83) , (84) , and (85) , proves (82) . □
Finally, we apply Lemmas 3.17 and 3.18 to prove the opening result of this subsection.
Proof of Lemma 3.16. We first prove (77) for a bounded function f f f λ > 0 \lambda> 0 λ > 0 m 1 ( X ) < ∞ \mathrm{m}_1(X) < \infty m 1 ( X ) < ∞ λ ≤ ∥ f ∥ L 1 ( m 1 ) / m 1 ( X ) \lambda\le\lVert f\rVert_{L^1(\mathrm{m}_1)}/\mathrm{m}_1(X) λ ≤ ∥ f ∥ L 1 ( m 1 ) / m 1 ( X ) (75) , followed by monotone convergence as r → ∞ r \to\infty r → ∞ m 2 ( X ) ≤ C m 1 ( X ) \mathrm{m}_2(X) \le C\mathrm{m}_1(X) m 2 ( X ) ≤ C m 1 ( X )
m 2 ( { ∣ S f ∣ > 2 λ } ) ≤ m 2 ( X ) ≤ C m 1 ( X ) ≤ C λ ∥ f ∥ L 1 ( m 1 ) . \mathrm{m}_2(\{\lvert Sf\rvert> 2\lambda\}) \le\mathrm{m}_2(X) \le C\mathrm{m}_1(X) \le\frac{C}{\lambda}\lVert f\rVert_{L^1(\mathrm{m}_1)}. m 2 ({∣ S f ∣ > 2 λ }) ≤ m 2 ( X ) ≤ C m 1 ( X ) ≤ λ C ∥ f ∥ L 1 ( m 1 ) . Thus, we can assume that λ ≫ ∥ f ∥ L 1 ( m 1 ) / m 1 ( X ) \lambda\gg\lVert f\rVert_{L^1(\mathrm{m}_1)}/\mathrm{m}_1(X) λ ≫ ∥ f ∥ L 1 ( m 1 ) / m 1 ( X ) m 1 ( X ) < ∞ \mathrm{m}_1(X) < \infty m 1 ( X ) < ∞
We perform the Calderón–Zygmund decomposition of [77 Lemma 2.14] as in [16 Theorem A.2] , on the adjacent dyadic lattices of [39 Theorem 4.1] , with the following choices. We say that a cell Q Q Q bad if it is associated with a ball B B B (79) , such that ∫ Q ∣ f ∣ d m 1 > λ m 1 ( A 2 B ) \int_Q \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 > \lambda\mathrm{m}_1(A^2B) ∫ Q ∣ f ∣ d m 1 > λ m 1 ( A 2 B ) f f f λ \lambda λ Q j Q_j Q j ℓ ( Q ) ≤ ℓ ( Q j ) \ell(Q) \le\ell(Q_j) ℓ ( Q ) ≤ ℓ ( Q j ) Q ⊃ Q j Q \supset Q_j Q ⊃ Q j { Q j } j \{Q_j\}_j { Q j } j j j j B j = B ( z j , r j ) B_j = B(z_j,r_j) B j = B ( z j , r j ) Q j Q_j Q j B j B_j B j
∫ Q j ∣ f ∣ d m 1 > λ m 1 ( A 2 B j ) . (86) \int_{Q_j} \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 > \lambda\mathrm{m}_1(A^2B_j). \tag*{(86)} ∫ Q j ∣ f ∣ d m 1 > λ m 1 ( A 2 B j ) . (86) By maximality, we also have
∫ Q ∣ f ∣ d m 1 ≤ λ m 1 ( A 2 B ) , if Q j ⊂ Q , ℓ ( Q j ) < ℓ ( Q ) , and Q is associated with B . (87) \int_Q \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 \le\lambda\mathrm{m}_1(A^2B), \qquad\text{if } Q_j \subset Q,\ \ell(Q_j) < \ell(Q),\ \text{and } Q\text{ is associated with }B. \tag*{(87)} ∫ Q ∣ f ∣ d m 1 ≤ λ m 1 ( A 2 B ) , if Q j ⊂ Q , ℓ ( Q j ) < ℓ ( Q ) , and Q is associated with B . (87) Moreover, by [37 Lemma 3.3] and the choice of β \beta β m 1 \mathrm{m}_1 m 1 x ∉ ⋃ j Q j x \notin\bigcup_j Q_j x ∈ / ⋃ j Q j m 1 \mathrm{m}_1 m 1 ( α , β ) (\alpha,\beta) ( α , β ) B B B x x x Q Q Q B B B x x x Q j Q_j Q j A 2 < α A^2 < \alpha A 2 < α
∫ B ∣ f ∣ d m 1 ≤ ∫ Q ∣ f ∣ d m 1 ≤ λ m 1 ( A 2 B ) ≤ β λ m 1 ( B ) , \int_B \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 \le\int_Q \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 \le\lambda\mathrm{m}_1(A^2B) \le\beta\lambda\mathrm{m}_1(B), ∫ B ∣ f ∣ d m 1 ≤ ∫ Q ∣ f ∣ d m 1 ≤ λ m 1 ( A 2 B ) ≤ β λ m 1 ( B ) , and, by the differentiation theorem along doubling balls [37 Corollary 3.6] , we obtain ∣ f ( y ) ∣ ≤ β λ \lvert f(y)\rvert\le\beta\lambda ∣ f ( y )∣ ≤ β λ m 1 \mathrm{m}_1 m 1 y ∈ X ∖ ⋃ j Q j y \in X \setminus\bigcup_j Q_j y ∈ X ∖ ⋃ j Q j
We denote
w j : = χ Q j ∑ k χ Q k , m j : = ∫ Q j ∣ f ∣ d m 1 , and t j : = max { 10 A r j , 10 Ξ ( z j ) } , w_j := \frac{\chi_{Q_j}}{\sum_k \chi_{Q_k}}, \qquad m_j := \int_{Q_j} \lvert f\rvert\, \mathrm{d}\mathrm{m}_1, \qquad\text{and}\qquad t_j := \max\{10Ar_j,10\Xi(z_j)\}, w j := ∑ k χ Q k χ Q j , m j := ∫ Q j ∣ f ∣ d m 1 , and t j := max { 10 A r j , 10Ξ ( z j )} , with w j ≡ 0 w_j \equiv0 w j ≡ 0 ⋃ k Q k \bigcup_k Q_k ⋃ k Q k k ≥ 0 k \ge0 k ≥ 0 D j , k : = B ( z j , α k t j ) D_{j,k} := B(z_j,\alpha^k t_j) D j , k := B ( z j , α k t j ) k j k_j k j k ≥ 0 k \ge0 k ≥ 0 D j , k D_{j,k} D j , k m 1 \mathrm{m}_1 m 1 ( α , β ) (\alpha,\beta) ( α , β ) t j ≥ Ξ ( z j ) t_j \ge\Xi(z_j) t j ≥ Ξ ( z j ) m 1 ( D j , 0 ) ≥ m 1 ( Q j ) > 0 \mathrm{m}_1(D_{j,0}) \ge\mathrm{m}_1(Q_j) > 0 m 1 ( D j , 0 ) ≥ m 1 ( Q j ) > 0 D j : = D j , k j D_j := D_{j,k_j} D j := D j , k j r ( D j ) = α k j t j r(D_j) = \alpha^{k_j}t_j r ( D j ) = α k j t j R j R_j R j D j D_j D j
m 1 ( D j , k ) ≤ β k − k j m 1 ( D j ) , for 0 ≤ k ≤ k j . (88) \mathrm{m}_1(D_{j,k}) \le\beta^{k-k_j}\mathrm{m}_1(D_j), \qquad\text{for }0 \le k \le k_j. \tag*{(88)} m 1 ( D j , k ) ≤ β k − k j m 1 ( D j ) , for 0 ≤ k ≤ k j . (88) If R k ∩ R j ≠ ∅ R_k \cap R_j \ne\varnothing R k ∩ R j = ∅ r ( D k ) ≤ r ( D j ) r(D_k) \le r(D_j) r ( D k ) ≤ r ( D j ) R k ⊂ 3 c D j R_k \subset3cD_j R k ⊂ 3 c D j Q Q Q 3 c D j 3cD_j 3 c D j Q j ⊂ Q Q_j \subset Q Q j ⊂ Q ℓ ( Q j ) ≤ c r j < 3 r ( D j ) ≤ ℓ ( Q ) \ell(Q_j) \le cr_j < 3r(D_j) \le\ell(Q) ℓ ( Q j ) ≤ c r j < 3 r ( D j ) ≤ ℓ ( Q ) r ( D j ) ≥ 10 A r j r(D_j) \ge10Ar_j r ( D j ) ≥ 10 A r j (87) , the choice of α \alpha α k j k_j k j
∫ 3 c D j ∣ f ∣ d m 1 ≤ λ m 1 ( 3 A 2 c D j ) ≤ β λ m 1 ( D j ) ≤ β λ m 1 ( R j ) . (89) \int_{3cD_j} \lvert f\rvert\, \mathrm{d}\mathrm{m}_1 \le\lambda\mathrm{m}_1(3A^2cD_j) \le\beta\lambda\mathrm{m}_1(D_j) \le\beta\lambda\mathrm{m}_1(R_j). \tag*{(89)} ∫ 3 c D j ∣ f ∣ d m 1 ≤ λ m 1 ( 3 A 2 c D j ) ≤ β λ m 1 ( D j ) ≤ β λ m 1 ( R j ) . (89) Arguing as in the proof of [77 Lemma 2.14] , these two properties provide functions φ j \varphi_j φ j
supp φ j ⊂ R j , ∫ φ j d m 1 = ∫ w j f d m 1 , ∥ φ j ∥ L ∞ ( m 1 ) ≤ 2 m j m 1 ( R j ) , and ∑ j ∣ φ j ∣ ≤ 4 β λ . (90) \begin{aligned}
\operatorname{supp}\varphi_j &\subset R_j,\qquad\int\varphi_j\,\mathrm{d}m_1=\int w_jf\,\mathrm{d}m_1,\\
\|\varphi_j\|_{L^\infty(m_1)}&\le\frac{2m_j}{m_1(R_j)},\qquad\text{and}\qquad\sum_j|\varphi_j|\le4\beta\lambda. \tag*{(90)}
\end{aligned} supp φ j ∥ φ j ∥ L ∞ ( m 1 ) ⊂ R j , ∫ φ j d m 1 = ∫ w j f d m 1 , ≤ m 1 ( R j ) 2 m j , and j ∑ ∣ φ j ∣ ≤ 4 β λ . (90) Indeed, if the cells are finitely many and ordered so that r ( D j ) r(D_j) r ( D j ) j j j φ j : = m 1 ( G j ) − 1 ( ∫ w j f d m 1 ) χ G j \varphi_j:=m_1(G_j)^{-1}\left(\int w_jf\,\mathrm{d}m_1\right)\chi_{G_j} φ j := m 1 ( G j ) − 1 ( ∫ w j f d m 1 ) χ G j G j : = R j ∩ { ∑ k < j ∣ φ k ∣ ≤ 2 β λ } G_j:=R_j\cap\{\sum_{k<j}|\varphi_k|\le2\beta\lambda\} G j := R j ∩ { ∑ k < j ∣ φ k ∣ ≤ 2 β λ }
∫ R j ∑ k < j ∣ φ k ∣ d m 1 ≤ ∫ 3 c D j ∣ f ∣ d m 1 ≤ β λ m 1 ( R j ) , \int_{R_j}\sum_{k<j}|\varphi_k|\,\mathrm{d}m_1\le\int_{3cD_j}|f|\,\mathrm{d}m_1\le\beta\lambda m_1(R_j), ∫ R j k < j ∑ ∣ φ k ∣ d m 1 ≤ ∫ 3 c D j ∣ f ∣ d m 1 ≤ β λ m 1 ( R j ) , by (89) , Chebyshev’s inequality gives m 1 ( G j ) ≥ m 1 ( R j ) / 2 m_1(G_j)\ge m_1(R_j)/2 m 1 ( G j ) ≥ m 1 ( R j ) /2
We now set
g : = f χ X ∖ ⋃ j Q j + ∑ j φ j and b j : = w j f − φ j , g:=f\chi_{X\setminus\bigcup_j Q_j}+\sum_j\varphi_j\qquad\text{and}\qquad b_j:=w_jf-\varphi_j, g := f χ X ∖ ⋃ j Q j + j ∑ φ j and b j := w j f − φ j , so that f = g + ∑ j b j f=g+\sum_jb_j f = g + ∑ j b j Q j Q_j Q j (90) imply
∥ g ∥ L 1 ( m 1 ) ≲ ∥ f ∥ L 1 ( m 1 ) , ∥ g ∥ L ∞ ( m 1 ) ≲ λ , ∑ j m j ≲ ∥ f ∥ L 1 ( m 1 ) , \|g\|_{L^1(m_1)}\lesssim\|f\|_{L^1(m_1)},\qquad\|g\|_{L^\infty(m_1)}\lesssim\lambda,\qquad\sum_jm_j\lesssim\|f\|_{L^1(m_1)}, ∥ g ∥ L 1 ( m 1 ) ≲ ∥ f ∥ L 1 ( m 1 ) , ∥ g ∥ L ∞ ( m 1 ) ≲ λ , j ∑ m j ≲ ∥ f ∥ L 1 ( m 1 ) , ∫ b j d m 1 = 0 , supp b j ⊂ c D j , and ∥ b j ∥ L 1 ( m 1 ) ≤ 2 m j . (91) \int b_j\,\mathrm{d}m_1=0,\qquad\operatorname{supp}b_j\subset cD_j,\qquad\text{and}\qquad\|b_j\|_{L^1(m_1)}\le2m_j. \tag*{(91)} ∫ b j d m 1 = 0 , supp b j ⊂ c D j , and ∥ b j ∥ L 1 ( m 1 ) ≤ 2 m j . (91) We set J : = { j : A r j ≥ Ξ ( z j ) } \mathcal{J}:=\{j:A_{r_j}\ge\Xi(z_j)\} J := { j : A r j ≥ Ξ ( z j )} E : = ⋃ j ∈ J A B j E:=\bigcup_{j\in\mathcal{J}}AB_j E := ⋃ j ∈ J A B j ∣ S h ( x ) ∣ ≤ C δ 0 − n ∥ h ∥ L 1 ( m 1 ) |Sh(x)|\le C\delta_0^{-n}\|h\|_{L^1(m_1)} ∣ S h ( x ) ∣ ≤ C δ 0 − n ∥ h ∥ L 1 ( m 1 ) x ∈ X x\in X x ∈ X h h h S S S L 1 ( m 1 ) L^1(m_1) L 1 ( m 1 ) m 2 m_2 m 2 S S S L 1 ( m 1 ) ∩ L 2 ( m 1 ) L^1(m_1)\cap L^2(m_1) L 1 ( m 1 ) ∩ L 2 ( m 1 ) S f = S g + ∑ j S b j Sf=Sg+\sum_jSb_j S f = S g + ∑ j S b j
m 2 ( { ∣ S f ∣ > 2 λ } ) ≤ m 2 ( E ) + ∥ S g ∥ L 2 ( m 2 ) 2 λ 2 + 1 λ ∑ j ∫ X ∖ E ∣ S b j ∣ d m 2 . (92) m_2(\{|Sf|>2\lambda\})\le m_2(E)+\frac{\|Sg\|_{L^2(m_2)}^2}{\lambda^2}+\frac{1}{\lambda}\sum_j\int_{X\setminus E}|Sb_j|\,\mathrm{d}m_2. \tag*{(92)} m 2 ({ ∣ S f ∣ > 2 λ }) ≤ m 2 ( E ) + λ 2 ∥ S g ∥ L 2 ( m 2 ) 2 + λ 1 j ∑ ∫ X ∖ E ∣ S b j ∣ d m 2 . (92) By (75) and the definition of J \mathcal{J} J (86) , and (91) , we have
m 2 ( E ) ≤ ∑ j ∈ J m 2 ( A B j ) ≤ C ∑ j ∈ J m 1 ( A 2 B j ) ≤ C λ ∑ j m j ≲ ∥ f ∥ L 1 ( m 1 ) λ , m_2(E)\le\sum_{j\in\mathcal{J}}m_2(AB_j)\le C\sum_{j\in\mathcal{J}}m_1(A^2B_j)\le\frac{C}{\lambda}\sum_jm_j\lesssim\frac{\|f\|_{L^1(m_1)}}{\lambda}, m 2 ( E ) ≤ j ∈ J ∑ m 2 ( A B j ) ≤ C j ∈ J ∑ m 1 ( A 2 B j ) ≤ λ C j ∑ m j ≲ λ ∥ f ∥ L 1 ( m 1 ) , and, moreover, (91) gives ∥ S g ∥ L 2 ( m 2 ) 2 ≤ ∥ S ∥ o p 2 ∥ g ∥ L 2 ( m 1 ) 2 ≲ ∥ S ∥ o p 2 λ ∥ f ∥ L 1 ( m 1 ) \|Sg\|_{L^2(m_2)}^2\le\|S\|_{\mathrm{op}}^2\|g\|_{L^2(m_1)}^2\lesssim\|S\|_{\mathrm{op}}^2\lambda\|f\|_{L^1(m_1)} ∥ S g ∥ L 2 ( m 2 ) 2 ≤ ∥ S ∥ op 2 ∥ g ∥ L 2 ( m 1 ) 2 ≲ ∥ S ∥ op 2 λ ∥ f ∥ L 1 ( m 1 ) (92) , we fix j j j k j ′ : = max { k j − 1 , 0 } k'_j:=\max\{k_j-1,0\} k j ′ := max { k j − 1 , 0 } ∣ S b j ∣ ≤ ∣ S ( w j f ) ∣ + ∣ S φ j ∣ |Sb_j|\le|S(w_jf)|+|S\varphi_j| ∣ S b j ∣ ≤ ∣ S ( w j f ) ∣ + ∣ S φ j ∣ D j , k j ′ D_{j,k'_j} D j , k j ′ D j , 0 D_{j,0} D j , 0 D j , k + 1 ∖ D j , k D_{j,k+1}\setminus D_{j,k} D j , k + 1 ∖ D j , k 0 ≤ k ≤ k j − 2 0\le k\le k_j-2 0 ≤ k ≤ k j − 2
∫ X ∖ E ∣ S b j ∣ d m 2 ≤ ∫ X ∖ 4 c D j ∣ S b j ∣ d m 2 + ∫ 4 c D j ∣ S φ j ∣ d m 2 + ∫ D j , 0 ∖ E ∣ S ( w j f ) ∣ d m 2 + ∑ k = 0 k j − 2 ∫ D j , k + 1 ∖ D j , k ∣ S ( w j f ) ∣ d m 2 + ∫ 4 c D j ∖ D j , k j ′ ∣ S ( w j f ) ∣ d m 2 = : I 1 + I 2 + I 3 + I 4 + I 5 , (93) \begin{aligned}
\int_{X\setminus E}|Sb_j|\,\mathrm{d}m_2
&\le\int_{X\setminus4cD_j}|Sb_j|\,\mathrm{d}m_2+\int_{4cD_j}|S\varphi_j|\,\mathrm{d}m_2+\int_{D_{j,0}\setminus E}|S(w_jf)|\,\mathrm{d}m_2\\
&\quad+\sum_{k=0}^{k_j-2}\int_{D_{j,k+1}\setminus D_{j,k}}|S(w_jf)|\,\mathrm{d}m_2+\int_{4cD_j\setminus D_{j,k'_j}}|S(w_jf)|\,\mathrm{d}m_2\\
&=:\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3+\mathcal{I}_4+\mathcal{I}_5, \tag*{(93)}
\end{aligned} ∫ X ∖ E ∣ S b j ∣ d m 2 ≤ ∫ X ∖ 4 c D j ∣ S b j ∣ d m 2 + ∫ 4 c D j ∣ S φ j ∣ d m 2 + ∫ D j , 0 ∖ E ∣ S ( w j f ) ∣ d m 2 + k = 0 ∑ k j − 2 ∫ D j , k + 1 ∖ D j , k ∣ S ( w j f ) ∣ d m 2 + ∫ 4 c D j ∖ D j , k j ′ ∣ S ( w j f ) ∣ d m 2 =: I 1 + I 2 + I 3 + I 4 + I 5 , (93) where the sum is understood as zero when $k_j<2.
The terms J 1 \mathcal{J}_1 J 1 J 2 \mathcal{J}_2 J 2 J 4 \mathcal{J}_4 J 4 J 5 \mathcal{J}_5 J 5 ( 1 , 1 ) (1,1) ( 1 , 1 ) [77 Chapter 2] , since (75) can be applied to balls centered at z j z_j z j t j ≥ 10 Ξ ( z j ) t_j \ge10\Xi(z_j) t j ≥ 10Ξ ( z j )
More precisely, cancellation, kernel regularity, and the growth of m 2 m_2 m 2 2 ℓ + 2 c r ( D j ) ≤ d ( x , z j ) < 2 ℓ + 3 c r ( D j ) 2^{\ell+2c}r(D_j) \le d(x,z_j) < 2^{\ell+3c}r(D_j) 2 ℓ + 2 c r ( D j ) ≤ d ( x , z j ) < 2 ℓ + 3 c r ( D j )
J 1 ≲ ∥ b j ∥ L 1 ( m 1 ) ≲ m j . \mathcal{J}_1 \lesssim\|b_j\|_{L^1(m_1)} \lesssim m_j. J 1 ≲ ∥ b j ∥ L 1 ( m 1 ) ≲ m j . By (75) , the choice of α \alpha α k j k_j k j m 2 ( 4 c D j ) ≤ C m 1 ( 4 A c D j ) ≤ C β m 1 ( R j ) m_2(4cD_j) \le C m_1(4AcD_j) \le C\beta m_1(R_j) m 2 ( 4 c D j ) ≤ C m 1 ( 4 A c D j ) ≤ C β m 1 ( R j ) (90) give
J 2 ≤ m 2 ( 4 c D j ) 1 / 2 ∥ S ∥ o p ∥ φ j ∥ L 2 ( m 1 ) \mathcal{J}_2 \le m_2(4cD_j)^{1/2}\|S\|_{\mathrm{op}}\|\varphi_j\|_{L^2(m_1)} J 2 ≤ m 2 ( 4 c D j ) 1/2 ∥ S ∥ op ∥ φ j ∥ L 2 ( m 1 ) ≲ ∥ S ∥ o p m 1 ( R j ) 1 / 2 ( ∥ φ j ∥ L ∞ ( m 1 ) ∥ φ j ∥ L 1 ( m 1 ) ) 1 / 2 ≲ ∥ S ∥ o p m j . \lesssim\|S\|_{\mathrm{op}}m_1(R_j)^{1/2}\bigl(\|\varphi_j\|_{L^\infty(m_1)}\|\varphi_j\|_{L^1(m_1)}\bigr)^{1/2}\lesssim\|S\|_{\mathrm{op}}m_j. ≲ ∥ S ∥ op m 1 ( R j ) 1/2 ( ∥ φ j ∥ L ∞ ( m 1 ) ∥ φ j ∥ L 1 ( m 1 ) ) 1/2 ≲ ∥ S ∥ op m j . Moreover, we have d ( x , y ) ≥ α k j t j / 2 d(x,y) \ge\alpha^{k_j}t_j/2 d ( x , y ) ≥ α k j t j /2 x ∉ D j , k x \notin D_{j,k} x ∈ / D j , k y ∈ Q j y \in Q_j y ∈ Q j m 2 ( D j , k + 1 ) ≤ C m 1 ( D j , k + 2 ) m_2(D_{j,k+1}) \le Cm_1(D_{j,k+2}) m 2 ( D j , k + 1 ) ≤ C m 1 ( D j , k + 2 ) A < α A < \alpha A < α (88) , (75) , and the fact that β > α n \beta> \alpha^n β > α n
J 4 + J 5 ≲ m j ∑ k = 0 k j − 2 m 1 ( D j , k + 2 ) ( α k t j ) n + m j m 2 ( 4 c D j ) ( α k j t j ) n \mathcal{J}_4+\mathcal{J}_5 \lesssim m_j\sum_{k=0}^{k_j-2}\frac{m_1(D_{j,k+2})}{(\alpha^k t_j)^n}+m_j\frac{m_2(4cD_j)}{(\alpha^{k_j}t_j)^n} J 4 + J 5 ≲ m j k = 0 ∑ k j − 2 ( α k t j ) n m 1 ( D j , k + 2 ) + m j ( α k j t j ) n m 2 ( 4 c D j ) ≲ m j m 1 ( D j ) r ( D j ) n α 2 n ∑ k = 0 k j − 2 ( α n β ) k j − k − 2 + α n m j ≲ m j . \lesssim m_j\frac{m_1(D_j)}{r(D_j)^n}\alpha^{2n}\sum_{k=0}^{k_j-2}\left(\frac{\alpha^n}{\beta}\right)^{k_j-k-2}+\alpha^n m_j\lesssim m_j. ≲ m j r ( D j ) n m 1 ( D j ) α 2 n k = 0 ∑ k j − 2 ( β α n ) k j − k − 2 + α n m j ≲ m j . The term J 3 \mathcal{J}_3 J 3 j ∉ J j\notin\mathcal{J} j ∈ / J t j = 10 Ξ ( z j ) t_j=10\Xi(z_j) t j = 10Ξ ( z j ) Ξ ( y ) ≥ Ξ ( z j ) − c r j ≥ 1 2 Ξ ( z j ) \Xi(y)\ge\Xi(z_j)-cr_j\ge\frac{1}{2}\Xi(z_j) Ξ ( y ) ≥ Ξ ( z j ) − c r j ≥ 2 1 Ξ ( z j ) y ∈ Q j y\in Q_j y ∈ Q j (76) and (75) give
J 3 ≤ ∫ D j , 0 ∣ S ( w j f ) ∣ d m 2 ≲ m j Ξ ( z j ) n m 2 ( B ( z j , 10 Ξ ( z j ) ) ) ≲ m j . \mathcal{J}_3\le\int_{D_{j,0}}|S(w_jf)|\,dm_2\lesssim\frac{m_j}{\Xi(z_j)^n}m_2\bigl(B(z_j,10\Xi(z_j))\bigr)\lesssim m_j. J 3 ≤ ∫ D j , 0 ∣ S ( w j f ) ∣ d m 2 ≲ Ξ ( z j ) n m j m 2 ( B ( z j , 10Ξ ( z j )) ) ≲ m j . If j ∈ J j\in\mathcal{J} j ∈ J t j = 10 A r j t_j=10Ar_j t j = 10 A r j A B j ⊂ E AB_j\subset E A B j ⊂ E d ( x , y ) ≥ ( A − c ) r j ≥ A r j / 2 d(x,y)\ge(A-c)r_j\ge Ar_j/2 d ( x , y ) ≥ ( A − c ) r j ≥ A r j /2 x ∈ D j , 0 ∖ A B j x\in D_{j,0}\setminus AB_j x ∈ D j , 0 ∖ A B j y ∈ Q j y\in Q_j y ∈ Q j (75) yield
J 3 ≤ ∫ D j , 0 ∖ A B j ∣ S ( w j f ) ∣ d m 2 ≲ m j ( A r j ) n m 2 ( B ( z j , 10 A r j ) ) ≲ m j . \mathcal{J}_3\le\int_{D_{j,0}\setminus AB_j}|S(w_jf)|\,dm_2\lesssim\frac{m_j}{(Ar_j)^n}m_2\bigl(B(z_j,10Ar_j)\bigr)\lesssim m_j. J 3 ≤ ∫ D j , 0 ∖ A B j ∣ S ( w j f ) ∣ d m 2 ≲ ( A r j ) n m j m 2 ( B ( z j , 10 A r j ) ) ≲ m j . Gathering the bounds for J 1 \mathcal{J}_1 J 1 J 2 \mathcal{J}_2 J 2 J 3 \mathcal{J}_3 J 3 J 4 \mathcal{J}_4 J 4 J 5 \mathcal{J}_5 J 5 (93) , we obtain
∫ X ∖ E ∣ S b j ∣ d m 2 ≲ ( 1 + ∥ S ∥ o p ) m j , \int_{X\setminus E}|Sb_j|\,dm_2\lesssim(1+\|S\|_{\mathrm{op}})m_j, ∫ X ∖ E ∣ S b j ∣ d m 2 ≲ ( 1 + ∥ S ∥ op ) m j , which, together with the bounds for m 2 ( E ) m_2(E) m 2 ( E ) ∥ S g ∥ L 2 ( m 2 ) \|Sg\|_{L^2(m_2)} ∥ S g ∥ L 2 ( m 2 ) (91) , plugged into (92) , yields
m 2 ( { ∣ S f ∣ > 2 λ } ) ≲ ( 1 + ∥ S ∥ o p ) 2 λ ∥ f ∥ L 1 ( m 1 ) . m_2(\{|Sf|>2\lambda\})\lesssim\frac{(1+\|S\|_{\mathrm{op}})^2}{\lambda}\|f\|_{L^1(m_1)}. m 2 ({ ∣ S f ∣ > 2 λ }) ≲ λ ( 1 + ∥ S ∥ op ) 2 ∥ f ∥ L 1 ( m 1 ) . This proves (77) for bounded functions with bounded support. The corresponding statement for functions in L 1 ( m 1 ) L^1(m_1) L 1 ( m 1 )
We now prove (78) . We assume without loss of generality that ∥ S ∥ o p ≤ 1 \|S\|_{\mathrm{op}}\le1 ∥ S ∥ op ≤ 1 S S S k k k ( 1 + ∥ S ∥ o p ) − 1 S (1+\|S\|_{\mathrm{op}})^{-1}S ( 1 + ∥ S ∥ op ) − 1 S ( 1 + ∥ S ∥ o p ) − 1 k (1+\|S\|_{\mathrm{op}})^{-1}k ( 1 + ∥ S ∥ op ) − 1 k (77) shows that (81) holds with a constant C w C_w C w n n n D D D γ \gamma γ C C C A A A
Let f f f (82) holds for every x ∈ supp m 3 x\in\operatorname{supp}m_3 x ∈ supp m 3 (80) with i = 2 i=2 i = 2 s = 2 s=2 s = 2
∥ M ~ Ξ , m 2 ( S f m 2 ) ∥ L 2 ( m 3 ) ≲ ∥ S f ∥ L 2 ( m 2 ) ≤ ∥ f ∥ L 2 ( m 1 ) . \|\widetilde{\mathcal{M}}_{\Xi,m_2}(Sf\,m_2)\|_{L^2(m_3)}\lesssim\|Sf\|_{L^2(m_2)}\le\|f\|_{L^2(m_1)}. ∥ M Ξ , m 2 ( S f m 2 ) ∥ L 2 ( m 3 ) ≲ ∥ S f ∥ L 2 ( m 2 ) ≤ ∥ f ∥ L 2 ( m 1 ) . Moreover, (80) with i = 1 i=1 i = 1 s = 2 s=2 s = 2 (82) , and we obtain ∥ S ♯ f ∥ L 2 ( m 3 ) ≲ ∥ f ∥ L 2 ( m 1 ) \|S_{\sharp}f\|_{L^2(m_3)}\lesssim\|f\|_{L^2(m_1)} ∥ S ♯ f ∥ L 2 ( m 3 ) ≲ ∥ f ∥ L 2 ( m 1 ) (78) for bounded functions with bounded support. The estimate extends to functions in L 2 ( m 1 ) L^2(m_1) L 2 ( m 1 ) □ \square □
The corona construction and the first reduction
Under the normalized measure-testing assumptions of Proposition 3.1 , we denote by B 0 ⊂ X B_0\subset X B 0 ⊂ X C 3 \mathcal{C}_3 C 3
μ ( Λ B 0 ) ≤ β μ ( δ B 0 ) . (94) \mu(\Lambda B_0)\leq\beta\mu(\delta B_0). \tag*{(94)} μ ( Λ B 0 ) ≤ β μ ( δ B 0 ) . (94) In particular, we assume that B 0 B_0 B 0 Definition 1.2 , and B 0 ∈ B B_0\in\mathcal{B} B 0 ∈ B B \mathcal{B} B B ∈ B ′ B\in\mathcal{B}' B ∈ B ′ (4) on B B B V B V_B V B
diam ( V B ) ≤ η ∗ r ( B ) and μ ( V B ) ≥ ζ 2 μ ( B ) . (95) \operatorname{diam}(V_B)\leq\eta_*r(B)\qquad\text{and}\qquad\mu(V_B)\geq\frac{\zeta}{2}\mu(B). \tag*{(95)} diam ( V B ) ≤ η ∗ r ( B ) and μ ( V B ) ≥ 2 ζ μ ( B ) . (95) We denote B ′ ′ : = B ∖ B ′ \mathcal{B}'' := \mathcal{B}\setminus\mathcal{B}' B ′′ := B ∖ B ′ T T T T ∗ T^* T ∗ (34) on every B ∈ B ′ ′ B\in\mathcal{B}'' B ∈ B ′′ η ∗ \eta_* η ∗ (101) .
Let D 0 : = D μ ( s ) \mathcal{D}_0:=\mathcal{D}^{(s)}_\mu D 0 := D μ ( s ) s : = 2 r ( B 0 ) s:=2r(B_0) s := 2 r ( B 0 ) D μ ( s ) \mathcal{D}^{(s)}_\mu D μ ( s ) Lemma 2.5 . We denote its doubling cells by D 0 d b \mathcal{D}^{\mathrm{db}}_0 D 0 db R ∈ D 0 R\in\mathcal{D}_0 R ∈ D 0
D 0 ( R ) : = { Q ∈ D 0 : Q ⊂ R } and D 0 d b ( R ) : = D 0 ( R ) ∩ D 0 d b . \mathcal{D}_0(R):=\{Q\in\mathcal{D}_0:Q\subset R\}\qquad\text{and}\qquad
\mathcal{D}^{\mathrm{db}}_0(R):=\mathcal{D}_0(R)\cap\mathcal{D}^{\mathrm{db}}_0. D 0 ( R ) := { Q ∈ D 0 : Q ⊂ R } and D 0 db ( R ) := D 0 ( R ) ∩ D 0 db . We denote μ 0 : = μ ∣ B 0 \mu_0:=\mu|_{B_0} μ 0 := μ ∣ B 0 B 0 B_0 B 0 μ 0 ( Q ) = μ ( Q ) \mu_0(Q)=\mu(Q) μ 0 ( Q ) = μ ( Q ) μ \mu μ § 9 .§ 1 . The construction in that appendix also applies to D 0 \mathcal{D}_0 D 0
The construction of the exceptional set and the suppression
The construction of the exceptional set and the suppression. Let R 1 , R 2 ∈ D 0 d b R_1,R_2\in\mathcal{D}^{\mathrm{db}}_0 R 1 , R 2 ∈ D 0 db B 0 B_0 B 0 B i : = B ~ ( R i ) B_i:=\widetilde{B}(R_i) B i := B ( R i ) i = 1 , 2 i=1,2 i = 1 , 2 § 9 .§ 1 . We assume that B 1 , B 2 ∈ B ′ ′ B_1,B_2\in\mathcal{B}'' B 1 , B 2 ∈ B ′′ ν ^ 1 \widehat{\nu}_1 ν 1 T T T B 1 B_1 B 1 ν ^ 2 \widehat{\nu}_2 ν 2 T ∗ T^* T ∗ B 2 B_2 B 2
ν i : = ν ^ i + μ ∣ X ∖ B i , i = 1 , 2. \nu_i:=\widehat{\nu}_i+\mu|_{X\setminus B_i},\qquad i=1,2. ν i := ν i + μ ∣ X ∖ B i , i = 1 , 2. The measures μ \mu μ ν ^ i \widehat{\nu}_i ν i (40) and (A1)–(A6) with B = B i B=B_i B = B i C 2 = 1 \mathcal{C}_2=1 C 2 = 1 T T T T ∗ T^* T ∗ i = 2 i=2 i = 2 (40) follows from B i ∈ B B_i\in\mathcal{B} B i ∈ B n n n μ \mu μ (2) , and (3) . To check (A6), we let U B i U_{B_i} U B i (34) for the corresponding operator, and we set
U ~ B i : = U B i ∪ ( B i ∖ ( 1 − τ ) B i ) . \widetilde{U}_{B_i}:=U_{B_i}\cup\bigl(B_i\setminus(1-\tau)B_i\bigr). U B i := U B i ∪ ( B i ∖ ( 1 − τ ) B i ) . By the small-boundary and doubling properties of B i B_i B i (8) , we have
μ ( U ~ B i ) ≤ ( ζ + C 3 β τ ) μ ( B i ) ≤ ε 0 μ ( B i ) , \mu(\widetilde{U}_{B_i})\leq(\zeta+C_3\beta\tau)\mu(B_i)\leq\varepsilon_0\mu(B_i), μ ( U B i ) ≤ ( ζ + C 3 β τ ) μ ( B i ) ≤ ε 0 μ ( B i ) , and (34) gives T ♯ ν ^ 1 ≤ 1 T_{\sharp}\widehat{\nu}_1\leq1 T ♯ ν 1 ≤ 1 B 1 ∖ U ~ B 1 B_1\setminus\widetilde{U}_{B_1} B 1 ∖ U B 1 T ∗ T^* T ∗ ν ^ 2 \widehat{\nu}_2 ν 2 U B : = U ~ B i U_B:=\widetilde{U}_{B_i} U B := U B i C 2 = 1 \mathcal{C}_2=1 C 2 = 1
Hence, we can apply the construction of § 3 to ν ^ 1 \widehat{\nu}_1 ν 1 ν ^ 2 \widehat{\nu}_2 ν 2 w 1 , w 2 ∈ Ω w_1,w_2\in\Omega w 1 , w 2 ∈ Ω W R 1 ( w 1 ) \mathcal{W}_{R_1}(w_1) W R 1 ( w 1 ) W R 2 ( w 2 ) \mathcal{W}_{R_2}(w_2) W R 2 ( w 2 ) (61) , and by J i , R i ( w i ) \mathcal{J}_{i,R_i}(w_i) J i , R i ( w i ) J i , R i m ^ ( w i ) \mathcal{J}_{i,R_i}^{\widehat{m}}(w_i) J i , R i m ( w i ) H i , R i \mathcal{H}_{i,R_i} H i , R i E i , R i \mathcal{E}_{i,R_i} E i , R i i = 1 , 2 i=1,2 i = 1 , 2 ( 1 − θ ) B i (1-\theta)B_i ( 1 − θ ) B i G i : = G R i ( w i ) ⊂ B i , θ G_i := G_{R_i}(w_i) \subset B_{i,\theta} G i := G R i ( w i ) ⊂ B i , θ δ 1 = 1 − 9 / ( 20 ε 1 ) ∈ ( 1 / 2 , 1 ) \delta_1 = 1 - 9/(20\varepsilon_1) \in(1/2,1) δ 1 = 1 − 9/ ( 20 ε 1 ) ∈ ( 1/2 , 1 ) i = 1 , 2 i=1,2 i = 1 , 2
By Lemma 3.5 , the restriction of ∣ ν i ^ ∣ |\widehat{\nu_i}| ∣ ν i ∣ B i ∖ S i , R i B_i \setminus\mathcal{S}_{i,R_i} B i ∖ S i , R i μ \mu μ σ \sigma σ K i ⊂ ( B i ∖ S i , R i ) ∩ supp μ K_i \subset(B_i \setminus\mathcal{S}_{i,R_i}) \cap\operatorname{supp}\mu K i ⊂ ( B i ∖ S i , R i ) ∩ supp μ ∣ ν i ^ ∣ ( ( B i ∖ S i , R i ) ∖ K i ) = 0 |\widehat{\nu_i}|((B_i \setminus\mathcal{S}_{i,R_i}) \setminus K_i)=0 ∣ ν i ∣ (( B i ∖ S i , R i ) ∖ K i ) = 0
F i ( w i ) : = G i ∩ K i . F_i(w_i) := G_i \cap K_i. F i ( w i ) := G i ∩ K i . Since G i ⊂ B i ∖ S i , R i G_i \subset B_i \setminus\mathcal{S}_{i,R_i} G i ⊂ B i ∖ S i , R i Lemma A.1 , B i ∩ supp μ ⊂ R i B_i \cap\operatorname{supp}\mu\subset R_i B i ∩ supp μ ⊂ R i F i ( w i ) ⊂ B i , θ ∩ R i F_i(w_i) \subset B_{i,\theta} \cap R_i F i ( w i ) ⊂ B i , θ ∩ R i
∣ ν i ^ ∣ ( F i ( w i ) ) = ∣ ν i ^ ∣ ( G i ) ≥ 1 − δ 1 2 ∣ ν i ^ ∣ ( B i ) , for w i ∈ Ω . |\widehat{\nu_i}|(F_i(w_i)) = |\widehat{\nu_i}|(G_i) \ge\frac{1-\delta_1}{2}|\widehat{\nu_i}|(B_i), \qquad\text{for } w_i \in\Omega. ∣ ν i ∣ ( F i ( w i )) = ∣ ν i ∣ ( G i ) ≥ 2 1 − δ 1 ∣ ν i ∣ ( B i ) , for w i ∈ Ω. In particular, F i ( w i ) ≠ ∅ F_i(w_i) \ne\varnothing F i ( w i ) = ∅
For i = 1 , 2 i=1,2 i = 1 , 2
Φ i , R i , w i ( x ) : = dist ( x , F i ( w i ) ) , x ∈ X . (96) \Phi_{i,R_i,w_i}(x) := \operatorname{dist}(x,F_i(w_i)), \qquad x \in X. \tag*{(96)} Φ i , R i , w i ( x ) := dist ( x , F i ( w i )) , x ∈ X . (96) Since F i ( w i ) ≠ ∅ F_i(w_i) \ne\varnothing F i ( w i ) = ∅ Φ i , R i , w i \Phi_{i,R_i,w_i} Φ i , R i , w i F i ( w i ) ‾ \overline{F_i(w_i)} F i ( w i ) w 1 w_1 w 1 w 2 w_2 w 2 Φ 1 , R 1 , w 1 \Phi_{1,R_1,w_1} Φ 1 , R 1 , w 1 Φ 2 , R 2 , w 2 \Phi_{2,R_2,w_2} Φ 2 , R 2 , w 2 Lemma 4.2 below, there is a subset of R i R_i R i μ \mu μ μ ( R i ) \mu(R_i) μ ( R i ) Φ i , R i , w i \Phi_{i,R_i,w_i} Φ i , R i , w i δ p r > 0 \delta_{\mathrm{pr}}>0 δ pr > 0
Lemma 4.2. Let R i ∈ D 0 d b R_i \in\mathcal{D}^{db}_0 R i ∈ D 0 d b i = 1 , 2 i=1,2 i = 1 , 2
δ p r : = 1 − δ 1 8 and δ ~ 1 : = 1 − δ 1 7 + δ 1 . \delta_{\mathrm{pr}} := \frac{1-\delta_1}{8}
\qquad\text{and}\qquad
\widetilde{\delta}_1 := \frac{1-\delta_1}{7+\delta_1}. δ pr := 8 1 − δ 1 and δ 1 := 7 + δ 1 1 − δ 1 . Then there exist sets G ~ i , R i ⊂ B i ∩ R i \widetilde{G}_{i,R_i} \subset B_i \cap R_i G i , R i ⊂ B i ∩ R i i = 1 , 2 i=1,2 i = 1 , 2
P ( { w ∈ Ω : Φ i , R i , w ( x ) = 0 } ) > δ p r , for all x ∈ G ~ i , R i . \mathbb{P}\bigl(\{w \in\Omega: \Phi_{i,R_i,w}(x)=0\}\bigr) > \delta_{\mathrm{pr}}, \qquad\text{for all } x \in\widetilde{G}_{i,R_i}. P ( { w ∈ Ω : Φ i , R i , w ( x ) = 0 } ) > δ pr , for all x ∈ G i , R i . Moreover, these sets satisfy
μ ( G ~ i , R i ) ≥ β ∗ − 1 ( m 1 β ) − 1 δ ~ 1 μ ( R i ) . (97) \mu(\widetilde{G}_{i,R_i}) \ge\beta_*^{-1}(m_1\beta)^{-1}\widetilde{\delta}_1\mu(R_i). \tag*{(97)} μ ( G i , R i ) ≥ β ∗ − 1 ( m 1 β ) − 1 δ 1 μ ( R i ) . (97) Proof. For x ∈ B i x \in B_i x ∈ B i
p i ( x ) : = P ( { w ∈ Ω : x ∈ F i ( w ) } ) . p_i(x) := \mathbb{P}\bigl(\{w \in\Omega: x \in F_i(w)\}\bigr). p i ( x ) := P ( { w ∈ Ω : x ∈ F i ( w )} ) . We have that
∫ B i p i ( x ) d ∣ ν i ^ ∣ ( x ) = ∫ w ∈ Ω ∫ χ F i ( w ) ( x ) d ∣ ν i ^ ∣ ( x ) d P ( w ) = ∫ w ∈ Ω ∣ ν i ^ ∣ ( F i ( w ) ) d P ( w ) ≥ 1 − δ 1 4 ∣ ν i ^ ∣ ( B i ) . (98) \int_{B_i} p_i(x)\,d|\widehat{\nu_i}|(x)
= \int_{w\in\Omega}\int\chi_{F_i(w)}(x)\,d|\widehat{\nu_i}|(x)\,d\mathbb{P}(w)
= \int_{w\in\Omega}|\widehat{\nu_i}|(F_i(w))\,d\mathbb{P}(w)
\ge\frac{1-\delta_1}{4}|\widehat{\nu_i}|(B_i). \tag*{(98)} ∫ B i p i ( x ) d ∣ ν i ∣ ( x ) = ∫ w ∈ Ω ∫ χ F i ( w ) ( x ) d ∣ ν i ∣ ( x ) d P ( w ) = ∫ w ∈ Ω ∣ ν i ∣ ( F i ( w )) d P ( w ) ≥ 4 1 − δ 1 ∣ ν i ∣ ( B i ) . (98) Thus, if we define
G ~ i , R i : = { x ∈ B i : p i ( x ) > 1 − δ 1 8 } , \widetilde{G}_{i,R_i} := \left\{x \in B_i : p_i(x) > \frac{1-\delta_1}{8}\right\}, G i , R i := { x ∈ B i : p i ( x ) > 8 1 − δ 1 } , in light of the latter estimate, we obtain that
∣ ν ^ i ∣ ( B i ∖ G ~ i , R i ) ≤ 8 7 + δ 1 ∫ B i ( 1 − p i ( x ) ) d ∣ ν ^ i ∣ ( x ) = 8 7 + δ 1 ( ∣ ν ^ i ∣ ( B i ) − ∫ B i p i ( x ) d ∣ ν ^ i ∣ ( x ) ) ≤ 6 + 2 δ 1 7 + δ 1 ∣ ν ^ i ∣ ( B i ) . \begin{aligned}
|\widehat{\nu}_i|(B_i \setminus\widetilde{G}_{i,R_i}) \le\frac{8}{7+\delta_1}\int_{B_i}(1-\mathbf{p}_i(x))\,d|\widehat{\nu}_i|(x) \\
&= \frac{8}{7+\delta_1}\left(|\widehat{\nu}_i|(B_i)-\int_{B_i}\mathbf{p}_i(x)\,d|\widehat{\nu}_i|(x)\right)\le\frac{6+2\delta_1}{7+\delta_1}|\widehat{\nu}_i|(B_i).
\end{aligned} ∣ ν i ∣ ( B i ∖ G i , R i ) ≤ 7 + δ 1 8 ∫ B i ( 1 − p i ( x )) d ∣ ν i ∣ ( x ) = 7 + δ 1 8 ( ∣ ν i ∣ ( B i ) − ∫ B i p i ( x ) d ∣ ν i ∣ ( x ) ) ≤ 7 + δ 1 6 + 2 δ 1 ∣ ν i ∣ ( B i ) . This implies that
∣ ν ^ i ∣ ( G ~ i , R i ) ≥ 1 − δ 1 7 + δ 1 ∣ ν ^ i ∣ ( B i ) . |\widehat{\nu}_i|(\widetilde{G}_{i,R_i})\ge\frac{1-\delta_1}{7+\delta_1}|\widehat{\nu}_i|(B_i). ∣ ν i ∣ ( G i , R i ) ≥ 7 + δ 1 1 − δ 1 ∣ ν i ∣ ( B i ) . Since p i ( x ) > 0 \mathbf{p}_i(x)>0 p i ( x ) > 0 x ∈ F i ( w ) x\in F_i(w) x ∈ F i ( w ) w w w F i ( w ) F_i(w) F i ( w ) L i , R i \mathcal{L}_{i,R_i} L i , R i
μ ( G ~ i , R i ) ≥ ( β m 1 ) − 1 1 − δ 1 7 + δ 1 μ ( B i ) . \mu(\widetilde{G}_{i,R_i})\ge(\beta m_1)^{-1}\frac{1-\delta_1}{7+\delta_1}\mu(B_i). μ ( G i , R i ) ≥ ( β m 1 ) − 1 7 + δ 1 1 − δ 1 μ ( B i ) . Moreover, if x ∉ F i ( w ) x\notin F_i(w) x ∈ / F i ( w ) w ∈ Ω w\in\Omega w ∈ Ω p i ( x ) = 0 \mathbf{p}_i(x)=0 p i ( x ) = 0
{#eq:4.5}
\widetilde{G}_{i,R_i}\subset\bigcup_{w\in\Omega}F_i(w)\subset B_{i,\theta}\setminus\bigcap_{w\in\Omega}\mathcal{W}_{R_i}(w).
Observe that if x ∈ G ~ i , R i x\in\widetilde{G}_{i,R_i} x ∈ G i , R i
P ( { w ∈ Ω : Φ i , R i , w ( x ) = 0 } ) = P ( { w ∈ Ω : dist ( x , F i ( w ) ) = 0 } ) ≥ p i ( x ) > δ p r . \mathbb{P}\left(\left\{w\in\Omega:\Phi_{i,R_i,w}(x)=0\right\}\right)=\mathbb{P}\left(\left\{w\in\Omega:\operatorname{dist}(x,F_i(w))=0\right\}\right)\ge\mathbf{p}_i(x)>\delta_{\mathrm{pr}}. P ( { w ∈ Ω : Φ i , R i , w ( x ) = 0 } ) = P ( { w ∈ Ω : dist ( x , F i ( w )) = 0 } ) ≥ p i ( x ) > δ pr . The property (A.2), then, implies that μ ( B i ) ≥ β ∗ − 1 μ ( R i ) \mu(B_i)\ge\beta_*^{-1}\mu(R_i) μ ( B i ) ≥ β ∗ − 1 μ ( R i )
For δ p r \delta_{\mathrm{pr}} δ pr
Φ i , R i ( x ) : = inf A ⊂ Ω , P ( A ) > δ p r sup w ∈ A Φ i , R i , w ( x ) . (99) \Phi_{i,R_i}(x):=\inf_{\substack{A\subset\Omega,\\ \mathbb{P}(A)>\delta_{\mathrm{pr}}}}\sup_{w\in A}\Phi_{i,R_i,w}(x). \tag*{(99)} Φ i , R i ( x ) := A ⊂ Ω , P ( A ) > δ pr inf w ∈ A sup Φ i , R i , w ( x ) . (99) The infimum is over measurable sets A A A F i ( w ) F_i(w) F i ( w ) B i B_i B i Φ i , R i \Phi_{i,R_i} Φ i , R i
Lemma 4.3. There exists c T ∈ ( 0 , 1 ) c_T\in(0,1) c T ∈ ( 0 , 1 ) β \beta β β ∗ \beta_* β ∗ D D D C 1 \mathcal{C}_1 C 1 ε 0 \varepsilon_0 ε 0
μ ( { x ∈ R i : Φ i , R i ( x ) = 0 } ) ≥ c T μ ( R i ) , for i = 1 , 2. \mu\left(\left\{x\in R_i:\Phi_{i,R_i}(x)=0\right\}\right)\ge c_T\mu(R_i),\qquad\text{for }i=1,2. μ ( { x ∈ R i : Φ i , R i ( x ) = 0 } ) ≥ c T μ ( R i ) , for i = 1 , 2. Proof. Let x ∈ G ~ i , R i x\in\widetilde{G}_{i,R_i} x ∈ G i , R i
A x i : = { w ∈ Ω : Φ i , R i , w ( x ) = 0 } A_x^i:=\left\{w\in\Omega:\Phi_{i,R_i,w}(x)=0\right\} A x i := { w ∈ Ω : Φ i , R i , w ( x ) = 0 } satisfies P ( A x i ) > δ p r \mathbb{P}(A_x^i)>\delta_{\mathrm{pr}} P ( A x i ) > δ pr x ∈ G ~ i , R i x\in\widetilde{G}_{i,R_i} x ∈ G i , R i
Φ i , R i ( x ) ≤ sup w ∈ A x i Φ i , R i , w ( x ) = 0. \Phi_{i,R_i}(x)\le\sup_{w\in A_x^i}\Phi_{i,R_i,w}(x)=0. Φ i , R i ( x ) ≤ w ∈ A x i sup Φ i , R i , w ( x ) = 0. Thus, G ~ i , R i ⊂ { x ∈ R i : Φ i , R i ( x ) = 0 } \widetilde{G}_{i,R_i}\subset\{x\in R_i:\Phi_{i,R_i}(x)=0\} G i , R i ⊂ { x ∈ R i : Φ i , R i ( x ) = 0 }
The corona decomposition
We define the stopping cells separately for the alternatives (1.4) and (1.5) in the definition of testing balls associated with doubling David-Mattila cells.
The families Stop′ ' ′ ′ ′ '' ′′ i = 1 , 2 i=1,2 i = 1 , 2 R i ∈ D 0 d b R_i\in\mathcal{D}_0^{\mathrm{db}} R i ∈ D 0 db B 0 B_0 B 0 B i : = B ~ ( R i ) B_i:=\widetilde{B}(R_i) B i := B ( R i ) B i ∈ B ′ B_i\in\mathcal{B}' B i ∈ B ′ V B i V_{B_i} V B i (95) . For k ≥ 0 k\geq0 k ≥ 0 ℓ k : = 56 K 0 s A 0 − k \ell_k:=56K_0sA_0^{-k} ℓ k := 56 K 0 s A 0 − k k k k k k k
η ∗ r ( B i ) A 0 < ℓ k ≤ η ∗ r ( B i ) . \frac{\eta_*r(B_i)}{A_0}<\ell_k\leq\eta_*r(B_i). A 0 η ∗ r ( B i ) < ℓ k ≤ η ∗ r ( B i ) . Since r ( B i ) ≤ r ( R i ) ≤ ℓ ( R i ) r(B_i)\leq r(R_i)\leq\ell(R_i) r ( B i ) ≤ r ( R i ) ≤ ℓ ( R i ) (29) , we have k > J ( R i ) k>J(R_i) k > J ( R i ) B i ∩ supp μ ⊂ R i B_i\cap\operatorname{supp}\mu\subset R_i B i ∩ supp μ ⊂ R i P P P k k k V B i ∩ supp μ V_{B_i}\cap\operatorname{supp}\mu V B i ∩ supp μ R i R_i R i v ∈ V B i ∩ supp μ v\in V_{B_i}\cap\operatorname{supp}\mu v ∈ V B i ∩ supp μ (29) and (26) , every such cell P P P
d ( z P , v ) ≤ diam ( P ) + diam ( V B i ) ≤ 2 η ∗ r ( B i ) and r ( P ) ≥ s A 0 − k > η ∗ r ( B i ) 56 K 0 A 0 . d(z_P,v)\leq\operatorname{diam}(P)+\operatorname{diam}(V_{B_i})\leq2\eta_*r(B_i)
\quad\text{and}\quad
r(P)\geq sA_0^{-k}>\frac{\eta_*r(B_i)}{56K_0A_0}. d ( z P , v ) ≤ diam ( P ) + diam ( V B i ) ≤ 2 η ∗ r ( B i ) and r ( P ) ≥ s A 0 − k > 56 K 0 A 0 η ∗ r ( B i ) . Note that the balls 5 B ( P ) 5B(P) 5 B ( P ) N N N N N N D D D K 0 K_0 K 0 A 0 A_0 A 0 V B i ∩ supp μ V_{B_i}\cap\operatorname{supp}\mu V B i ∩ supp μ (95) and {(A.2)} give a cell P ~ B i ⊂ R i \widetilde{P}_{B_i}\subset R_i P B i ⊂ R i k k k
μ ( P ~ B i ) ≥ ζ 2 N μ ( B i ) ≥ α ∗ μ ( R i ) , where α ∗ : = ζ 2 N β ∗ . (100) \mu(\widetilde{P}_{B_i})\geq\frac{\zeta}{2N}\mu(B_i)\geq\alpha_*\mu(R_i),
\qquad\text{where}\qquad
\alpha_*:=\frac{\zeta}{2N\beta_*}. \tag*{(100)} μ ( P B i ) ≥ 2 N ζ μ ( B i ) ≥ α ∗ μ ( R i ) , where α ∗ := 2 N β ∗ ζ . (100) Moreover, ℓ ( P ~ B i ) = ℓ k ≤ η ∗ r ( B i ) ≤ η ∗ ℓ ( R i ) \ell(\widetilde{P}_{B_i})=\ell_k\leq\eta_*r(B_i)\leq\eta_*\ell(R_i) ℓ ( P B i ) = ℓ k ≤ η ∗ r ( B i ) ≤ η ∗ ℓ ( R i ) (30) , ℓ ( P ~ B i ) ≈ η ∗ ℓ ( R i ) \ell(\widetilde{P}_{B_i})\approx\eta_*\ell(R_i) ℓ ( P B i ) ≈ η ∗ ℓ ( R i ) P B i P_{B_i} P B i P ~ B i \widetilde{P}_{B_i} P B i P B i = P ~ B i P_{B_i}=\widetilde{P}_{B_i} P B i = P B i R i R_i R i P ~ B i ⊂ P B i ⊂ R i \widetilde{P}_{B_i}\subset P_{B_i}\subset R_i P B i ⊂ P B i ⊂ R i Q ∈ D 0 Q\in\mathcal{D}_0 Q ∈ D 0 Θ μ ( Q ) : = μ ( Q ) / ℓ ( Q ) n \Theta_\mu(Q):=\mu(Q)/\ell(Q)^n Θ μ ( Q ) := μ ( Q ) / ℓ ( Q ) n
Lemma 4.4. For R i ∈ D 0 d b R_i\in\mathcal{D}_0^{\mathrm{db}} R i ∈ D 0 db B i ∈ B ′ B_i\in\mathcal{B}' B i ∈ B ′
Θ μ ( P B i ) ≳ Θ μ ( P ~ B i ) ≥ α ∗ η ∗ − n Θ μ ( R i ) . \Theta_\mu(P_{B_i})\gtrsim\Theta_\mu(\widetilde{P}_{B_i})\geq\alpha_*\eta_*^{-n}\Theta_\mu(R_i). Θ μ ( P B i ) ≳ Θ μ ( P B i ) ≥ α ∗ η ∗ − n Θ μ ( R i ) . Proof. If P B i ≠ P ~ B i P_{B_i}\neq\widetilde{P}_{B_i} P B i = P B i S S S P ~ B i ⊂ S ⊊ P B i \widetilde{P}_{B_i}\subset S\subsetneq P_{B_i} P B i ⊂ S ⊊ P B i Θ μ ( 100 B ( P ~ B i ) ) ≲ Θ μ ( 100 B ( P B i ) ) \Theta_\mu(100B(\widetilde{P}_{B_i}))\lesssim\Theta_\mu(100B(P_{B_i})) Θ μ ( 100 B ( P B i )) ≲ Θ μ ( 100 B ( P B i )) P B i = P ~ B i P_{B_i}=\widetilde{P}_{B_i} P B i = P B i P ~ B i ⊂ 100 B ( P ~ B i ) \widetilde{P}_{B_i}\subset100B(\widetilde{P}_{B_i}) P B i ⊂ 100 B ( P B i ) r ( P ~ B i ) ≤ ℓ ( P ~ B i ) r(\widetilde{P}_{B_i})\leq\ell(\widetilde{P}_{B_i}) r ( P B i ) ≤ ℓ ( P B i ) P B i P_{B_i} P B i ℓ ( P B i ) ≲ r ( P B i ) \ell(P_{B_i})\lesssim r(P_{B_i}) ℓ ( P B i ) ≲ r ( P B i )
Θ μ ( P ~ B i ) ≤ 100 n Θ μ ( 100 B ( P ~ B i ) ) ≲ Θ μ ( 100 B ( P B i ) ) ≲ Θ μ ( P B i ) . \Theta_\mu(\widetilde{P}_{B_i})\leq100^n\Theta_\mu(100B(\widetilde{P}_{B_i}))\lesssim\Theta_\mu(100B(P_{B_i}))\lesssim\Theta_\mu(P_{B_i}). Θ μ ( P B i ) ≤ 10 0 n Θ μ ( 100 B ( P B i )) ≲ Θ μ ( 100 B ( P B i )) ≲ Θ μ ( P B i ) . Moreover, since ℓ ( P ~ B i ) ≤ η ∗ ℓ ( R i ) \ell(\widetilde{P}_{B_i})\leq\eta_*\ell(R_i) ℓ ( P B i ) ≤ η ∗ ℓ ( R i ) (100) gives
Θ μ ( P ~ B i ) = μ ( P ~ B i ) ℓ ( P ~ B i ) n ≥ α ∗ η ∗ − n Θ μ ( R i ) . \Theta_\mu(\widetilde{P}_{B_i})
=\frac{\mu(\widetilde{P}_{B_i})}{\ell(\widetilde{P}_{B_i})^n}
\geq\alpha_*\eta_*^{-n}\Theta_\mu(R_i). Θ μ ( P B i ) = ℓ ( P B i ) n μ ( P B i ) ≥ α ∗ η ∗ − n Θ μ ( R i ) . Since μ ( P B i ) ≥ μ ( P ~ B i ) ≥ α ∗ μ ( R i ) \mu(P_{B_i})\geq\mu(\widetilde{P}_{B_i})\geq\alpha_*\mu(R_i) μ ( P B i ) ≥ μ ( P B i ) ≥ α ∗ μ ( R i )
Θ μ ( P B i ) μ ( P B i ) ≥ c 0 α ∗ 2 η ∗ − n Θ μ ( R i ) μ ( R i ) , \Theta_\mu(P_{B_i})\mu(P_{B_i})\geq c_0\alpha_*^2\eta_*^{-n}\Theta_\mu(R_i)\mu(R_i), Θ μ ( P B i ) μ ( P B i ) ≥ c 0 α ∗ 2 η ∗ − n Θ μ ( R i ) μ ( R i ) , where c 0 > 0 c_0>0 c 0 > 0 n n n η ∗ ∈ ( 0 , 1 / 10 ) \eta_*\in(0,1/10) η ∗ ∈ ( 0 , 1/10 ) c 0 α ∗ 2 η ∗ − n ≥ 2 c_0\alpha_*^2\eta_*^{-n}\geq2 c 0 α ∗ 2 η ∗ − n ≥ 2 n n n ζ \zeta ζ
Θ μ ( P B i ) μ ( P B i ) ≥ 2 Θ μ ( R i ) μ ( R i ) . (101) \Theta_\mu(P_{B_i})\mu(P_{B_i})\geq2\Theta_\mu(R_i)\mu(R_i). \tag*{(101)} Θ μ ( P B i ) μ ( P B i ) ≥ 2 Θ μ ( R i ) μ ( R i ) . (101) In particular, P B i ⊊ R i P_{B_i}\subsetneq R_i P B i ⊊ R i Stop ′ ( R i ) \operatorname{Stop}'(R_i) Stop ′ ( R i ) S ⊂ R i S\subset R_i S ⊂ R i P B i P_{B_i} P B i R i R_i R i R i R_i R i P B i P_{B_i} P B i i = 1 , 2 i=1,2 i = 1 , 2 B i = B ~ ( R i ) ∈ B ′ ′ B_i=\widetilde{B}(R_i)\in\mathcal{B}'' B i = B ( R i ) ∈ B ′′ (99) , Φ i , R i \Phi_{i,R_i} Φ i , R i x ∈ R i x\in R_i x ∈ R i Q x : = { x } Q_x:=\{x\} Q x := { x } Φ i , R i ( x ) = 0 \Phi_{i,R_i}(x)=0 Φ i , R i ( x ) = 0 Q x Q_x Q x D 0 ( R i ) \mathcal{D}_0(R_i) D 0 ( R i ) x x x
ℓ ( Q x ) < 1 480 A 0 inf y ∈ Q x Φ i , R i ( y ) . (102) \ell(Q_x)<\frac{1}{480A_0}\inf_{y\in Q_x}\Phi_{i,R_i}(y). \tag*{(102)} ℓ ( Q x ) < 480 A 0 1 y ∈ Q x inf Φ i , R i ( y ) . (102) which we refer to as a regularized cell . By Lemma 4.3, there exists a point z i ∈ R i z_i\in R_i z i ∈ R i Φ i , R i ( z i ) = 0 \Phi_{i,R_i}(z_i)=0 Φ i , R i ( z i ) = 0 R i R_i R i (102) . On the other hand, sufficiently small descendants P ∋ x P\ni x P ∋ x
inf P Φ i , R i ≥ Φ i , R i ( x ) − diam ( P ) ≥ Φ i , R i ( x ) − ℓ ( P ) > 480 A 0 ℓ ( P ) \inf_{P}\Phi_{i,R_i}\geq\Phi_{i,R_i}(x)-\operatorname{diam}(P)\geq\Phi_{i,R_i}(x)-\ell(P)>480A_0\ell(P) P inf Φ i , R i ≥ Φ i , R i ( x ) − diam ( P ) ≥ Φ i , R i ( x ) − ℓ ( P ) > 480 A 0 ℓ ( P ) whenever ℓ ( P ) < Φ i , R i ( x ) / ( 480 A 0 + 1 ) \ell(P)<\Phi_{i,R_i}(x)/(480A_0+1) ℓ ( P ) < Φ i , R i ( x ) / ( 480 A 0 + 1 ) Q x Q_x Q x R i R_i R i D 0 ( R i ) \mathcal{D}_0(R_i) D 0 ( R i ) (102) .
We define Stop i ′ ′ ( R i ) \operatorname{Stop}''_i(R_i) Stop i ′′ ( R i ) Q x Q_x Q x x ∈ R i x\in R_i x ∈ R i Φ i , R i ( x ) > 0 \Phi_{i,R_i}(x)>0 Φ i , R i ( x ) > 0 R i ∩ { Φ i , R i > 0 } R_i\cap\{\Phi_{i,R_i}>0\} R i ∩ { Φ i , R i > 0 }
Lemma 4.5. The regularized cells satisfy the following properties:
(1) If Q ∈ Stop i ′ ′ ( R i ) Q\in\operatorname{Stop}''_i(R_i) Q ∈ Stop i ′′ ( R i ) x ∈ B ( z Q , 50 ℓ ( Q ) ) x\in B(z_Q,50\ell(Q)) x ∈ B ( z Q , 50 ℓ ( Q )) Φ i , R i ( x ) ≈ ℓ ( Q ) \Phi_{i,R_i}(x)\approx\ell(Q) Φ i , R i ( x ) ≈ ℓ ( Q )
430 A 0 ℓ ( Q ) < Φ i , R i ( x ) ≤ 481 A 0 2 ℓ ( Q ) . (103) 430A_0\ell(Q)<\Phi_{i,R_i}(x)\leq481A_0^2\ell(Q). \tag*{(103)} 430 A 0 ℓ ( Q ) < Φ i , R i ( x ) ≤ 481 A 0 2 ℓ ( Q ) . (103) (2) If Q , Q ′ ∈ Stop i ′ ′ ( R i ) Q,Q'\in\operatorname{Stop}''_i(R_i) Q , Q ′ ∈ Stop i ′′ ( R i ) B ( z Q , 50 ℓ ( Q ) ) ∩ B ( z Q ′ , 50 ℓ ( Q ′ ) ) ≠ ∅ B(z_Q,50\ell(Q))\cap B(z_{Q'},50\ell(Q'))\neq\varnothing B ( z Q , 50 ℓ ( Q )) ∩ B ( z Q ′ , 50 ℓ ( Q ′ )) = ∅ ℓ ( Q ) ≈ ℓ ( Q ′ ) \ell(Q)\approx\ell(Q') ℓ ( Q ) ≈ ℓ ( Q ′ )
1 2 A 0 ℓ ( Q ′ ) ≤ ℓ ( Q ) ≤ 2 A 0 ℓ ( Q ′ ) . \frac{1}{2A_0}\ell(Q')\leq\ell(Q)\leq2A_0\ell(Q'). 2 A 0 1 ℓ ( Q ′ ) ≤ ℓ ( Q ) ≤ 2 A 0 ℓ ( Q ′ ) . (3) It holds that
∑ P ∈ Stop i ′ ′ ( R i ) μ ( P ) ≤ ( 1 − c T ) μ ( R i ) . (104) \sum_{P\in\operatorname{Stop}''_i(R_i)}\mu(P)\leq(1-c_T)\mu(R_i). \tag*{(104)} P ∈ Stop i ′′ ( R i ) ∑ μ ( P ) ≤ ( 1 − c T ) μ ( R i ) . (104) Proof. Let Q ∈ Stop i ′ ′ ( R i ) Q\in\operatorname{Stop}''_i(R_i) Q ∈ Stop i ′′ ( R i ) x ∈ B ( z Q , 50 ℓ ( Q ) ) x\in B(z_Q,50\ell(Q)) x ∈ B ( z Q , 50 ℓ ( Q )) (102) yields Φ i , R i ( z Q ) > 480 A 0 ℓ ( Q ) \Phi_{i,R_i}(z_Q)>480A_0\ell(Q) Φ i , R i ( z Q ) > 480 A 0 ℓ ( Q )
Φ i , R i ( x ) ≥ Φ i , R i ( z Q ) − d ( x , z Q ) > ( 480 A 0 − 50 ) ℓ ( Q ) ≥ 430 A 0 ℓ ( Q ) . \Phi_{i,R_i}(x)\geq\Phi_{i,R_i}(z_Q)-d(x,z_Q)>(480A_0-50)\ell(Q)\geq430A_0\ell(Q). Φ i , R i ( x ) ≥ Φ i , R i ( z Q ) − d ( x , z Q ) > ( 480 A 0 − 50 ) ℓ ( Q ) ≥ 430 A 0 ℓ ( Q ) . By definition of regularized cell, the David–Mattila parent Q ^ \widehat{Q} Q Q Q Q (102) . Hence, we have
inf y ∈ Q ^ Φ i , R i ( y ) ≤ 480 A 0 ℓ ( Q ^ ) = 480 A 0 2 ℓ ( Q ) . (105) \inf_{y\in\widehat{Q}}\Phi_{i,R_i}(y)\leq480A_0\ell(\widehat{Q})=480A_0^2\ell(Q). \tag*{(105)} y ∈ Q inf Φ i , R i ( y ) ≤ 480 A 0 ℓ ( Q ) = 480 A 0 2 ℓ ( Q ) . (105) For k ∈ N k\in\mathbb{N} k ∈ N y k ∈ Q ^ y_k\in\widehat{Q} y k ∈ Q Φ i , R i ( y k ) < inf Q ^ Φ i , R i + 1 / k \Phi_{i,R_i}(y_k)<\inf_{\widehat{Q}}\Phi_{i,R_i}+1/k Φ i , R i ( y k ) < inf Q Φ i , R i + 1/ k z Q ∈ Q ⊂ Q ^ z_Q\in Q\subset\widehat{Q} z Q ∈ Q ⊂ Q
d ( x , y k ) ≤ d ( x , z Q ) + diam ( Q ^ ) < ( 50 + A 0 ) ℓ ( Q ) < 2 A 0 ℓ ( Q ) . d(x,y_k)\leq d(x,z_Q)+\operatorname{diam}(\widehat{Q})<(50+A_0)\ell(Q)<2A_0\ell(Q). d ( x , y k ) ≤ d ( x , z Q ) + diam ( Q ) < ( 50 + A 0 ) ℓ ( Q ) < 2 A 0 ℓ ( Q ) . Thus Lipschitz continuity and (105) yield
Φ i , R i ( x ) ≤ Φ i , R i ( y k ) + d ( x , y k ) \Phi_{i,R_i}(x)\leq\Phi_{i,R_i}(y_k)+d(x,y_k) Φ i , R i ( x ) ≤ Φ i , R i ( y k ) + d ( x , y k ) < ( 480 A 0 2 + 2 A 0 ) ℓ ( Q ) + 1 / k < 481 A 0 2 ℓ ( Q ) + 1 / k . (106) < (480A_0^2+2A_0)\ell(Q)+1/k<481A_0^2\ell(Q)+1/k. \tag*{(106)} < ( 480 A 0 2 + 2 A 0 ) ℓ ( Q ) + 1/ k < 481 A 0 2 ℓ ( Q ) + 1/ k . (106) Thus, the upper bound in (103) follows from (105) and taking the limit as k → ∞ k\to\infty k → ∞ (106) .
To prove (2), choose x ∈ B ( z Q , 50 ℓ ( Q ) ) ∩ B ( z Q ′ , 50 ℓ ( Q ′ ) ) x\in B(z_Q,50\ell(Q))\cap B(z_{Q'},50\ell(Q')) x ∈ B ( z Q , 50 ℓ ( Q )) ∩ B ( z Q ′ , 50 ℓ ( Q ′ )) (103) ,
430 A 0 ℓ ( Q ) < Φ i , R i ( x ) ≤ 481 A 0 2 ℓ ( Q ′ ) 430A_0\ell(Q)<\Phi_{i,R_i}(x)\leq481A_0^2\ell(Q') 430 A 0 ℓ ( Q ) < Φ i , R i ( x ) ≤ 481 A 0 2 ℓ ( Q ′ ) and
ℓ ( Q ) < 481 430 A 0 ℓ ( Q ′ ) < 2 A 0 ℓ ( Q ′ ) . \ell(Q)<\frac{481}{430}A_0\ell(Q')<2A_0\ell(Q'). ℓ ( Q ) < 430 481 A 0 ℓ ( Q ′ ) < 2 A 0 ℓ ( Q ′ ) . Interchanging Q Q Q Q ′ Q' Q ′ Stop i ′ ′ ( R i ) \operatorname{Stop}_{i}^{\prime\prime}(R_i) Stop i ′′ ( R i )
∑ P ∈ Stop i ′ ′ ( R i ) μ ( P ) = μ ( ⋃ S ∈ Stop i ′ ′ ( R i ) S ) ≤ μ ( { x ∈ R i : Φ i , R i ( x ) > 0 } ) , (107) \sum_{P\in\operatorname{Stop}_{i}^{\prime\prime}(R_i)}\mu(P)=\mu\left(\bigcup_{S\in\operatorname{Stop}_{i}^{\prime\prime}(R_i)}S\right)\leq\mu\left(\{x\in R_i:\Phi_{i,R_i}(x)>0\}\right), \tag*{(107)} P ∈ Stop i ′′ ( R i ) ∑ μ ( P ) = μ S ∈ Stop i ′′ ( R i ) ⋃ S ≤ μ ( { x ∈ R i : Φ i , R i ( x ) > 0 } ) , (107) and conclude by Lemma 4.3. □
The coronization. We now iterate the construction of Stop i ′ ( R i ) \operatorname{Stop}_{i}^{\prime}(R_i) Stop i ′ ( R i ) Stop i ′ ′ ( R i ) \operatorname{Stop}_{i}^{\prime\prime}(R_i) Stop i ′′ ( R i ) Top 0 0 \operatorname{Top}_{0}^{0} Top 0 0 Q ∈ D 0 Q\in\mathcal{D}_{0} Q ∈ D 0 B 0 B_{0} B 0 Top 0 0 \operatorname{Top}_{0}^{0} Top 0 0 B 0 B_{0} B 0 μ \mu μ
Next, for Q ∈ D 0 Q\in\mathcal{D}_{0} Q ∈ D 0 MDb ( Q ) \operatorname{MDb}(Q) MDb ( Q ) D 0 d b \mathcal{D}_{0}^{db} D 0 d b Q Q Q Q Q Q μ \mu μ MDb ( Q ) = { Q } \operatorname{MDb}(Q)=\{Q\} MDb ( Q ) = { Q } Q Q Q
For R i ∈ D 0 d b R_i\in\mathcal{D}_{0}^{db} R i ∈ D 0 d b
Next i ( R i ) : = { ⋃ Q ∈ Stop i ′ ( R i ) MDb ( Q ) , if B ~ ( R i ) ∈ B ′ , ⋃ Q ∈ Stop i ′ ′ ( R i ) MDb ( Q ) , if B ~ ( R i ) ∈ B ′ ′ . \operatorname{Next}_{i}(R_i):=
\begin{cases}
\bigcup_{Q\in\operatorname{Stop}_{i}^{\prime}(R_i)}\operatorname{MDb}(Q), & \text{if }\widetilde{B}(R_i)\in\mathcal{B}^{\prime},\\
\bigcup_{Q\in\operatorname{Stop}_{i}^{\prime\prime}(R_i)}\operatorname{MDb}(Q), & \text{if }\widetilde{B}(R_i)\in\mathcal{B}^{\prime\prime}.
\end{cases} Next i ( R i ) := { ⋃ Q ∈ Stop i ′ ( R i ) MDb ( Q ) , ⋃ Q ∈ Stop i ′′ ( R i ) MDb ( Q ) , if B ( R i ) ∈ B ′ , if B ( R i ) ∈ B ′′ . Denoting Top i 0 : = Top i 0 : = Top 0 \operatorname{Top}_{i}^{0}:=\operatorname{Top}_{i}^{0}:=\operatorname{Top}^{0} Top i 0 := Top i 0 := Top 0 k ≥ 0 k\geq0 k ≥ 0
Top i k + 1 : = ⋃ P ∈ Top i k Next i ( P ) . \operatorname{Top}_{i}^{k+1}:=\bigcup_{P\in\operatorname{Top}_{i}^{k}}\operatorname{Next}_{i}(P). Top i k + 1 := P ∈ Top i k ⋃ Next i ( P ) . Thus, we finally denote Top i : = ⋃ k = 0 ∞ Top i k \operatorname{Top}_{i}:=\bigcup_{k=0}^{\infty}\operatorname{Top}_{i}^{k} Top i := ⋃ k = 0 ∞ Top i k
Top i ′ : = { Q ∈ Top i : B ~ ( Q ) ∈ B ′ } \operatorname{Top}_{i}^{\prime}:=\{Q\in\operatorname{Top}_{i}:\widetilde{B}(Q)\in\mathcal{B}^{\prime}\} Top i ′ := { Q ∈ Top i : B ( Q ) ∈ B ′ } and
Top i ′ ′ : = { Q ∈ Top i : B ~ ( Q ) ∈ B ′ ′ } = Top i ∖ Top i ′ . \operatorname{Top}_{i}^{\prime\prime}:=\{Q\in\operatorname{Top}_{i}:\widetilde{B}(Q)\in\mathcal{B}^{\prime\prime}\}=\operatorname{Top}_{i}\setminus\operatorname{Top}_{i}^{\prime}. Top i ′′ := { Q ∈ Top i : B ( Q ) ∈ B ′′ } = Top i ∖ Top i ′ . In the following lemma, we prove that Top i ′ ′ \operatorname{Top}_{i}^{\prime\prime} Top i ′′
Lemma 4.6. For every S ∈ D 0 S\in\mathcal{D}_{0} S ∈ D 0
∑ Q ∈ Top i ′ ′ Q ⊂ S μ ( Q ) ≤ c T − 1 μ ( S ) . (108) \sum_{\substack{Q\in\operatorname{Top}_{i}^{\prime\prime}\\Q\subset S}}\mu(Q)\leq c_{T}^{-1}\mu(S). \tag*{(108)} Q ∈ Top i ′′ Q ⊂ S ∑ μ ( Q ) ≤ c T − 1 μ ( S ) . (108) Proof. Let R ∈ Top i ′ ′ R\in\operatorname{Top}_{i}^{\prime\prime} R ∈ Top i ′′ Stop i ′ ′ ( R ) \operatorname{Stop}_{i}^{\prime\prime}(R) Stop i ′′ ( R ) μ \mu μ (104) we have
∑ Q ∈ Next i ( R ) μ ( Q ) = ∑ P ∈ Stop i ′ ′ ( R ) μ ( P ) ≤ ( 1 − c T ) μ ( R ) . (109) \sum_{Q\in\operatorname{Next}_{i}(R)}\mu(Q)=\sum_{P\in\operatorname{Stop}_{i}^{\prime\prime}(R)}\mu(P)\leq(1-c_{T})\mu(R). \tag*{(109)} Q ∈ Next i ( R ) ∑ μ ( Q ) = P ∈ Stop i ′′ ( R ) ∑ μ ( P ) ≤ ( 1 − c T ) μ ( R ) . (109) Next, for R ∈ Top i ′ ′ R\in\operatorname{Top}_{i}^{\prime\prime} R ∈ Top i ′′
Succ ( R ) : = { Q ∈ Top i ′ ′ : Q ⊊ R and there is no P ∈ Top i ′ ′ s.t. Q ⊊ P ⊊ R } , \operatorname{Succ}(R):=\{Q\in\operatorname{Top}_{i}^{\prime\prime}:Q\subsetneq R\text{ and there is no }P\in\operatorname{Top}_{i}^{\prime\prime}\text{ s.t. }Q\subsetneq P\subsetneq R\}, Succ ( R ) := { Q ∈ Top i ′′ : Q ⊊ R and there is no P ∈ Top i ′′ s.t. Q ⊊ P ⊊ R } , which is a family of pairwise disjoint cells. Moreover, by construction, every cell in Succ ( R ) \operatorname{Succ}(R) Succ ( R ) Next i ( R ) \operatorname{Next}_{i}(R) Next i ( R )
∑ Q ∈ Succ ( R ) μ ( Q ) ≤ ∑ Q ∈ Next i ( R ) μ ( Q ) ≤ ( 1 − c T ) μ ( R ) . (110) \sum_{Q\in\operatorname{Succ}(R)}\mu(Q)\leq\sum_{Q\in\operatorname{Next}_{i}(R)}\mu(Q)\leq(1-c_{T})\mu(R). \tag*{(110)} Q ∈ Succ ( R ) ∑ μ ( Q ) ≤ Q ∈ Next i ( R ) ∑ μ ( Q ) ≤ ( 1 − c T ) μ ( R ) . (110) To prove the lemma, we apply (110) iteratively. Namely, for S ∈ D 0 S \in\mathcal{D}_0 S ∈ D 0 F 0 ( S ) \mathcal{F}_0(S) F 0 ( S ) Q ∈ Top i ′ ′ Q \in\operatorname{Top}^{\prime\prime}_i Q ∈ Top i ′′ S S S
F k ( S ) : = ⋃ Q ∈ F k − 1 ( S ) Succ ( Q ) for k ≥ 1. \mathcal{F}_k(S) := \bigcup_{Q \in\mathcal{F}_{k-1}(S)} \operatorname{Succ}(Q) \qquad\text{for } k \ge1. F k ( S ) := Q ∈ F k − 1 ( S ) ⋃ Succ ( Q ) for k ≥ 1. Hence, cells in F k ( S ) \mathcal{F}_k(S) F k ( S )
{ Q ∈ Top i ′ ′ : Q ⊂ S } = ⋃ k ≥ 0 F k ( S ) , \{Q \in\operatorname{Top}^{\prime\prime}_i : Q \subset S\} = \bigcup_{k \ge0} \mathcal{F}_k(S), { Q ∈ Top i ′′ : Q ⊂ S } = k ≥ 0 ⋃ F k ( S ) , so by (110) we obtain
∑ Q ∈ Top i ′ ′ Q ⊂ S μ ( Q ) = ∑ k ≥ 0 ∑ Q ∈ F k ( S ) μ ( Q ) ≤ ∑ k ≥ 0 ( 1 − c T ) k μ ( S ) = c T − 1 μ ( S ) , \sum_{\substack{Q \in\operatorname{Top}^{\prime\prime}_i\\ Q \subset S}} \mu(Q)
= \sum_{k \ge0} \sum_{Q \in\mathcal{F}_k(S)} \mu(Q)
\le\sum_{k \ge0} (1-c_T)^k \mu(S)
= c_T^{-1}\mu(S), Q ∈ Top i ′′ Q ⊂ S ∑ μ ( Q ) = k ≥ 0 ∑ Q ∈ F k ( S ) ∑ μ ( Q ) ≤ k ≥ 0 ∑ ( 1 − c T ) k μ ( S ) = c T − 1 μ ( S ) , where in the last equality we used that c T ∈ ( 0 , 1 ) c_T \in(0,1) c T ∈ ( 0 , 1 )
Then, we apply (101) and Lemma 4.6 to obtain a packing condition for cells in Top i \operatorname{Top}_i Top i
Lemma 4.7. With η ∗ \eta_* η ∗ (101) , there exists c ′ > 0 c' > 0 c ′ > 0 n n n C 0 \mathcal{C}_0 C 0 c T c_T c T S ∈ D 0 S \in\mathcal{D}_0 S ∈ D 0
∑ Q ∈ Top i Q ⊂ S Θ μ ( Q ) μ ( Q ) ≤ c ′ μ ( S ) . (111) \sum_{\substack{Q \in\operatorname{Top}_i\\ Q \subset S}} \Theta_\mu(Q)\mu(Q) \le c'\mu(S). \tag*{(111)} Q ∈ Top i Q ⊂ S ∑ Θ μ ( Q ) μ ( Q ) ≤ c ′ μ ( S ) . (111) Proof. It suffices to prove (111) for S ∈ Top i S \in\operatorname{Top}_i S ∈ Top i S S S Top i \operatorname{Top}_i Top i S S S k ≥ 0 k \ge0 k ≥ 0
Top i , ( k ) : = ⋃ j = 0 k Top i j and Top i , ( k ) ′ : = ⋃ j = 0 k ( Top i ′ ∩ Top i j ) \operatorname{Top}_{i,(k)} := \bigcup_{j=0}^{k}\operatorname{Top}_i^j
\qquad\text{and}\qquad
\operatorname{Top}'_{i,(k)} := \bigcup_{j=0}^{k}\left(\operatorname{Top}'_i \cap\operatorname{Top}_i^j\right) Top i , ( k ) := j = 0 ⋃ k Top i j and Top i , ( k ) ′ := j = 0 ⋃ k ( Top i ′ ∩ Top i j ) and Top i , ( k ) ( S ) : = { P ∈ Top i , ( k ) : P ⊂ S } \operatorname{Top}_{i,(k)}(S) := \{P \in\operatorname{Top}_{i,(k)} : P \subset S\} Top i , ( k ) ( S ) := { P ∈ Top i , ( k ) : P ⊂ S } Top i , ( k ) ′ ( S ) : = Top i , ( k ) ( S ) ∩ Top i ′ \operatorname{Top}'_{i,(k)}(S) := \operatorname{Top}_{i,(k)}(S) \cap\operatorname{Top}'_i Top i , ( k ) ′ ( S ) := Top i , ( k ) ( S ) ∩ Top i ′ Top i , ( k ) ′ ′ \operatorname{Top}^{\prime\prime}_{i,(k)} Top i , ( k ) ′′ Top i , ( k ) ′ ′ ( S ) \operatorname{Top}^{\prime\prime}_{i,(k)}(S) Top i , ( k ) ′′ ( S )
Let η ∗ \eta_* η ∗ (101) . Every cell P ∈ Top i P \in\operatorname{Top}_i P ∈ Top i n n n μ \mu μ
Θ μ ( P ) ≤ μ ( 28 B ( P ) ) ℓ ( P ) n ≤ C 0 ( 28 r ( P ) ℓ ( P ) ) n ≤ C 0 2 n . (112) \Theta_\mu(P) \le\frac{\mu(28B(P))}{\ell(P)^n}
\le\mathcal{C}_0\left(\frac{28r(P)}{\ell(P)}\right)^n
\le\frac{\mathcal{C}_0}{2^n}. \tag*{(112)} Θ μ ( P ) ≤ ℓ ( P ) n μ ( 28 B ( P )) ≤ C 0 ( ℓ ( P ) 28 r ( P ) ) n ≤ 2 n C 0 . (112) Moreover, if P ∈ Top i ′ P \in\operatorname{Top}'_i P ∈ Top i ′ P B ~ ( P ) ∈ Stop i ′ ( P ) P_{\widetilde{B}(P)} \in\operatorname{Stop}'_i(P) P B ( P ) ∈ Stop i ′ ( P ) P B ~ ( P ) ∈ Next i ( P ) P_{\widetilde{B}(P)} \in\operatorname{Next}_i(P) P B ( P ) ∈ Next i ( P )
∑ Q ∈ Next i ( P ) Θ μ ( Q ) μ ( Q ) ≥ Θ μ ( P B ~ ( P ) ) μ ( P B ~ ( P ) ) ≥ 2 Θ μ ( P ) μ ( P ) . (113) \sum_{Q \in\operatorname{Next}_i(P)} \Theta_\mu(Q)\mu(Q)
\ge\Theta_\mu\left(P_{\widetilde{B}(P)}\right)\mu\left(P_{\widetilde{B}(P)}\right)
\ge2\Theta_\mu(P)\mu(P). \tag*{(113)} Q ∈ Next i ( P ) ∑ Θ μ ( Q ) μ ( Q ) ≥ Θ μ ( P B ( P ) ) μ ( P B ( P ) ) ≥ 2 Θ μ ( P ) μ ( P ) . (113) Hence, noting that Top i , ( k ) ′ ′ ( S ) ⊆ Top i ′ ′ ( S ) \operatorname{Top}^{\prime\prime}_{i,(k)}(S) \subseteq\operatorname{Top}^{\prime\prime}_i(S) Top i , ( k ) ′′ ( S ) ⊆ Top i ′′ ( S )
∑ P ∈ Top i , ( k ) ( S ) Θ μ ( P ) μ ( P ) = ∑ P ∈ Top i , ( k ) ′ ( S ) Θ μ ( P ) μ ( P ) + ∑ P ∈ Top i , ( k ) ′ ′ ( S ) Θ μ ( P ) μ ( P ) ≤ 1 2 ∑ P ∈ Top i , ( k ) ′ ( S ) Q ∈ Next i ( P ) Θ μ ( Q ) μ ( Q ) + ∑ P ∈ Top i , ( k ) ′ ′ ( S ) Θ μ ( P ) μ ( P ) ≤ 1 2 ∑ P ∈ Top i , ( k ) ( S ) Θ μ ( P ) μ ( P ) + C 0 2 n μ ( S ) + ∑ P ∈ Top i ′ ′ ( S ) Θ μ ( P ) μ ( P ) . (114) \begin{aligned}
\sum_{P \in\operatorname{Top}_{i,(k)}(S)} \Theta_\mu(P)\mu(P)
&= \sum_{P \in\operatorname{Top}'_{i,(k)}(S)} \Theta_\mu(P)\mu(P)
+ \sum_{P \in\operatorname{Top}^{\prime\prime}_{i,(k)}(S)} \Theta_\mu(P)\mu(P) \\
&\le\frac{1}{2}\sum_{\substack{P \in\operatorname{Top}'_{i,(k)}(S)\\ Q \in\operatorname{Next}_i(P)}} \Theta_\mu(Q)\mu(Q)
+ \sum_{P \in\operatorname{Top}^{\prime\prime}_{i,(k)}(S)} \Theta_\mu(P)\mu(P) \\
&\le\frac{1}{2}\sum_{P \in\operatorname{Top}_{i,(k)}(S)} \Theta_\mu(P)\mu(P)
+ \frac{\mathcal{C}_0}{2^n}\mu(S)
+ \sum_{P \in\operatorname{Top}^{\prime\prime}_i(S)} \Theta_\mu(P)\mu(P). \tag*{(114)}
\end{aligned} P ∈ Top i , ( k ) ( S ) ∑ Θ μ ( P ) μ ( P ) = P ∈ Top i , ( k ) ′ ( S ) ∑ Θ μ ( P ) μ ( P ) + P ∈ Top i , ( k ) ′′ ( S ) ∑ Θ μ ( P ) μ ( P ) ≤ 2 1 P ∈ Top i , ( k ) ′ ( S ) Q ∈ Next i ( P ) ∑ Θ μ ( Q ) μ ( Q ) + P ∈ Top i , ( k ) ′′ ( S ) ∑ Θ μ ( P ) μ ( P ) ≤ 2 1 P ∈ Top i , ( k ) ( S ) ∑ Θ μ ( P ) μ ( P ) + 2 n C 0 μ ( S ) + P ∈ Top i ′′ ( S ) ∑ Θ μ ( P ) μ ( P ) . (114) where the new successors at level ( k + 1 ) (k+1) ( k + 1 ) S S S 2 − n c 0 μ ( S ) 2^{-n}\mathfrak{c}_0\mu(S) 2 − n c 0 μ ( S )
By Lemma 4.6 , T o p i ′ \mathrm{Top}'_i Top i ′ (112) implies
∑ P ∈ T o p i ′ ′ ( S ) Θ μ ( P ) μ ( P ) ≤ c 0 2 n ∑ P ∈ T o p i ′ ′ ( S ) μ ( P ) ≤ c 0 2 n c T − 1 μ ( S ) . (115) \sum_{P\in\mathrm{Top}''_i(S)}\Theta_\mu(P)\mu(P)\leq\frac{\mathfrak{c}_0}{2^n}\sum_{P\in\mathrm{Top}''_i(S)}\mu(P)\leq\frac{\mathfrak{c}_0}{2^n}c_T^{-1}\mu(S). \tag*{(115)} P ∈ Top i ′′ ( S ) ∑ Θ μ ( P ) μ ( P ) ≤ 2 n c 0 P ∈ Top i ′′ ( S ) ∑ μ ( P ) ≤ 2 n c 0 c T − 1 μ ( S ) . (115) Thus, we gather (114) , (115) , and obtain
∑ P ∈ T o p i , ( k ) ( S ) Θ μ ( P ) μ ( P ) ≤ 2 c 0 2 n ( 1 + c T − 1 ) μ ( S ) = : c ′ μ ( S ) . \sum_{P\in\mathrm{Top}_{i,(k)}(S)}\Theta_\mu(P)\mu(P)\leq\frac{2\mathfrak{c}_0}{2^n}(1+c_T^{-1})\mu(S)=:c'\mu(S). P ∈ Top i , ( k ) ( S ) ∑ Θ μ ( P ) μ ( P ) ≤ 2 n 2 c 0 ( 1 + c T − 1 ) μ ( S ) =: c ′ μ ( S ) . Taking the limit as k → ∞ k\to\infty k → ∞
To simplify notation, for R i ∈ T o p i R_i\in\mathrm{Top}_i R i ∈ Top i
S t o p i ( R i ) : = { S t o p i ′ ( R i ) , if R i ∈ T o p i ′ , S t o p i ′ ′ ( R i ) , if R i ∈ T o p i ′ ′ . \mathrm{Stop}_i(R_i):=
\begin{cases}
\mathrm{Stop}'_i(R_i), & \text{if }R_i\in\mathrm{Top}'_i,\\
\mathrm{Stop}''_i(R_i), & \text{if }R_i\in\mathrm{Top}''_i.
\end{cases} Stop i ( R i ) := { Stop i ′ ( R i ) , Stop i ′′ ( R i ) , if R i ∈ Top i ′ , if R i ∈ Top i ′′ . For R i ∈ T o p i R_i\in\mathrm{Top}_i R i ∈ Top i
T r e e i ( R i ) : = { Q ∈ D 0 ( R i ) : ∄ S ∈ S t o p i ( R i ) such that Q ⊆ S } . (116) \mathrm{Tree}_i(R_i):=\{Q\in\mathcal{D}_0(R_i):\nexists S\in\mathrm{Stop}_i(R_i)\text{ such that }Q\subseteq S\}. \tag*{(116)} Tree i ( R i ) := { Q ∈ D 0 ( R i ) : ∄ S ∈ Stop i ( R i ) such that Q ⊆ S } . (116) For R i ∈ T o p i R_i\in\mathrm{Top}_i R i ∈ Top i (116) satisfies the following properties:
The cell R i R_i R i T r e e i ( R i ) \mathrm{Tree}_i(R_i) Tree i ( R i ) Q ⊂ R i Q\subset R_i Q ⊂ R i Q ∈ T r e e i ( R i ) Q\in\mathrm{Tree}_i(R_i) Q ∈ Tree i ( R i )
If Q , Q ′ ∈ T r e e i ( R i ) Q,Q'\in\mathrm{Tree}_i(R_i) Q , Q ′ ∈ Tree i ( R i ) Q ⊂ Q ′ Q\subset Q' Q ⊂ Q ′ P ∈ D 0 ( R i ) P\in\mathcal{D}_0(R_i) P ∈ D 0 ( R i ) Q ⊂ P ⊂ Q ′ Q\subset P\subset Q' Q ⊂ P ⊂ Q ′ P ∈ T r e e i ( R i ) P\in\mathrm{Tree}_i(R_i) P ∈ Tree i ( R i )
If Q ∈ T r e e i ( R i ) Q\in\mathrm{Tree}_i(R_i) Q ∈ Tree i ( R i ) Q Q Q T r e e i ( R i ) \mathrm{Tree}_i(R_i) Tree i ( R i ) S t o p i ( R i ) \mathrm{Stop}_i(R_i) Stop i ( R i )
Finally, for R i ∈ T o p i R_i\in\mathrm{Top}_i R i ∈ Top i Q ∈ S t o p i ( R i ) Q\in\mathrm{Stop}_i(R_i) Q ∈ Stop i ( R i )
T i , Q : = { P ∈ D 0 ( Q ) : ∄ S ∈ M D b ( Q ) such that P ⊆ S } . \mathcal{T}_{i,Q}:=\{P\in\mathcal{D}_0(Q):\nexists S\in\mathrm{MDb}(Q)\text{ such that }P\subseteq S\}. T i , Q := { P ∈ D 0 ( Q ) : ∄ S ∈ MDb ( Q ) such that P ⊆ S } . If Q Q Q T i , Q = ∅ \mathcal{T}_{i,Q}=\varnothing T i , Q = ∅ T i , Q \mathcal{T}_{i,Q} T i , Q (116) and the definition of T i , Q \mathcal{T}_{i,Q} T i , Q T r e e i ( R i ) \mathrm{Tree}_i(R_i) Tree i ( R i ) T i , Q \mathcal{T}_{i,Q} T i , Q R i ∈ T o p i R_i\in\mathrm{Top}_i R i ∈ Top i Q ∈ S t o p i ( R i ) Q\in\mathrm{Stop}_i(R_i) Q ∈ Stop i ( R i )
The mixed corona construction
Since the cells in T o p 0 \mathrm{Top}^0 Top 0 B 0 B_0 B 0 μ \mu μ
∑ Q ∈ T o p i ′ ′ μ ( Q ) ≤ c T − 1 μ ( B 0 ) \sum_{Q\in\mathrm{Top}''_i}\mu(Q)\leq c_T^{-1}\mu(B_0) Q ∈ Top i ′′ ∑ μ ( Q ) ≤ c T − 1 μ ( B 0 ) and
∑ Q ∈ T o p i Θ μ ( Q ) μ ( Q ) ≤ c ′ μ ( B 0 ) . \sum_{Q\in\mathrm{Top}_i}\Theta_\mu(Q)\mu(Q)\leq c'\mu(B_0). Q ∈ Top i ∑ Θ μ ( Q ) μ ( Q ) ≤ c ′ μ ( B 0 ) . For R 1 ∈ T o p 1 ′ R_1\in\mathrm{Top}'_1 R 1 ∈ Top 1 ′ R 2 ∈ T o p 2 ′ ′ R_2\in\mathrm{Top}''_2 R 2 ∈ Top 2 ′′
T r e e ~ ( R 1 , R 2 ) : = T r e e 1 ( R 1 ) ∩ T r e e 2 ( R 2 ) , \widetilde{\mathrm{Tree}}(R_1,R_2):=\mathrm{Tree}_1(R_1)\cap\mathrm{Tree}_2(R_2), Tree ( R 1 , R 2 ) := Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 ) , which we also refer to as a mixed tree . If T r e e 1 ( R 1 ) ∩ T r e e 2 ( R 2 ) ≠ ∅ \mathrm{Tree}_1(R_1)\cap\mathrm{Tree}_2(R_2)\neq\varnothing Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 ) = ∅ T r e e ~ ( R 1 , R 2 ) \widetilde{\mathrm{Tree}}(R_1,R_2) Tree ( R 1 , R 2 ) R R R R 1 R_1 R 1 R 2 R_2 R 2 Q ∈ Tree ~ ( R 1 , R 2 ) Q \in\widetilde{\operatorname{Tree}}(R_1,R_2) Q ∈ Tree ( R 1 , R 2 ) P P P Q Q Q Tree ~ ( R 1 , R 2 ) \widetilde{\operatorname{Tree}}(R_1,R_2) Tree ( R 1 , R 2 ) P ∈ Stop 1 ( R 1 ) ∪ Stop 2 ( R 2 ) P \in\operatorname{Stop}_1(R_1) \cup\operatorname{Stop}_2(R_2) P ∈ Stop 1 ( R 1 ) ∪ Stop 2 ( R 2 )
Tree ~ ( R ) : = Tree ~ ( R 1 , R 2 ) . \widetilde{\operatorname{Tree}}(R):=\widetilde{\operatorname{Tree}}(R_1,R_2). Tree ( R ) := Tree ( R 1 , R 2 ) . Finally, we define the initial intermediate family
T 0 : = { Q ∈ D 0 : Q ⊄ R for every R ∈ Top 0 } . (117) \mathcal{T}_0:=\{Q\in\mathcal{D}_0: Q\not\subset R\text{ for every }R\in\operatorname{Top}^0\}. \tag*{(117)} T 0 := { Q ∈ D 0 : Q ⊂ R for every R ∈ Top 0 } . (117) We denote by Top ~ \widetilde{\operatorname{Top}} Top R ∈ Top ~ R\in\widetilde{\operatorname{Top}} R ∈ Top ( R 1 , R 2 ) ∈ Top 1 ′ ′ × Top 2 ′ ′ (R_1,R_2)\in\operatorname{Top}_1^{\prime\prime}\times\operatorname{Top}_2^{\prime\prime} ( R 1 , R 2 ) ∈ Top 1 ′′ × Top 2 ′′ R ∈ Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 ) R\in\operatorname{Tree}_1(R_1)\cap\operatorname{Tree}_2(R_2) R ∈ Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 )
Top ~ ⊂ Top 1 ′ ′ ∪ Top 2 ′ ′ . \widetilde{\operatorname{Top}}\subset\operatorname{Top}_1^{\prime\prime}\cup\operatorname{Top}_2^{\prime\prime}. Top ⊂ Top 1 ′′ ∪ Top 2 ′′ . Hence Lemma 4.6 implies the following estimate.
Lemma 4.8. The collection Top ~ \widetilde{\operatorname{Top}} Top S ∈ D 0 S\in\mathcal{D}_0 S ∈ D 0
∑ Q ∈ Top ~ Q ⊂ S μ ( Q ) ≤ 2 c T − 1 μ ( S ) , \sum_{\substack{Q\in\widetilde{\operatorname{Top}}\\Q\subset S}}\mu(Q)\le2c_T^{-1}\mu(S), Q ∈ Top Q ⊂ S ∑ μ ( Q ) ≤ 2 c T − 1 μ ( S ) , where c T c_T c T
Proof. By the definition of Top ~ \widetilde{\operatorname{Top}} Top Lemma 4.6 , for S ∈ D 0 S\in\mathcal{D}_0 S ∈ D 0
∑ Q ∈ Top ~ Q ⊂ S μ ( Q ) ≤ ∑ Q ∈ Top 1 ′ ′ Q ⊂ S μ ( Q ) + ∑ Q ∈ Top 2 ′ ′ Q ⊂ S μ ( Q ) ≤ 2 c T − 1 μ ( S ) . \sum_{\substack{Q\in\widetilde{\operatorname{Top}}\\Q\subset S}}\mu(Q)
\le
\sum_{\substack{Q\in\operatorname{Top}_1^{\prime\prime}\\Q\subset S}}\mu(Q)
+
\sum_{\substack{Q\in\operatorname{Top}_2^{\prime\prime}\\Q\subset S}}\mu(Q)
\le2c_T^{-1}\mu(S). Q ∈ Top Q ⊂ S ∑ μ ( Q ) ≤ Q ∈ Top 1 ′′ Q ⊂ S ∑ μ ( Q ) + Q ∈ Top 2 ′′ Q ⊂ S ∑ μ ( Q ) ≤ 2 c T − 1 μ ( S ) . Finally, we gather all the families of cells defined in the mixed corona construction, and note that
D 0 = T 0 ∪ [ ⋃ R ∈ Top ~ Tree ~ ( R ) ] ∪ [ ⋃ i = 1 , 2 ⋃ R i ∈ Top i ′ ( Tree i ( R i ) ∪ ⋃ Q ∈ Stop i ′ ( R i ) T i , Q ) ] ∪ [ ⋃ i = 1 , 2 ⋃ R i ∈ Top i ′ ′ ⋃ Q ∈ Stop i ′ ′ ( R i ) T i , Q ] . (118) \mathcal{D}_0=\mathcal{T}_0\cup\left[\bigcup_{R\in\widetilde{\operatorname{Top}}}\widetilde{\operatorname{Tree}}(R)\right]\cup\left[\bigcup_{i=1,2}\bigcup_{R_i\in\operatorname{Top}_i'}\left(\operatorname{Tree}_i(R_i)\cup\bigcup_{Q\in\operatorname{Stop}_i'(R_i)}\mathcal{T}_{i,Q}\right)\right]\cup\left[\bigcup_{i=1,2}\bigcup_{R_i\in\operatorname{Top}_i''}\bigcup_{Q\in\operatorname{Stop}_i''(R_i)}\mathcal{T}_{i,Q}\right]. \tag*{(118)} D 0 = T 0 ∪ R ∈ Top ⋃ Tree ( R ) ∪ i = 1 , 2 ⋃ R i ∈ Top i ′ ⋃ Tree i ( R i ) ∪ Q ∈ Stop i ′ ( R i ) ⋃ T i , Q ∪ i = 1 , 2 ⋃ R i ∈ Top i ′′ ⋃ Q ∈ Stop i ′′ ( R i ) ⋃ T i , Q . (118) SIOs associated with the mixed corona decomposition
The families Tree ~ ( R ) \widetilde{\operatorname{Tree}}(R) Tree ( R ) R ∈ Top ~ R\in\widetilde{\operatorname{Top}} R ∈ Top (118) . Moreover, a cell may belong to the remaining families for both i = 1 i=1 i = 1 i = 2 i=2 i = 2
Now, we define the SIO associated with each family on the right-hand side of (118) . To do so, for Q ∈ D 0 Q\in\mathcal{D}_0 Q ∈ D 0
ρ ( Q ) : = s A 0 − J ( Q ) = ℓ ( Q ) 56 K 0 , (119) \rho(Q):=sA_0^{-J(Q)}=\frac{\ell(Q)}{56K_0}, \tag*{(119)} ρ ( Q ) := s A 0 − J ( Q ) = 56 K 0 ℓ ( Q ) , (119) and note that
ρ ( Q ) ≤ r ( Q ) ≤ K 0 ρ ( Q ) . (120) \rho(Q)\le r(Q)\le K_0\rho(Q). \tag*{(120)} ρ ( Q ) ≤ r ( Q ) ≤ K 0 ρ ( Q ) . (120) If Q ∈ D 0 ∖ D 0 d b Q\in\mathcal{D}_0\setminus\mathcal{D}_0^{db} Q ∈ D 0 ∖ D 0 d b r ( Q ) = ρ ( Q ) r(Q)=\rho(Q) r ( Q ) = ρ ( Q )
Then, for ε > 0 \varepsilon>0 ε > 0 Q ∈ D 0 Q\in\mathcal{D}_0 Q ∈ D 0 f ∈ L l o c 1 ( μ ) f\in L^1_{\mathrm{loc}}(\mu) f ∈ L loc 1 ( μ )
T ε , Q ( f μ ) ( x ) : = χ Q ( x ) ∫ max ( ε , 2 ρ ( Q ) / A 0 ) < d ( x , y ) ≤ 2 ρ ( Q ) K ( x , y ) f ( y ) d μ ( y ) , x ∈ X . (121) T_{\varepsilon,Q}(f\mu)(x):=\chi_Q(x)\int_{\max(\varepsilon,2\rho(Q)/A_0)<d(x,y)\le2\rho(Q)}K(x,y)f(y)\,d\mu(y),\qquad x\in X. \tag*{(121)} T ε , Q ( f μ ) ( x ) := χ Q ( x ) ∫ m a x ( ε , 2 ρ ( Q ) / A 0 ) < d ( x , y ) ≤ 2 ρ ( Q ) K ( x , y ) f ( y ) d μ ( y ) , x ∈ X . (121) We claim that, for every f ∈ L l o c 1 ( μ ) f \in L^1_{\mathrm{loc}}(\mu) f ∈ L loc 1 ( μ ) B 0 B_0 B 0 μ \mu μ x ∈ B 0 x \in B_0 x ∈ B 0
T μ , ε f ( x ) = ∑ Q ∈ D 0 T ε , Q ( f μ ) ( x ) . (122) T_{\mu,\varepsilon}f(x)=\sum_{Q\in\mathcal{D}_0}T_{\varepsilon,Q}(f\mu)(x). \tag*{(122)} T μ , ε f ( x ) = Q ∈ D 0 ∑ T ε , Q ( f μ ) ( x ) . (122) To prove (122) , for μ \mu μ x ∈ supp ( μ ∣ B 0 ) x\in\operatorname{supp}(\mu|_{B_0}) x ∈ supp ( μ ∣ B 0 ) Q k ( x ) Q_k(x) Q k ( x ) D 0 \mathcal{D}_0 D 0 k k k x x x ρ ( Q k + 1 ( x ) ) = ρ ( Q k ( x ) ) / A 0 \rho(Q_{k+1}(x))=\rho(Q_k(x))/A_0 ρ ( Q k + 1 ( x )) = ρ ( Q k ( x )) / A 0
d ( x , y ) ≤ 2 r ( B 0 ) = s < 2 ρ ( Q 0 ( x ) ) , for x , y ∈ supp ( μ ∣ B 0 ) , d(x,y)\leq2r(B_0)=s<2\rho(Q_0(x)),\qquad\text{for }x,y\in\operatorname{supp}(\mu|_{B_0}), d ( x , y ) ≤ 2 r ( B 0 ) = s < 2 ρ ( Q 0 ( x )) , for x , y ∈ supp ( μ ∣ B 0 ) , the half-open annuli in (121) form a disjoint partition of { y ∈ supp ( μ ∣ B 0 ) : d ( x , y ) > ε } \{y\in\operatorname{supp}(\mu|_{B_0}):d(x,y)>\varepsilon\} { y ∈ supp ( μ ∣ B 0 ) : d ( x , y ) > ε }
∣ T ε , Q ( f μ ) ( x ) ∣ ≲ C A 0 n μ ( 30 B ( Q ) ∩ B 0 ) ρ ( Q ) n ∥ f ∥ L ∞ ( μ ) χ Q ( x ) , (123) |T_{\varepsilon,Q}(f\mu)(x)|\lesssim C A_0^n\frac{\mu(30B(Q)\cap B_0)}{\rho(Q)^n}\|f\|_{L^\infty(\mu)}\chi_Q(x), \tag*{(123)} ∣ T ε , Q ( f μ ) ( x ) ∣ ≲ C A 0 n ρ ( Q ) n μ ( 30 B ( Q ) ∩ B 0 ) ∥ f ∥ L ∞ ( μ ) χ Q ( x ) , (123) since x ∈ Q ⊂ 28 B ( Q ) x\in Q\subset28B(Q) x ∈ Q ⊂ 28 B ( Q ) ρ ( Q ) ≤ r ( Q ) \rho(Q)\leq r(Q) ρ ( Q ) ≤ r ( Q ) B ( x , 2 ρ ( Q ) ) ∩ supp ( μ ∣ B 0 ) ⊂ 30 B ( Q ) B(x,2\rho(Q))\cap\operatorname{supp}(\mu|_{B_0})\subset30B(Q) B ( x , 2 ρ ( Q )) ∩ supp ( μ ∣ B 0 ) ⊂ 30 B ( Q )
For R ∈ Top R\in\operatorname{Top} R ∈ Top mixed tree operators
K ~ ε , R ( f μ ) ( x ) : = ∑ Q ∈ Tree ( R ) T ε , Q ( f μ ) ( x ) (124) \widetilde{K}_{\varepsilon,R}(f\mu)(x):=\sum_{Q\in\operatorname{Tree}(R)}T_{\varepsilon,Q}(f\mu)(x) \tag*{(124)} K ε , R ( f μ ) ( x ) := Q ∈ Tree ( R ) ∑ T ε , Q ( f μ ) ( x ) (124) and
Σ ~ ε f ( x ) : = ∑ R ∈ Top ∣ K ~ ε , R ( f μ ) ( x ) ∣ . (125) \widetilde{\Sigma}_{\varepsilon}f(x):=\sum_{R\in\operatorname{Top}}|\widetilde{K}_{\varepsilon,R}(f\mu)(x)|. \tag*{(125)} Σ ε f ( x ) := R ∈ Top ∑ ∣ K ε , R ( f μ ) ( x ) ∣. (125) For R i ∈ Top i R_i\in\operatorname{Top}_i R i ∈ Top i P ∈ Stop i ( R i ) P\in\operatorname{Stop}_i(R_i) P ∈ Stop i ( R i )
K ε , i , P ( f μ ) ( x ) : = ∑ Q ∈ T i , P T ε , Q ( f μ ) ( x ) (126) K_{\varepsilon,i,P}(f\mu)(x):=\sum_{Q\in\mathcal{T}_{i,P}}T_{\varepsilon,Q}(f\mu)(x) \tag*{(126)} K ε , i , P ( f μ ) ( x ) := Q ∈ T i , P ∑ T ε , Q ( f μ ) ( x ) (126) the SIO associated with the intermediate trees between stopping cells and the following top cells. We also define
Σ 0 , ε f ( x ) : = ∑ Q ∈ T 0 ∣ T ε , Q ( f μ ) ( x ) ∣ , \Sigma_{0,\varepsilon}f(x):=\sum_{Q\in\mathcal{T}_0}|T_{\varepsilon,Q}(f\mu)(x)|, Σ 0 , ε f ( x ) := Q ∈ T 0 ∑ ∣ T ε , Q ( f μ ) ( x ) ∣ , Σ i , ε ′ f ( x ) : = ∑ R i ∈ Top i ′ ( ∑ Q ∈ Tree i ( R i ) ∣ T ε , Q ( f μ ) ( x ) ∣ + ∑ P ∈ Stop i ′ ( R i ) ∑ Q ∈ T i , P ∣ T ε , Q ( f μ ) ( x ) ∣ ) , \Sigma'_{i,\varepsilon}f(x):=\sum_{R_i\in\operatorname{Top}'_i}\left(\sum_{Q\in\operatorname{Tree}_i(R_i)}|T_{\varepsilon,Q}(f\mu)(x)|+\sum_{P\in\operatorname{Stop}'_i(R_i)}\sum_{Q\in\mathcal{T}_{i,P}}|T_{\varepsilon,Q}(f\mu)(x)|\right), Σ i , ε ′ f ( x ) := R i ∈ Top i ′ ∑ Q ∈ Tree i ( R i ) ∑ ∣ T ε , Q ( f μ ) ( x ) ∣ + P ∈ Stop i ′ ( R i ) ∑ Q ∈ T i , P ∑ ∣ T ε , Q ( f μ ) ( x ) ∣ , Σ i , ε ′ ′ f ( x ) : = ∑ R i ∈ Top i ′ ′ ∑ P ∈ Stop i ′ ′ ( R i ) ∑ Q ∈ T i , P ∣ T ε , Q ( f μ ) ( x ) ∣ , i = 1 , 2. \Sigma''_{i,\varepsilon}f(x):=\sum_{R_i\in\operatorname{Top}''_i}\sum_{P\in\operatorname{Stop}''_i(R_i)}\sum_{Q\in\mathcal{T}_{i,P}}|T_{\varepsilon,Q}(f\mu)(x)|,\qquad i=1,2. Σ i , ε ′′ f ( x ) := R i ∈ Top i ′′ ∑ P ∈ Stop i ′′ ( R i ) ∑ Q ∈ T i , P ∑ ∣ T ε , Q ( f μ ) ( x ) ∣ , i = 1 , 2. Thus, combining (118) and (122) , we obtain the pointwise bound
∣ T μ , ε f ( x ) ∣ ≤ Σ ~ ε f ( x ) + Σ 0 , ε f ( x ) + ∑ i = 1 , 2 ( Σ i , ε ′ f ( x ) + Σ i , ε ′ ′ f ( x ) ) . (127) |T_{\mu,\varepsilon}f(x)|\leq\widetilde{\Sigma}_{\varepsilon}f(x)+\Sigma_{0,\varepsilon}f(x)+\sum_{i=1,2}\left(\Sigma'_{i,\varepsilon}f(x)+\Sigma''_{i,\varepsilon}f(x)\right). \tag*{(127)} ∣ T μ , ε f ( x ) ∣ ≤ Σ ε f ( x ) + Σ 0 , ε f ( x ) + i = 1 , 2 ∑ ( Σ i , ε ′ f ( x ) + Σ i , ε ′′ f ( x ) ) . (127) In the following lemmas, we estimate the L 1 ( μ ∣ B 0 ) L^1(\mu|_{B_0}) L 1 ( μ ∣ B 0 ) (127) , assuming that the operators are applied to L ∞ ( μ ) L^\infty(\mu) L ∞ ( μ ) B 0 B_0 B 0
We recall that μ 0 = μ ∣ B 0 \mu_0=\mu|_{B_0} μ 0 = μ ∣ B 0 μ 0 ≤ μ \mu_0\leq\mu μ 0 ≤ μ Q ⊂ B 0 Q\subset B_0 Q ⊂ B 0 μ 0 ( Q ) = μ ( Q ) \mu_0(Q)=\mu(Q) μ 0 ( Q ) = μ ( Q ) (120) and the n n n μ \mu μ
μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ≲ K 0 , n Θ μ ( 100 B ( Q ) ) ≲ C 0 . (128) \frac{\mu_0(30B(Q))}{\rho(Q)^n}\lesssim_{K_0,n}\Theta_\mu(100B(Q))\lesssim C_0. \tag*{(128)} ρ ( Q ) n μ 0 ( 30 B ( Q )) ≲ K 0 , n Θ μ ( 100 B ( Q )) ≲ C 0 . (128) If Q ⊂ B 0 Q \subset B_0 Q ⊂ B 0 (128) to
μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ≲ K 0 , n Θ μ ( 100 B ( Q ) ) ≲ K 0 , n Θ μ ( Q ) . (129) \frac{\mu_0(30B(Q))}{\rho(Q)^n} \lesssim_{K_0,n} \Theta_\mu(100B(Q)) \lesssim_{K_0,n} \Theta_\mu(Q). \tag*{(129)} ρ ( Q ) n μ 0 ( 30 B ( Q )) ≲ K 0 , n Θ μ ( 100 B ( Q )) ≲ K 0 , n Θ μ ( Q ) . (129) We first bound the operator associated with cells in T 0 \mathcal{T}_0 T 0
Lemma 4.9. For every g ∈ L ∞ ( μ ) g \in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0
∫ B 0 ∣ Σ 0 , ε g ( x ) ∣ d μ ( x ) ≤ c Σ 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , \int_{B_0} \lvert\Sigma_{0,\varepsilon}g(x)\rvert\,d\mu(x) \leq c_\Sigma^0 \lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0), ∫ B 0 ∣ Σ 0 , ε g ( x )∣ d μ ( x ) ≤ c Σ 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , where c Σ 0 > 0 c_\Sigma^0>0 c Σ 0 > 0 n n n C C C A 0 A_0 A 0 K 0 K_0 K 0 C 0 \mathcal{C}_0 C 0 β \beta β C 3 \mathcal{C}_3 C 3
Proof. Let x ∈ B 0 ∩ supp μ x \in B_0 \cap\operatorname{supp}\mu x ∈ B 0 ∩ supp μ μ \mu μ x ∈ B 0 x \in B_0 x ∈ B 0 R x R_x R x Top 0 \operatorname{Top}^0 Top 0 x x x P x P_x P x x x x B 0 B_0 B 0 B 0 B_0 B 0 Top 0 \operatorname{Top}^0 Top 0 x x x T 0 \mathcal{T}_0 T 0 R x R_x R x
Σ 0 , ε g ( x ) = ∑ Q ∋ x , Q ⊄ B 0 ∣ T ε , Q ( g μ ) ( x ) ∣ + ∑ R x ⊊ Q ⊂ P x ∣ T ε , Q ( g μ ) ( x ) ∣ = : I 1 ( x ) + I 2 ( x ) . \Sigma_{0,\varepsilon}g(x)
=
\sum_{\substack{Q\ni x,\,Q\not\subset B_0}} \lvert T_{\varepsilon,Q}(g\mu)(x)\rvert
+
\sum_{R_x\subsetneq Q\subset P_x} \lvert T_{\varepsilon,Q}(g\mu)(x)\rvert
=: \mathcal{I}_1(x)+\mathcal{I}_2(x). Σ 0 , ε g ( x ) = Q ∋ x , Q ⊂ B 0 ∑ ∣ T ε , Q ( g μ ) ( x )∣ + R x ⊊ Q ⊂ P x ∑ ∣ T ε , Q ( g μ ) ( x )∣ =: I 1 ( x ) + I 2 ( x ) . To estimate I 1 ( x ) \mathcal{I}_1(x) I 1 ( x ) t ( x ) : = dist ( x , X ∖ B 0 ) t(x):=\operatorname{dist}(x,X\setminus B_0) t ( x ) := dist ( x , X ∖ B 0 ) I 1 ( x ) = 0 \mathcal{I}_1(x)=0 I 1 ( x ) = 0 X = B 0 X=B_0 X = B 0 Q Q Q B 0 B_0 B 0 ℓ ( Q ) ≥ diam ( Q ) ≥ t ( x ) \ell(Q)\geq\operatorname{diam}(Q)\geq t(x) ℓ ( Q ) ≥ diam ( Q ) ≥ t ( x ) ℓ ( Q ) ≤ 56 K 0 s = 112 K 0 r ( B 0 ) \ell(Q)\leq56K_0s=112K_0r(B_0) ℓ ( Q ) ≤ 56 K 0 s = 112 K 0 r ( B 0 ) A 0 A_0 A 0 C A 0 , K 0 ( 1 + log + ( r ( B 0 ) / t ( x ) ) ) C_{A_0,K_0}(1+\log^+(r(B_0)/t(x))) C A 0 , K 0 ( 1 + log + ( r ( B 0 ) / t ( x ))) (123) and (128) give
I 1 ( x ) ≲ ( 1 + log + r ( B 0 ) t ( x ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . \mathcal{I}_1(x)\lesssim\left(1+\log^+\frac{r(B_0)}{t(x)}\right)\lVert g\rVert_{L^\infty(\mu|_{B_0})}. I 1 ( x ) ≲ ( 1 + log + t ( x ) r ( B 0 ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . To estimate I 2 ( x ) \mathcal{I}_2(x) I 2 ( x ) Q Q Q R x ⊊ Q ⊂ P x R_x\subsetneq Q\subset P_x R x ⊊ Q ⊂ P x R x R_x R x (123) , (128) , and Lemma 2.8, applied to the parent of R x R_x R x P x P_x P x
I 2 ( x ) ≲ ∑ R x ⊊ Q ⊂ P x Θ μ ( 100 B ( Q ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≲ Θ μ ( 100 B ( P x ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) . \mathcal{I}_2(x)\lesssim\sum_{R_x\subsetneq Q\subset P_x}\Theta_\mu(100B(Q))\lVert g\rVert_{L^\infty(\mu|_{B_0})}
\lesssim\Theta_\mu(100B(P_x))\lVert g\rVert_{L^\infty(\mu|_{B_0})}
\leq\mathcal{C}_0\lVert g\rVert_{L^\infty(\mu|_{B_0})}. I 2 ( x ) ≲ R x ⊊ Q ⊂ P x ∑ Θ μ ( 100 B ( Q )) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≲ Θ μ ( 100 B ( P x )) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) . Finally, we write B 0 = B ( z 0 , r 0 ) B_0=B(z_0,r_0) B 0 = B ( z 0 , r 0 ) t ( x ) ≥ r 0 − d ( x , z 0 ) t(x)\geq r_0-d(x,z_0) t ( x ) ≥ r 0 − d ( x , z 0 ) u > 0 u>0 u > 0 C 3 \mathcal{C}_3 C 3 B 0 B_0 B 0 (94) give
μ ( { x ∈ B 0 : t ( x ) < r 0 e − u } ) ≤ C 3 e − u μ ( 5 B 0 ) ≤ C 3 β e − u μ ( B 0 ) , \mu\bigl(\{x\in B_0:t(x)<r_0e^{-u}\}\bigr)\leq\mathcal{C}_3e^{-u}\mu(5B_0)\leq\mathcal{C}_3\beta e^{-u}\mu(B_0), μ ( { x ∈ B 0 : t ( x ) < r 0 e − u } ) ≤ C 3 e − u μ ( 5 B 0 ) ≤ C 3 β e − u μ ( B 0 ) , and hence
∫ B 0 log + ( r 0 t ( x ) ) d μ ( x ) = ∫ 0 ∞ μ ( { x ∈ B 0 : t ( x ) < r 0 e − u } ) d u ≤ C 3 β μ ( B 0 ) . \int_{B_0}\log^+\left(\frac{r_0}{t(x)}\right)\,d\mu(x)
=
\int_0^\infty\mu\bigl(\{x\in B_0:t(x)<r_0e^{-u}\}\bigr)\,du
\leq\mathcal{C}_3\beta\mu(B_0). ∫ B 0 log + ( t ( x ) r 0 ) d μ ( x ) = ∫ 0 ∞ μ ( { x ∈ B 0 : t ( x ) < r 0 e − u } ) d u ≤ C 3 β μ ( B 0 ) . Integrating the estimates for I 1 ( x ) \mathcal{I}_1(x) I 1 ( x ) I 2 ( x ) \mathcal{I}_2(x) I 2 ( x ) B 0 B_0 B 0
∫ B 0 Σ 0 , ε g d μ ≲ ( 1 + C 3 β ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . \int_{B_0}\Sigma_{0,\varepsilon}g\,d\mu
\lesssim(1+\mathcal{C}_3\beta)\lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0). ∫ B 0 Σ 0 , ε g d μ ≲ ( 1 + C 3 β ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . We next estimate Σ i , ε ′ \Sigma'_{i,\varepsilon} Σ i , ε ′ Σ i , ε ′ ′ \Sigma''_{i,\varepsilon} Σ i , ε ′′ c T c_{\mathcal{T}} c T c ′ c' c ′ (104) and (111) , respectively.
Lemma 4.10. If g ∈ L ∞ ( μ ) g\in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0 c Σ ′ > 0 c'_\Sigma>0 c Σ ′ > 0
∫ B 0 ∣ Σ 1 , ε ′ g ( x ) ∣ d μ ( x ) + ∫ B 0 ∣ Σ 2 , ε ′ ′ g ( x ) ∣ d μ ( x ) ≤ c Σ c ′ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . (130) \int_{B_0}\lvert\Sigma'_{1,\varepsilon}g(x)\rvert\,d\mu(x)
+
\int_{B_0}\lvert\Sigma''_{2,\varepsilon}g(x)\rvert\,d\mu(x)
\leq c_\Sigma c' \lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0). \tag*{(130)} ∫ B 0 ∣ Σ 1 , ε ′ g ( x )∣ d μ ( x ) + ∫ B 0 ∣ Σ 2 , ε ′′ g ( x )∣ d μ ( x ) ≤ c Σ c ′ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . (130) The constant c Σ ′ c'_\Sigma c Σ ′ n n n C C C A 0 A_0 A 0 K 0 K_0 K 0 C 0 \mathcal{C}_0 C 0 η ∗ \eta_* η ∗
Proof. Let R i ∈ Top i ′ R_i \in\operatorname{Top}'_i R i ∈ Top i ′ i = 1 , 2 i=1,2 i = 1 , 2 R i R_i R i (129) yields
μ 0 ( 30 B ( R i ) ) ρ ( R i ) n ≲ K 0 , n Θ μ ( R i ) . \frac{\mu_0(30B(R_i))}{\rho(R_i)^n} \lesssim_{K_0,n} \Theta_\mu(R_i). ρ ( R i ) n μ 0 ( 30 B ( R i )) ≲ K 0 , n Θ μ ( R i ) . Moreover, if Q ∈ Tree i ( R i ) Q \in\operatorname{Tree}_i(R_i) Q ∈ Tree i ( R i ) Q ≠ R i Q \ne R_i Q = R i Stop ′ \operatorname{Stop}' Stop ′ ρ ( Q ) ≳ η ∗ ρ ( R i ) \rho(Q) \gtrsim\eta_* \rho(R_i) ρ ( Q ) ≳ η ∗ ρ ( R i ) ρ ( Q ) ≤ ρ ( R i ) \rho(Q) \le\rho(R_i) ρ ( Q ) ≤ ρ ( R i )
μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ≲ η ∗ , n μ ( 100 B ( R i ) ) ρ ( R i ) n ≲ K 0 , n Θ μ ( R i ) . \frac{\mu_0(30B(Q))}{\rho(Q)^n} \lesssim_{\eta_*,n} \frac{\mu(100B(R_i))}{\rho(R_i)^n} \lesssim_{K_0,n} \Theta_\mu(R_i). ρ ( Q ) n μ 0 ( 30 B ( Q )) ≲ η ∗ , n ρ ( R i ) n μ ( 100 B ( R i )) ≲ K 0 , n Θ μ ( R i ) . Also, ρ ( Q ) ≳ η ∗ ρ ( R i ) \rho(Q) \gtrsim\eta_* \rho(R_i) ρ ( Q ) ≳ η ∗ ρ ( R i ) J ( Q ) − J ( R i ) ≲ 1 + log ( 1 / η ∗ ) J(Q)-J(R_i) \lesssim1+\log(1/\eta_*) J ( Q ) − J ( R i ) ≲ 1 + log ( 1/ η ∗ ) Q ∈ Tree i ( R i ) Q \in\operatorname{Tree}_i(R_i) Q ∈ Tree i ( R i ) (123) , then, yields
∫ B 0 ∑ Q ∈ Tree i ( R i ) ∣ T ε , Q ( g μ ) ∣ d μ ≲ ∫ B 0 ∑ Q ∈ Tree i ( R i ) μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ Q ( x ) d μ ( x ) ≲ η ∗ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (131) \begin{aligned}
\int_{B_0}\sum_{Q\in\operatorname{Tree}_i(R_i)} |T_{\varepsilon,Q}(g\mu)|\,d\mu
&\lesssim\int_{B_0}\sum_{Q\in\operatorname{Tree}_i(R_i)}
\frac{\mu_0(30B(Q))}{\rho(Q)^n}\lVert g\rVert_{L^\infty(\mu|_{B_0})}\chi_Q(x)\,d\mu(x)\\
&\lesssim_{\eta_*}\Theta_\mu(R_i)\mu(R_i)\lVert g\rVert_{L^\infty(\mu|_{B_0})}. \tag*{(131)}
\end{aligned} ∫ B 0 Q ∈ Tree i ( R i ) ∑ ∣ T ε , Q ( g μ ) ∣ d μ ≲ ∫ B 0 Q ∈ Tree i ( R i ) ∑ ρ ( Q ) n μ 0 ( 30 B ( Q )) ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ Q ( x ) d μ ( x ) ≲ η ∗ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (131) If P ∈ Stop i ′ ( R i ) P \in\operatorname{Stop}'_i(R_i) P ∈ Stop i ′ ( R i ) T i , P \mathcal{T}_{i,P} T i , P μ \mu μ x ∈ P x \in P x ∈ P
∑ Q ∈ T i , P x ∈ Q μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ≲ Θ μ ( 100 B ( P ) ) ≲ η ∗ , n Θ μ ( R i ) , (132) \sum_{\substack{Q\in\mathcal{T}_{i,P}\\x\in Q}}
\frac{\mu_0(30B(Q))}{\rho(Q)^n}
\lesssim\Theta_\mu(100B(P)) \lesssim_{\eta_*,n} \Theta_\mu(R_i), \tag*{(132)} Q ∈ T i , P x ∈ Q ∑ ρ ( Q ) n μ 0 ( 30 B ( Q )) ≲ Θ μ ( 100 B ( P )) ≲ η ∗ , n Θ μ ( R i ) , (132) where the last bound uses ρ ( P ) ≳ η ∗ ρ ( R i ) \rho(P) \gtrsim\eta_*\rho(R_i) ρ ( P ) ≳ η ∗ ρ ( R i ) 100 B ( P ) ⊂ 100 B ( R i ) 100B(P) \subset100B(R_i) 100 B ( P ) ⊂ 100 B ( R i ) R i R_i R i P P P
So, since cells in Stop i ′ ( R i ) \operatorname{Stop}'_i(R_i) Stop i ′ ( R i )
∫ B 0 ∑ P ∈ Stop i ′ ( R i ) ∑ Q ∈ T i , P ∣ T ε , Q ( g μ ) ∣ d μ ≲ {xref}‘(4.30)‘ ∫ B 0 ∑ P ∈ Stop i ′ ( R i ) Θ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ P ( x ) d μ ( x ) ≲ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (133) \begin{aligned}
\int_{B_0}\sum_{P\in\operatorname{Stop}'_i(R_i)}
\sum_{Q\in\mathcal{T}_{i,P}} |T_{\varepsilon,Q}(g\mu)|\,d\mu
&\mathrel{\overset{\text{\{xref\}`(4.30)`}}{\lesssim}}
\int_{B_0}\sum_{P\in\operatorname{Stop}'_i(R_i)}
\Theta_\mu(R_i)\lVert g\rVert_{L^\infty(\mu|_{B_0})}\chi_P(x)\,d\mu(x)\\
&\lesssim\Theta_\mu(R_i)\mu(R_i)\lVert g\rVert_{L^\infty(\mu|_{B_0})}. \tag*{(133)}
\end{aligned} ∫ B 0 P ∈ Stop i ′ ( R i ) ∑ Q ∈ T i , P ∑ ∣ T ε , Q ( g μ ) ∣ d μ ≲ {xref}‘(4.30)‘ ∫ B 0 P ∈ Stop i ′ ( R i ) ∑ Θ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ P ( x ) d μ ( x ) ≲ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (133) Thus, we gather (131) , (133) , the packing condition (111) , and obtain
∫ B 0 ∣ Σ i , ε ′ g ( x ) ∣ d μ ( x ) ≲ ∑ R i ∈ Top i ′ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ c ′ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . \int_{B_0} |\Sigma'_{i,\varepsilon}g(x)|\,d\mu(x)
\lesssim\sum_{R_i\in\operatorname{Top}'_i}\Theta_\mu(R_i)\mu(R_i)\lVert g\rVert_{L^\infty(\mu|_{B_0})}
\le c' \lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0). ∫ B 0 ∣ Σ i , ε ′ g ( x ) ∣ d μ ( x ) ≲ R i ∈ Top i ′ ∑ Θ μ ( R i ) μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ c ′ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . □ \square □
Lemma 4.11
Lemma 4.11. If g ∈ L ∞ ( μ ) g \in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0
∫ B 0 ∣ Σ 1 , ε ′ ′ g ( x ) ∣ d μ ( x ) + ∫ B 0 ∣ Σ 2 , ε ′ ′ g ( x ) ∣ d μ ( x ) ≤ c Σ ′ ′ C T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (134) \int_{B_0}|\Sigma''_{1,\varepsilon}g(x)|\,d\mu(x)+\int_{B_0}|\Sigma''_{2,\varepsilon}g(x)|\,d\mu(x)
\le c''_\Sigma C_T^{-1}\lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0), \tag*{(134)} ∫ B 0 ∣ Σ 1 , ε ′′ g ( x ) ∣ d μ ( x ) + ∫ B 0 ∣ Σ 2 , ε ′′ g ( x ) ∣ d μ ( x ) ≤ c Σ ′′ C T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (134) where c Σ ′ ′ > 0 c''_\Sigma>0 c Σ ′′ > 0 n n n C C C A 0 A_0 A 0 K 0 K_0 K 0 C 0 C_0 C 0
Proof. The bound (134) can be proved similarly to (133) . Indeed, let i = 1 , 2 i=1,2 i = 1 , 2 R i ∈ Top i ′ ′ R_i\in\operatorname{Top}''_i R i ∈ Top i ′′ P ∈ Stop i ′ ′ ( R i ) P\in\operatorname{Stop}''_i(R_i) P ∈ Stop i ′′ ( R i ) T i , P \mathcal{T}_{i,P} T i , P (132) , we can apply once again the summation argument in Lemma 4.9 and, if P ∈ Stop i ′ ′ ( R i ) P\in\operatorname{Stop}''_i(R_i) P ∈ Stop i ′′ ( R i ) (128) , for μ \mu μ x ∈ P x\in P x ∈ P
∑ Q ∈ T i , P x ∈ Q μ 0 ( 30 B ( Q ) ) ρ ( Q ) n ≲ μ ( 100 B ( P ) ) ρ ( P ) n ≲ C 0 . (135) \sum_{\substack{Q\in\mathcal{T}_{i,P}\\x\in Q}}
\frac{\mu_0(30B(Q))}{\rho(Q)^n}
\lesssim\frac{\mu(100B(P))}{\rho(P)^n}\lesssim C_0. \tag*{(135)} Q ∈ T i , P x ∈ Q ∑ ρ ( Q ) n μ 0 ( 30 B ( Q )) ≲ ρ ( P ) n μ ( 100 B ( P )) ≲ C 0 . (135) Since cells in Stop ′ ′ ( R i ) \operatorname{Stop}^{\prime\prime}(R_i) Stop ′′ ( R i ) R i R_i R i (135) and (123) we have
∫ B 0 ∑ P ∈ Stop i ′ ′ ( R i ) ∑ Q ∈ T i , P ∣ T ε , Q ( g μ ) ∣ d μ ≲ ∫ B 0 ∑ P ∈ Stop i ′ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ P ( x ) d μ ( x ) ≤ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (136) \int_{B_0}\sum_{P\in\operatorname{Stop}^{\prime\prime}_i(R_i)}\sum_{Q\in\mathcal{T}_{i,P}} |T_{\varepsilon,Q}(g\mu)|\,d\mu\lesssim\int_{B_0}\sum_{P\in\operatorname{Stop}^{\prime}_i(R_i)} \|g\|_{L^\infty(\mu|_{B_0})}\chi_P(x)\,d\mu(x)
\leq\mu(R_i)\|g\|_{L^\infty(\mu|_{B_0})}. \tag*{(136)} ∫ B 0 P ∈ Stop i ′′ ( R i ) ∑ Q ∈ T i , P ∑ ∣ T ε , Q ( g μ ) ∣ d μ ≲ ∫ B 0 P ∈ Stop i ′ ( R i ) ∑ ∥ g ∥ L ∞ ( μ ∣ B 0 ) χ P ( x ) d μ ( x ) ≤ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) . (136) Recall that, by Lemma 4.6 , Top ′ ′ \operatorname{Top}^{\prime\prime} Top ′′
∫ B 0 ∣ Σ ~ ε , g ( x ) ∣ d μ ( x ) ≲ ∑ R i ∈ Top ′ ′ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ c T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . \int_{B_0}|\widetilde{\Sigma}_{\varepsilon,g}(x)|\,d\mu(x)\lesssim\sum_{R_i\in\operatorname{Top}^{\prime\prime}}\mu(R_i)\|g\|_{L^\infty(\mu|_{B_0})}\leq c_{\mathrm{T}}^{-1}\|g\|_{L^\infty(\mu|_{B_0})}\mu(B_0). ∫ B 0 ∣ Σ ε , g ( x ) ∣ d μ ( x ) ≲ R i ∈ Top ′′ ∑ μ ( R i ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ c T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . We now estimate Σ ~ ε \widetilde{\Sigma}_{\varepsilon} Σ ε (139) . Let R i ∈ Top ′ ′ R_i\in\operatorname{Top}^{\prime\prime} R i ∈ Top ′′ i = 1 , 2 i=1,2 i = 1 , 2 Tree ( R 1 , R 2 ) ≠ ∅ \operatorname{Tree}(R_1,R_2)\ne\varnothing Tree ( R 1 , R 2 ) = ∅ R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 R 2 ⊂ R 1 R_2\subset R_1 R 2 ⊂ R 1
Φ ~ R 1 , R 2 ( x ) : = max ( Φ 1 , R 1 ( x ) , Φ 2 , R 2 ( x ) ) . (137) \widetilde{\Phi}_{R_1,R_2}(x):=\max\bigl(\Phi_{1,R_1}(x),\Phi_{2,R_2}(x)\bigr). \tag*{(137)} Φ R 1 , R 2 ( x ) := max ( Φ 1 , R 1 ( x ) , Φ 2 , R 2 ( x ) ) . (137) With yet another minor abuse of notation, for R ∈ Top ~ R\in\widetilde{\operatorname{Top}} R ∈ Top
Φ ~ R ( x ) : = Φ ~ R 1 , R 2 ( x ) , (138) \widetilde{\Phi}_R(x):=\widetilde{\Phi}_{R_1,R_2}(x), \tag*{(138)} Φ R ( x ) := Φ R 1 , R 2 ( x ) , (138) where we recall that R R R R 1 R_1 R 1 R 2 R_2 R 2 Φ ~ R \widetilde{\Phi}_R Φ R 1 1 1
Lemma 4.12. Let ε > 0 \varepsilon>0 ε > 0
sup R ∈ Top ~ ∥ T Φ ~ R , μ ∣ 84 B ( R ) ∥ L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≤ C T . (139) \sup_{R\in\widetilde{\operatorname{Top}}}\|T_{\widetilde{\Phi}_R,\mu|_{84B(R)}}\|_{L^2(\mu|_{84B(R)})\to L^2(\mu|_{84B(R)})}\leq\mathcal{C}_{\mathrm{T}}. \tag*{(139)} R ∈ Top sup ∥ T Φ R , μ ∣ 84 B ( R ) ∥ L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≤ C T . (139) Then, for every g ∈ L ∞ ( μ ) g\in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0
∫ B 0 ∣ Σ ~ ε g ( x ) ∣ d μ ( x ) ≤ c Σ ′ ′ ′ c T − 1 ( 1 + C T ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (140) \int_{B_0}|\widetilde{\Sigma}_{\varepsilon}g(x)|\,d\mu(x)\leq c_{\Sigma}^{\prime\prime\prime}c_{\mathrm{T}}^{-1}(1+\mathcal{C}_{\mathrm{T}})\|g\|_{L^\infty(\mu|_{B_0})}\mu(B_0), \tag*{(140)} ∫ B 0 ∣ Σ ε g ( x ) ∣ d μ ( x ) ≤ c Σ ′′′ c T − 1 ( 1 + C T ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (140) where c Σ ′ ′ ′ > 0 c_{\Sigma}^{\prime\prime\prime}>0 c Σ ′′′ > 0 n n n C C C γ \gamma γ C 0 C_0 C 0 A 0 A_0 A 0 K 0 K_0 K 0
Proof. We fix R ∈ Top ~ R\in\widetilde{\operatorname{Top}} R ∈ Top Tree ( R 1 , R 2 ) \operatorname{Tree}(R_1,R_2) Tree ( R 1 , R 2 ) x ∈ R x\in R x ∈ R
∣ K ~ ε , R ( g μ ) ( x ) ∣ ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) + ∣ T Φ ~ R , ε ( χ 84 B ( R ) g μ ) ( x ) ∣ . (141) |\widetilde{K}_{\varepsilon,R}(g\mu)(x)|\lesssim\|g\|_{L^\infty(\mu|_{B_0})}+|T_{\widetilde{\Phi}_R,\varepsilon}(\chi_{84B(R)}g\mu)(x)|. \tag*{(141)} ∣ K ε , R ( g μ ) ( x ) ∣ ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) + ∣ T Φ R , ε ( χ 84 B ( R ) g μ ) ( x ) ∣. (141) We denote by Stop ~ ( R ) \widetilde{\operatorname{Stop}}(R) Stop ( R )
{ Q ∈ Stop 1 ′ ′ ( R 1 ) ∪ Stop 2 ′ ′ ( R 2 ) : Q ⊂ R } , \{Q\in\operatorname{Stop}^{\prime\prime}_1(R_1)\cup\operatorname{Stop}^{\prime\prime}_2(R_2):Q\subset R\}, { Q ∈ Stop 1 ′′ ( R 1 ) ∪ Stop 2 ′′ ( R 2 ) : Q ⊂ R } , which are proper subcells of R R R R ∈ Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 ) R\in\operatorname{Tree}_1(R_1)\cap\operatorname{Tree}_2(R_2) R ∈ Tree 1 ( R 1 ) ∩ Tree 2 ( R 2 ) i = 1 , 2 i=1,2 i = 1 , 2 x ∈ R x\in R x ∈ R Φ i , R i ( x ) > 0 \Phi_{i,R_i}(x)>0 Φ i , R i ( x ) > 0 Stop i ′ ′ ( R i ) \operatorname{Stop}^{\prime\prime}_i(R_i) Stop i ′′ ( R i ) R R R R ∈ Tree i ( R i ) R\in\operatorname{Tree}_i(R_i) R ∈ Tree i ( R i ) Stop ~ ( R ) \widetilde{\operatorname{Stop}}(R) Stop ( R )
Let x ∈ R x\in R x ∈ R x x x Q ∈ Stop ~ ( R ) Q\in\widetilde{\operatorname{Stop}}(R) Q ∈ Stop ( R ) Q Q Q Tree ~ ( R ) \widetilde{\operatorname{Tree}}(R) Tree ( R ) x x x P P P Q ⊊ P ⊂ R Q\subsetneq P\subset R Q ⊊ P ⊂ R T ε , P T_{\varepsilon,P} T ε , P
K ~ ε , R ( g μ ) ( x ) = ∫ max ( ε , a ( x ) ) < d ( x , y ) ≤ 2 ρ ( R ) K ( x , y ) g ( y ) d μ ( y ) , where a ( x ) : = 2 ρ ( Q ) . (142) \widetilde{K}_{\varepsilon,R}(g\mu)(x)=\int_{\max(\varepsilon,a(x))<d(x,y)\leq2\rho(R)}K(x,y)g(y)\,d\mu(y),\qquad\text{where }a(x):=2\rho(Q). \tag*{(142)} K ε , R ( g μ ) ( x ) = ∫ m a x ( ε , a ( x )) < d ( x , y ) ≤ 2 ρ ( R ) K ( x , y ) g ( y ) d μ ( y ) , where a ( x ) := 2 ρ ( Q ) . (142) Moreover, if Q ∈ Stop ′ ′ ( R 1 ) Q \in\operatorname{Stop}^{\prime\prime}(R_1) Q ∈ Stop ′′ ( R 1 ) (102) gives Φ 1 , R 1 ( x ) > 480 A 0 ℓ ( Q ) \Phi_{1,R_1}(x)>480A_0\ell(Q) Φ 1 , R 1 ( x ) > 480 A 0 ℓ ( Q ) Φ 2 , R 2 ( x ) > 0 \Phi_{2,R_2}(x)>0 Φ 2 , R 2 ( x ) > 0 x x x S ∈ Stop ′ ′ ( R 2 ) S\in\operatorname{Stop}^{\prime\prime}(R_2) S ∈ Stop ′′ ( R 2 ) S ⊂ Q S\subset Q S ⊂ Q Q Q Q (103) , applied to Q Q Q S S S Q ∈ Stop ′ ′ ( R 2 ) Q\in\operatorname{Stop}^{\prime\prime}(R_2) Q ∈ Stop ′′ ( R 2 )
480 A 0 ℓ ( Q ) < Φ ~ R ( x ) ≤ 481 A 0 2 ℓ ( Q ) . (143) 480A_0\ell(Q)<\widetilde{\Phi}_R(x)\le481A_0^2\ell(Q). \tag*{(143)} 480 A 0 ℓ ( Q ) < Φ R ( x ) ≤ 481 A 0 2 ℓ ( Q ) . (143) If x x x Stop ~ ( R ) \widetilde{\operatorname{Stop}}(R) Stop ( R ) Φ ~ R ( x ) = 0 \widetilde{\Phi}_R(x)=0 Φ R ( x ) = 0 D 0 ( R ) \mathcal{D}_0(R) D 0 ( R ) x x x Tree ~ ( R ) \widetilde{\operatorname{Tree}}(R) Tree ( R ) (142) holds with a ( x ) : = 0 a(x):=0 a ( x ) := 0
In both cases, we set ε x : = max ( ε , Φ ~ R ( x ) ) \varepsilon_x:=\max(\varepsilon,\widetilde{\Phi}_R(x)) ε x := max ( ε , Φ R ( x ))
K ~ ε , R ( g μ ) ( x ) = T Φ ~ R , ε x ( χ 84 B ( R ) g μ ) ( x ) + ( T ε x − T Φ ~ R , ε x ) ( χ 84 B ( R ) g μ ) ( x ) \widetilde{K}_{\varepsilon,R}(g\mu)(x)=T_{\widetilde{\Phi}_R,\varepsilon_x}(\chi_{84B(R)}g\mu)(x)+(T_{\varepsilon_x}-T_{\widetilde{\Phi}_R,\varepsilon_x})(\chi_{84B(R)}g\mu)(x) K ε , R ( g μ ) ( x ) = T Φ R , ε x ( χ 84 B ( R ) g μ ) ( x ) + ( T ε x − T Φ R , ε x ) ( χ 84 B ( R ) g μ ) ( x ) + ( K ~ ε , R ( g μ ) ( x ) − T ε x ( χ 84 B ( R ) g μ ) ( x ) ) = : J ~ 1 + J ~ 2 + J ~ 3 . \qquad+\bigl(\widetilde{K}_{\varepsilon,R}(g\mu)(x)-T_{\varepsilon_x}(\chi_{84B(R)}g\mu)(x)\bigr)=:\widetilde{J}_1+\widetilde{J}_2+\widetilde{J}_3. + ( K ε , R ( g μ ) ( x ) − T ε x ( χ 84 B ( R ) g μ ) ( x ) ) =: J 1 + J 2 + J 3 . If ε x = ε \varepsilon_x=\varepsilon ε x = ε J ~ 1 = T Φ ~ R , ε ( χ 84 B ( R ) g μ ) ( x ) \widetilde{J}_1=T_{\widetilde{\Phi}_R,\varepsilon}(\chi_{84B(R)}g\mu)(x) J 1 = T Φ R , ε ( χ 84 B ( R ) g μ ) ( x ) Φ ~ R ( x ) > ε \widetilde{\Phi}_R(x)>\varepsilon Φ R ( x ) > ε (70) and the n n n μ \mu μ
∣ J ~ 1 − T Φ ~ R , ε ( χ 84 B ( R ) g μ ) ( x ) ∣ ≤ ∫ ε < d ( x , y ) ≤ Φ ~ R ( x ) ∣ K Φ ~ R ( x , y ) ∣ ∣ g ( y ) ∣ d μ ( y ) ≤ C μ ( B ( x , 2 Φ ~ R ( x ) ) ) Φ ~ R ( x ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ C C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) . \begin{aligned}
\left|\widetilde{J}_1-T_{\widetilde{\Phi}_R,\varepsilon}(\chi_{84B(R)}g\mu)(x)\right|
\le\int_{\varepsilon<d(x,y)\le\widetilde{\Phi}_R(x)}\left|K_{\widetilde{\Phi}_R}(x,y)\right||g(y)|\,d\mu(y) \\
&\le C\frac{\mu(B(x,2\widetilde{\Phi}_R(x)))}{\widetilde{\Phi}_R(x)}\|g\|_{L^\infty(\mu|_{B_0})}
\le C\mathcal{C}_0\|g\|_{L^\infty(\mu|_{B_0})}.
\end{aligned} J 1 − T Φ R , ε ( χ 84 B ( R ) g μ ) ( x ) ≤ ∫ ε < d ( x , y ) ≤ Φ R ( x ) K Φ R ( x , y ) ∣ g ( y ) ∣ d μ ( y ) ≤ C Φ R ( x ) μ ( B ( x , 2 Φ R ( x ))) ∥ g ∥ L ∞ ( μ ∣ B 0 ) ≤ C C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) . If Φ ~ R ( x ) > 0 \widetilde{\Phi}_R(x)>0 Φ R ( x ) > 0 (71) and the n n n μ \mu μ ∣ J ~ 2 ∣ ≲ C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) |\widetilde{J}_2|\lesssim\mathcal{C}_0\|g\|_{L^\infty(\mu|_{B_0})} ∣ J 2 ∣ ≲ C 0 ∥ g ∥ L ∞ ( μ ∣ B 0 ) J ~ 2 = 0 \widetilde{J}_2=0 J 2 = 0 Φ ~ R ( x ) = 0 \widetilde{\Phi}_R(x)=0 Φ R ( x ) = 0 K Φ ~ R ( x , ⋅ ) = K ( x , ⋅ ) K_{\widetilde{\Phi}_R}(x,\cdot)=K(x,\cdot) K Φ R ( x , ⋅ ) = K ( x , ⋅ ) J ~ 3 \widetilde{J}_3 J 3 x ∈ R ⊂ 28 B ( R ) x\in R\subset28B(R) x ∈ R ⊂ 28 B ( R ) B ( x , 2 ρ ( R ) ) ⊂ 84 B ( R ) ⊂ B ( x , 112 r ( R ) ) B(x,2\rho(R))\subset84B(R)\subset B(x,112r(R)) B ( x , 2 ρ ( R )) ⊂ 84 B ( R ) ⊂ B ( x , 112 r ( R )) (142) , J ~ 3 \widetilde{J}_3 J 3 K ( x , ⋅ ) g K(x,\cdot)g K ( x , ⋅ ) g
{ y : a ( x ) < d ( x , y ) ≤ Φ ~ R ( x ) } ∪ { y : 2 ρ ( R ) < d ( x , y ) < 112 r ( R ) } . \{y:a(x)<d(x,y)\le\widetilde{\Phi}_R(x)\}\cup\{y:2\rho(R)<d(x,y)<112r(R)\}. { y : a ( x ) < d ( x , y ) ≤ Φ R ( x )} ∪ { y : 2 ρ ( R ) < d ( x , y ) < 112 r ( R )} . By (143) and (120) , the ratio between the outer and the inner radius of each of these annuli is bounded by a constant depending only on A 0 A_0 A 0 K 0 K_0 K 0 K K K n n n μ \mu μ ∣ J ~ 3 ∣ ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) |\widetilde{J}_3|\lesssim\|g\|_{L^\infty(\mu|_{B_0})} ∣ J 3 ∣ ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) (141) .
Since K ~ ε , R ( g μ ) \widetilde{K}_{\varepsilon,R}(g\mu) K ε , R ( g μ ) R R R (141) gives
∫ B 0 ∣ Σ ~ ε g ∣ d μ ≤ ∑ R ∈ Top ∫ R ∣ K ~ ε , R ( g μ ) ∣ d μ \int_{B_0}|\widetilde{\Sigma}_{\varepsilon}g|\,d\mu
\le\sum_{R\in\operatorname{Top}}\int_R|\widetilde{K}_{\varepsilon,R}(g\mu)|\,d\mu ∫ B 0 ∣ Σ ε g ∣ d μ ≤ R ∈ Top ∑ ∫ R ∣ K ε , R ( g μ ) ∣ d μ ≲ ∑ R ∈ Top ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( R ) + ∑ R ∈ Top ∫ R ∣ T Φ ~ R , ε ( χ 84 B ( R ) g μ ) ∣ d μ = : J ~ 1 + J ~ 2 . (144) \lesssim\sum_{R\in\operatorname{Top}}\|g\|_{L^\infty(\mu|_{B_0})}\mu(R)
+\sum_{R\in\operatorname{Top}}\int_R|T_{\widetilde{\Phi}_R,\varepsilon}(\chi_{84B(R)}g\mu)|\,d\mu
=:\widetilde{J}_1+\widetilde{J}_2. \tag*{(144)} ≲ R ∈ Top ∑ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( R ) + R ∈ Top ∑ ∫ R ∣ T Φ R , ε ( χ 84 B ( R ) g μ ) ∣ d μ =: J 1 + J 2 . (144) To bound J ~ 1 \widetilde{J}_1 J 1
J ~ 1 ≤ 2 C T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . (145) \widetilde{J}_1\le2\mathcal{C}_T^{-1}\|g\|_{L^\infty(\mu|_{B_0})}\mu(B_0). \tag*{(145)} J 1 ≤ 2 C T − 1 ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . (145) To estimate J ~ 2 \widetilde{J}_2 J 2 R ∈ Top R\in\operatorname{Top} R ∈ Top σ : = μ ∣ 84 B ( R ) \sigma:=\mu|_{84B(R)} σ := μ ∣ 84 B ( R ) R R R
μ ( 84 B ( R ) ) ≤ μ ( 100 B ( R ) ) ≤ K 0 μ ( B ( R ) ) ≤ K 0 μ ( R ) , \mu(84B(R))\le\mu(100B(R))\le K_0\mu(B(R))\le K_0\mu(R), μ ( 84 B ( R )) ≤ μ ( 100 B ( R )) ≤ K 0 μ ( B ( R )) ≤ K 0 μ ( R ) , so the Cauchy–Schwarz inequality and (139) give
∫ R ∣ T Φ ~ R , ε ( χ 84 B ( R ) g μ ) ∣ d μ ≤ μ ( R ) 1 / 2 ∥ T Φ ~ R , σ , ε g ∥ L 2 ( σ ) ≤ C T μ ( R ) 1 / 2 ∥ g ∥ L 2 ( σ ) ≤ K 0 1 / 2 C T ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( R ) . \begin{aligned}
\int_R|T_{\widetilde{\Phi}_R,\varepsilon}(\chi_{84B(R)}g\mu)|\,d\mu
\le\mu(R)^{1/2}\|T_{\widetilde{\Phi}_R,\sigma,\varepsilon}g\|_{L^2(\sigma)}
\le\mathcal{C}_T\mu(R)^{1/2}\|g\|_{L^2(\sigma)} \\
&\le K_0^{1/2}\mathcal{C}_T\|g\|_{L^\infty(\mu|_{B_0})}\mu(R).
\end{aligned} ∫ R ∣ T Φ R , ε ( χ 84 B ( R ) g μ ) ∣ d μ ≤ μ ( R ) 1/2 ∥ T Φ R , σ , ε g ∥ L 2 ( σ ) ≤ C T μ ( R ) 1/2 ∥ g ∥ L 2 ( σ ) ≤ K 0 1/2 C T ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( R ) . Summing over R ∈ Top ~ R \in\widetilde{\operatorname{Top}} R ∈ Top
J ~ 2 ≤ 2 K 0 1 / 2 c T − 1 E T ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , \widetilde{J}_2 \leq2K_0^{1/2}c_T^{-1}\mathcal{E}_T\lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0), J 2 ≤ 2 K 0 1/2 c T − 1 E T ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , which, together with (144) and (145) , concludes the proof of the lemma. □
Finally, gathering all the estimates of this section, we obtain the following result.
Corollary 4.13. Let B 0 B_0 B 0 C 3 C_3 C 3 μ \mu μ (94) . Assume that there exists E T > 0 \mathcal{E}_T>0 E T > 0 ε \varepsilon ε
sup R ∈ Top ( B 0 ) ∥ T Φ ~ R , μ ∣ 84 B ( R ) , ε ∥ L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≤ E T . (146) \sup_{R\in\operatorname{Top}(B_0)}
\left\lVert T_{\widetilde{\Phi}_{R,\mu|_{84B(R)},\varepsilon}} \right\rVert_{L^2(\mu|_{84B(R)})\to L^2(\mu|_{84B(R)})}\leq\mathcal{E}_T. \tag*{(146)} R ∈ Top ( B 0 ) sup T Φ R , μ ∣ 84 B ( R ) , ε L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≤ E T . (146) Then for every g ∈ L ∞ ( μ ) g\in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0 ε > 0 \varepsilon>0 ε > 0
∥ T μ , ε g ∥ L 1 ( μ ∣ B 0 ) + ∥ T μ , ε ∗ g ∥ L 1 ( μ ∣ B 0 ) ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (147) \lVert T_{\mu,\varepsilon}g\rVert_{L^1(\mu|_{B_0})}+\lVert T_{\mu,\varepsilon}^*g\rVert_{L^1(\mu|_{B_0})}\lesssim\lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0), \tag*{(147)} ∥ T μ , ε g ∥ L 1 ( μ ∣ B 0 ) + ∥ T μ , ε ∗ g ∥ L 1 ( μ ∣ B 0 ) ≲ ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) , (147) where the implicit constant depends on A 0 A_0 A 0 K 0 K_0 K 0 n n n C C C γ \gamma γ C 0 \mathcal{C}_0 C 0 C 3 \mathcal{C}_3 C 3 β \beta β η ∗ \eta_* η ∗ c T c_T c T c ′ c' c ′ E T \mathcal{E}_T E T ε \varepsilon ε
Proof. By (127) , Lemma 4.9, Lemma 4.10, Lemma 4.11, and Lemma 4.12, for every g ∈ L ∞ ( μ ) g\in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0
∫ B 0 ∣ T μ , ε g ∣ d μ ≤ ( c Σ 0 + c Σ ′ + c Σ ′ ′ c T − 1 + c Σ ′ ′ c T − 1 ( 1 + E T ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . \int_{B_0}|T_{\mu,\varepsilon}g|\,d\mu\leq
\left(c_{\Sigma}^{0}+c_{\Sigma}'+c_{\Sigma}''c_T^{-1}+c_{\Sigma}''c_T^{-1}(1+\mathcal{E}_T)\right)\lVert g\rVert_{L^\infty(\mu|_{B_0})}\mu(B_0). ∫ B 0 ∣ T μ , ε g ∣ d μ ≤ ( c Σ 0 + c Σ ′ + c Σ ′′ c T − 1 + c Σ ′′ c T − 1 ( 1 + E T ) ) ∥ g ∥ L ∞ ( μ ∣ B 0 ) μ ( B 0 ) . The same arguments apply to T ε ∗ T_\varepsilon^* T ε ∗ L 2 L^2 L 2
Bounds of different norms for the suppressed operator
Let R 1 ∈ Top 1 ′ ′ R_1\in\operatorname{Top}_1'' R 1 ∈ Top 1 ′′ R 2 ∈ Top 2 ′ ′ R_2\in\operatorname{Top}_2'' R 2 ∈ Top 2 ′′ Tree ~ ( R 1 , R 2 ) ≠ ∅ \widetilde{\operatorname{Tree}}(R_1,R_2)\neq\varnothing Tree ( R 1 , R 2 ) = ∅ R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 R 2 ⊂ R 1 R_2\subset R_1 R 2 ⊂ R 1
Let i = 1 , 2 i=1,2 i = 1 , 2
The testing ball B i = B ~ ( R i ) B_i=\widetilde{B}(R_i) B i = B ( R i )
The testing measures ν ^ 1 \widehat{\nu}_1 ν 1 T T T B 1 B_1 B 1 ν ^ 2 \widehat{\nu}_2 ν 2 T ∗ T^* T ∗ B 2 B_2 B 2
The measure ν i = ν ^ i + μ ∣ X ∖ B i \nu_i=\widehat{\nu}_i+\mu|_{X\setminus B_i} ν i = ν i + μ ∣ X ∖ B i
The sets F i ( w i ) ⊂ ( 1 − θ ) B i F_i(w_i)\subset(1-\theta)B_i F i ( w i ) ⊂ ( 1 − θ ) B i w i ∈ Ω w_i\in\Omega w i ∈ Ω
The suppressions Φ i , R i , w i \Phi_{i,R_i,w_i} Φ i , R i , w i Φ i , R i \Phi_{i,R_i} Φ i , R i (96) and (99) , respectively.
We set μ i : = μ ∣ B i \mu_i:=\mu|_{B_i} μ i := μ ∣ B i ∣ ν i ∣ = ∣ ν ^ i ∣ + μ ∣ X ∖ B i |\nu_i|=|\widehat{\nu}_i|+\mu|_{X\setminus B_i} ∣ ν i ∣ = ∣ ν i ∣ + μ ∣ X ∖ B i ∣ ν ^ i ∣ ( X ∖ B i ) = 0 |\widehat{\nu}_i|(X\setminus B_i)=0 ∣ ν i ∣ ( X ∖ B i ) = 0 w = ( w 1 , w 2 ) ∈ Ω 2 w=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2
Ξ i ( x ) : = Ξ i , R i , w i ( x ) : = max { Φ i , R i ( x ) , Φ i , R i , w i ( x ) } , i = 1 , 2 , (148) \Xi_i(x):=\Xi_{i,R_i,w_i}(x):=\max\{\Phi_{i,R_i}(x),\Phi_{i,R_i,w_i}(x)\},\qquad i=1,2, \tag*{(148)} Ξ i ( x ) := Ξ i , R i , w i ( x ) := max { Φ i , R i ( x ) , Φ i , R i , w i ( x )} , i = 1 , 2 , (148) and set
Ξ w ( x ) : = max { Ξ 1 , R 1 , w 1 ( x ) , Ξ 2 , R 2 , w 2 ( x ) } . (149) \Xi_w(x):=\max\{\Xi_{1,R_1,w_1}(x),\Xi_{2,R_2,w_2}(x)\}. \tag*{(149)} Ξ w ( x ) := max { Ξ 1 , R 1 , w 1 ( x ) , Ξ 2 , R 2 , w 2 ( x )} . (149) These functions are non-negative and 1-Lipschitz. In this section, the implicit constants depend only on n n n D D D γ \gamma γ ε 0 \varepsilon_0 ε 0 C C C C 0 \mathcal{C}_0 C 0 C 1 \mathcal{C}_1 C 1 C 3 \mathcal{C}_3 C 3 β \beta β R 1 R_1 R 1 R 2 R_2 R 2 w w w ε > 0 \varepsilon>0 ε > 0
Growth and comparison of the measures
Let i = 1 , 2 i=1,2 i = 1 , 2 r 1 , i r_{1,i} r 1 , i r 2 , i r_{2,i} r 2 , i (44) , and by R ~ i \widetilde{R}_i R i (52) , all associated with ν ^ i \widehat{\nu}_i ν i B i B_i B i (58) and (61) , the set F i ( w i ) ⊂ G i F_i(w_i)\subset G_i F i ( w i ) ⊂ G i S i , R ~ i ∪ H i , R ~ i S_{i,\widetilde{R}_i}\cup H_{i,\widetilde{R}_i} S i , R i ∪ H i , R i B ( x , r j , i ( x ) ) B(x,r_{j,i}(x)) B ( x , r j , i ( x )) x ∈ B i x \in B_i x ∈ B i j = 1 , 2 j=1,2 j = 1 , 2 B ( x , R i ( x ) ) B(x,\mathcal{R}_i(x)) B ( x , R i ( x )) x ∈ X x \in X x ∈ X (96) and (148) , we have
r j , i ( x ) ≤ dist ( x , F i ( w i ) ) ≤ Ξ i ( x ) ≤ Ξ w ( x ) , x ∈ B i , j = 1 , 2 , (150) r_{j,i}(x) \le\operatorname{dist}(x,F_i(w_i)) \le\Xi_i(x) \le\Xi_w(x), \qquad x \in B_i,\quad j=1,2, \tag*{(150)} r j , i ( x ) ≤ dist ( x , F i ( w i )) ≤ Ξ i ( x ) ≤ Ξ w ( x ) , x ∈ B i , j = 1 , 2 , (150) and R i ( x ) ≤ dist ( x , F i ( w i ) ) ≤ Ξ w ( x ) \mathcal{R}_i(x) \le\operatorname{dist}(x,F_i(w_i)) \le\Xi_w(x) R i ( x ) ≤ dist ( x , F i ( w i )) ≤ Ξ w ( x ) x ∈ X x \in X x ∈ X x ∈ B i x \in B_i x ∈ B i
∣ ν ^ i ∣ ( B ( x , t ) ) ≤ m 1 μ ( B ( x , 5 t ) ) , for t > r 1 , i ( x ) , (151) |\widehat{\nu}_i|(B(x,t)) \le m_1\mu(B(x,5t)), \qquad\text{for }t>r_{1,i}(x), \tag*{(151)} ∣ ν i ∣ ( B ( x , t )) ≤ m 1 μ ( B ( x , 5 t )) , for t > r 1 , i ( x ) , (151) μ i ( B ( x , t ) ) ≤ m 2 ∣ ν ^ i ∣ ( B ( x , 5 t ) ) , for t > r 2 , i ( x ) , (152) \mu_i(B(x,t)) \le m_2|\widehat{\nu}_i|(B(x,5t)), \qquad\text{for }t>r_{2,i}(x), \tag*{(152)} μ i ( B ( x , t )) ≤ m 2 ∣ ν i ∣ ( B ( x , 5 t )) , for t > r 2 , i ( x ) , (152) and, for x ∈ X x \in X x ∈ X t ≥ Ξ w ( x ) t \ge\Xi_w(x) t ≥ Ξ w ( x )
∣ ν ^ i ∣ ( B ( x , t ) ) ≤ m t n . (153) |\widehat{\nu}_i|(B(x,t)) \le\mathbf{m}t^n. \tag*{(153)} ∣ ν i ∣ ( B ( x , t )) ≤ m t n . (153) If x ∈ X x \in X x ∈ X t > 0 t>0 t > 0 t + Ξ w ( x ) > Ξ w ( x ) t+\Xi_w(x)>\Xi_w(x) t + Ξ w ( x ) > Ξ w ( x ) (153) and the n n n μ \mu μ m ∈ { μ , μ i , ∣ ν ^ i ∣ , ∣ ν i ∣ } \mathbf{m}\in\{\mu,\mu_i,|\widehat{\nu}_i|,|\nu_i|\} m ∈ { μ , μ i , ∣ ν i ∣ , ∣ ν i ∣ }
m ( B ( x , t ) ) ≤ ( m + C 0 ) ( t + Ξ w ( x ) ) n , x ∈ X , t > 0. (154) \mathbf{m}(B(x,t)) \le(\mathbf{m}+\mathcal{C}_0)(t+\Xi_w(x))^n,\qquad x\in X,t>0. \tag*{(154)} m ( B ( x , t )) ≤ ( m + C 0 ) ( t + Ξ w ( x ) ) n , x ∈ X , t > 0. (154) The measures μ \mu μ ∣ ν 1 ∣ |\nu_1| ∣ ν 1 ∣ ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ m , m ′ ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣ } \mathbf{m},\mathbf{m}'\in\{\mu,|\nu_1|,|\nu_2|\} m , m ′ ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣ } x ∈ X x\in X x ∈ X t > 0 t>0 t > 0 t ≥ Ξ w ( x ) t\ge\Xi_w(x) t ≥ Ξ w ( x )
m ( B ( x , t ) ) ≤ ( 1 + m 1 ) ( 1 + m 2 ) m ′ ( B ( x , 121 t ) ) . (155) \mathbf{m}(B(x,t))\le(1+m_1)(1+m_2)\mathbf{m}'(B(x,121t)). \tag*{(155)} m ( B ( x , t )) ≤ ( 1 + m 1 ) ( 1 + m 2 ) m ′ ( B ( x , 121 t )) . (155) To prove (155) , let x ∈ X x\in X x ∈ X t > 0 t>0 t > 0 t ≥ Ξ w ( x ) t\ge\Xi_w(x) t ≥ Ξ w ( x )
∣ ν ^ i ∣ ( B ( x , t ) ) ≤ m 1 μ ( B ( x , 11 t ) ) and μ i ( B ( x , t ) ) ≤ m 2 ∣ ν ^ i ∣ ( B ( x , 11 t ) ) . |\widehat{\nu}_i|(B(x,t))\le m_1\mu(B(x,11t))\qquad\text{and}\qquad\mu_i(B(x,t))\le m_2|\widehat{\nu}_i|(B(x,11t)). ∣ ν i ∣ ( B ( x , t )) ≤ m 1 μ ( B ( x , 11 t )) and μ i ( B ( x , t )) ≤ m 2 ∣ ν i ∣ ( B ( x , 11 t )) . If B ( x , t ) ∩ B i = ∅ B(x,t)\cap B_i=\varnothing B ( x , t ) ∩ B i = ∅ z ∈ B ( x , t ) ∩ B i z\in B(x,t)\cap B_i z ∈ B ( x , t ) ∩ B i Ξ w \Xi_w Ξ w (150) gives r j , i ( z ) ≤ Ξ w ( z ) < 2 t r_{j,i}(z)\le\Xi_w(z)<2t r j , i ( z ) ≤ Ξ w ( z ) < 2 t j = 1 , 2 j=1,2 j = 1 , 2 B ( x , t ) ⊂ B ( z , 2 t ) B(x,t)\subset B(z,2t) B ( x , t ) ⊂ B ( z , 2 t ) B ( z , 10 t ) ⊂ B ( x , 11 t ) B(z,10t)\subset B(x,11t) B ( z , 10 t ) ⊂ B ( x , 11 t ) (151) and (152) , applied with z z z 2 t 2t 2 t x x x t t t
μ ( B ( x , t ) ∖ B i ) ≤ min { μ ( B ( x , 11 t ) ) , ∣ ν i ∣ ( B ( x , 11 t ) ) } , \mu(B(x,t)\setminus B_i)\le\min\{\mu(B(x,11t)),|\nu_i|(B(x,11t))\}, μ ( B ( x , t ) ∖ B i ) ≤ min { μ ( B ( x , 11 t )) , ∣ ν i ∣ ( B ( x , 11 t ))} , we deduce that
∣ ν i ∣ ( B ( x , t ) ) ≤ ( 1 + m 1 ) μ ( B ( x , 11 t ) ) and μ ( B ( x , t ) ) ≤ ( 1 + m 2 ) ∣ ν i ∣ ( B ( x , 11 t ) ) . |\nu_i|(B(x,t))\le(1+m_1)\mu(B(x,11t))\qquad\text{and}\qquad\mu(B(x,t))\le(1+m_2)|\nu_i|(B(x,11t)). ∣ ν i ∣ ( B ( x , t )) ≤ ( 1 + m 1 ) μ ( B ( x , 11 t )) and μ ( B ( x , t )) ≤ ( 1 + m 2 ) ∣ ν i ∣ ( B ( x , 11 t )) . These inequalities yield (155) for m = ∣ ν i ∣ \mathbf{m}=|\nu_i| m = ∣ ν i ∣ m ′ ∈ { ∣ ν 1 ∣ , ∣ ν 2 ∣ } \mathbf{m}'\in\{|\nu_1|,|\nu_2|\} m ′ ∈ { ∣ ν 1 ∣ , ∣ ν 2 ∣ }
The localized operator
Let i = 1 , 2 i=1,2 i = 1 , 2 z B i z_{B_i} z B i B i B_i B i
χ i ( x ) : = min { 1 , max { 0 , 2 ( r ( B i ) − d ( x , z B i ) ) θ r ( B i ) } } . \chi_i(x):=\min\left\{1,\max\left\{0,\frac{2(r(B_i)-d(x,z_{B_i}))}{\theta r(B_i)}\right\}\right\}. χ i ( x ) := min { 1 , max { 0 , θ r ( B i ) 2 ( r ( B i ) − d ( x , z B i )) } } . Then χ i = 1 \chi_i=1 χ i = 1 ( 1 − θ / 2 ) B i (1-\theta/2)B_i ( 1 − θ /2 ) B i χ i = 0 \chi_i=0 χ i = 0 X ∖ B i X\setminus B_i X ∖ B i
∣ χ i ( x ) − χ i ( x ′ ) ∣ ≤ 2 d ( x , x ′ ) θ r ( B i ) . |\chi_i(x)-\chi_i(x')|\le2\frac{d(x,x')}{\theta r(B_i)}. ∣ χ i ( x ) − χ i ( x ′ ) ∣ ≤ 2 θ r ( B i ) d ( x , x ′ ) . Moreover, if χ i ( x ) ≠ 1 \chi_i(x)\ne1 χ i ( x ) = 1 d ( x , z B i ) > ( 1 − θ / 2 ) r ( B i ) d(x,z_{B_i})>(1-\theta/2)r(B_i) d ( x , z B i ) > ( 1 − θ /2 ) r ( B i ) F i ( w i ) ⊂ ( 1 − θ ) B i F_i(w_i)\subset(1-\theta)B_i F i ( w i ) ⊂ ( 1 − θ ) B i
Ξ w ( x ) ≥ dist ( x , F i ( w i ) ) ≥ d ( x , z B i ) − ( 1 − θ ) r ( B i ) ≥ θ 2 r ( B i ) . (156) \Xi_w(x)\ge\operatorname{dist}(x,F_i(w_i))\ge d(x,z_{B_i})-(1-\theta)r(B_i)\ge\frac{\theta}{2}r(B_i). \tag*{(156)} Ξ w ( x ) ≥ dist ( x , F i ( w i )) ≥ d ( x , z B i ) − ( 1 − θ ) r ( B i ) ≥ 2 θ r ( B i ) . (156) Set r 0 : = min { r ( B 1 ) , r ( B 2 ) } r_0 := \min\{r(B_1),r(B_2)\} r 0 := min { r ( B 1 ) , r ( B 2 )} φ \varphi φ ψ \psi ψ § 3.1 . For ε > 0 \varepsilon> 0 ε > 0
L ε ( x , y ) : = χ 2 ( x ) χ 1 ( y ) ψ ( d ( x , y ) r 0 ) φ ( d ( x , y ) ε ) K Ξ w ( x , y ) (157) L_\varepsilon(x,y):=\chi_2(x)\chi_1(y)\psi\left(\frac{d(x,y)}{r_0}\right)\varphi\left(\frac{d(x,y)}{\varepsilon}\right)K_{\Xi_w}(x,y) \tag*{(157)} L ε ( x , y ) := χ 2 ( x ) χ 1 ( y ) ψ ( r 0 d ( x , y ) ) φ ( ε d ( x , y ) ) K Ξ w ( x , y ) (157) and we write L ε ( x , y ) = ϱ ε ( x , y ) K Ξ w ( x , y ) L_\varepsilon(x,y)=\varrho_\varepsilon(x,y)K_{\Xi_w}(x,y) L ε ( x , y ) = ϱ ε ( x , y ) K Ξ w ( x , y )
ϱ ε ( x , y ) : = χ 2 ( x ) χ 1 ( y ) a ε ( d ( x , y ) ) and a ε ( t ) : = φ ( t / ε ) ψ ( t / r 0 ) , t ≥ 0. \varrho_\varepsilon(x,y):=\chi_2(x)\chi_1(y)a_\varepsilon(d(x,y))
\qquad\text{and}\qquad
a_\varepsilon(t):=\varphi(t/\varepsilon)\psi(t/r_0),\quad t\geq0. ϱ ε ( x , y ) := χ 2 ( x ) χ 1 ( y ) a ε ( d ( x , y )) and a ε ( t ) := φ ( t / ε ) ψ ( t / r 0 ) , t ≥ 0. Then, for a positive measure m \mathbf{m} m
S m , ε f ( x ) : = ∫ L ε ( x , y ) f ( y ) d m ( y ) , f ∈ L l o c 1 ( m ) . S_{\mathbf{m},\varepsilon}f(x):=\int L_\varepsilon(x,y)f(y)\,d\mathbf{m}(y),\qquad f\in L^1_{\mathrm{loc}}(\mathbf{m}). S m , ε f ( x ) := ∫ L ε ( x , y ) f ( y ) d m ( y ) , f ∈ L loc 1 ( m ) . We note that, since φ ′ ( t / ε ) \varphi'(t/\varepsilon) φ ′ ( t / ε ) ψ ′ ( t / r 0 ) \psi'(t/r_0) ψ ′ ( t / r 0 ) ε ≤ t ≤ 2 ε \varepsilon\leq t\leq2\varepsilon ε ≤ t ≤ 2 ε r 0 ≤ t ≤ 2 r 0 r_0\leq t\leq2r_0 r 0 ≤ t ≤ 2 r 0 ∣ a ε ′ ( t ) ∣ ≲ 1 / t \lvert a_\varepsilon'(t)\rvert\lesssim1/t ∣ a ε ′ ( t )∣ ≲ 1/ t 0 ≤ a ε ≤ 1 0\leq a_\varepsilon\leq1 0 ≤ a ε ≤ 1 a ε ( t ) = 0 a_\varepsilon(t)=0 a ε ( t ) = 0 t ≥ 2 r 0 t\geq2r_0 t ≥ 2 r 0
Let x , x ′ , y ∈ X x,x',y\in X x , x ′ , y ∈ X d ( x , x ′ ) ≤ d ( x , y ) / 2 d(x,x')\leq d(x,y)/2 d ( x , x ′ ) ≤ d ( x , y ) /2 d ( x ′ , y ) ≥ d ( x , y ) / 2 d(x',y)\geq d(x,y)/2 d ( x ′ , y ) ≥ d ( x , y ) /2 0 ≤ χ 1 , χ 2 ≤ 1 0\leq\chi_1,\chi_2\leq1 0 ≤ χ 1 , χ 2 ≤ 1 r ( B 2 ) ≥ r 0 r(B_2)\geq r_0 r ( B 2 ) ≥ r 0 χ 2 \chi_2 χ 2
∣ ϱ ε ( x , y ) − ϱ ε ( x ′ , y ) ∣ ≤ ∣ χ 2 ( x ) − χ 2 ( x ′ ) ∣ a ε ( d ( x , y ) ) + ∣ a ε ( d ( x , y ) ) − a ε ( d ( x ′ , y ) ) ∣ ≲ d ( x , x ′ ) r 0 χ { d ( x , y ) < 2 r 0 } + d ( x , x ′ ) d ( x , y ) ≲ d ( x , x ′ ) d ( x , y ) . \begin{aligned}
\lvert\varrho_\varepsilon(x,y)-\varrho_\varepsilon(x',y)\rvert
&\leq\lvert\chi_2(x)-\chi_2(x')\rvert a_\varepsilon(d(x,y))\\
&\quad+\lvert a_\varepsilon(d(x,y))-a_\varepsilon(d(x',y))\rvert\\
&\lesssim\frac{d(x,x')}{r_0}\chi_{\{d(x,y)<2r_0\}}+\frac{d(x,x')}{d(x,y)}
\lesssim\frac{d(x,x')}{d(x,y)}.
\end{aligned} ∣ ϱ ε ( x , y ) − ϱ ε ( x ′ , y )∣ ≤ ∣ χ 2 ( x ) − χ 2 ( x ′ )∣ a ε ( d ( x , y )) + ∣ a ε ( d ( x , y )) − a ε ( d ( x ′ , y ))∣ ≲ r 0 d ( x , x ′ ) χ { d ( x , y ) < 2 r 0 } + d ( x , y ) d ( x , x ′ ) ≲ d ( x , y ) d ( x , x ′ ) . Hence, by Lemma 3.13 , ∣ ϱ ε ∣ ≤ 1 \lvert\varrho_\varepsilon\rvert\leq1 ∣ ϱ ε ∣ ≤ 1 0 < γ ≤ 1 0<\gamma\leq1 0 < γ ≤ 1
∣ L ε ( x , y ) − L ε ( x ′ , y ) ∣ ≤ ∣ K Ξ w ( x , y ) − K Ξ w ( x ′ , y ) ∣ + ∣ ϱ ε ( x , y ) − ϱ ε ( x ′ , y ) ∣ ∣ K Ξ w ( x ′ , y ) ∣ ≲ d ( x , x ′ ) γ d ( x , y ) n + γ + d ( x , x ′ ) d ( x , y ) n + 1 ≲ d ( x , x ′ ) γ d ( x , y ) n + γ . \begin{aligned}
\lvert L_\varepsilon(x,y)-L_\varepsilon(x',y)\rvert
&\leq\lvert K_{\Xi_w}(x,y)-K_{\Xi_w}(x',y)\rvert\\
&\quad+\lvert\varrho_\varepsilon(x,y)-\varrho_\varepsilon(x',y)\rvert\lvert K_{\Xi_w}(x',y)\rvert\\
&\lesssim\frac{d(x,x')^\gamma}{d(x,y)^{n+\gamma}}+\frac{d(x,x')}{d(x,y)^{n+1}}
\lesssim\frac{d(x,x')^\gamma}{d(x,y)^{n+\gamma}}.
\end{aligned} ∣ L ε ( x , y ) − L ε ( x ′ , y )∣ ≤ ∣ K Ξ w ( x , y ) − K Ξ w ( x ′ , y )∣ + ∣ ϱ ε ( x , y ) − ϱ ε ( x ′ , y )∣ ∣ K Ξ w ( x ′ , y )∣ ≲ d ( x , y ) n + γ d ( x , x ′ ) γ + d ( x , y ) n + 1 d ( x , x ′ ) ≲ d ( x , y ) n + γ d ( x , x ′ ) γ . Since r ( B 1 ) ≥ r 0 r(B_1)\geq r_0 r ( B 1 ) ≥ r 0
∣ L ε ( x , y ) − L ε ( x ′ , y ) ∣ + ∣ L ε ( y , x ) − L ε ( y , x ′ ) ∣ ≲ d ( x , x ′ ) γ d ( x , y ) n + γ , d ( x , x ′ ) ≤ 1 2 d ( x , y ) . (158) \lvert L_\varepsilon(x,y)-L_\varepsilon(x',y)\rvert+\lvert L_\varepsilon(y,x)-L_\varepsilon(y,x')\rvert
\lesssim\frac{d(x,x')^\gamma}{d(x,y)^{n+\gamma}},
\qquad d(x,x')\leq\frac{1}{2}d(x,y). \tag*{(158)} ∣ L ε ( x , y ) − L ε ( x ′ , y )∣ + ∣ L ε ( y , x ) − L ε ( y , x ′ )∣ ≲ d ( x , y ) n + γ d ( x , x ′ ) γ , d ( x , x ′ ) ≤ 2 1 d ( x , y ) . (158) Let Ψ ε : = Ξ w + ε \Psi_\varepsilon:=\Xi_w+\varepsilon Ψ ε := Ξ w + ε L ε ( x , y ) = 0 L_\varepsilon(x,y)=0 L ε ( x , y ) = 0 d ( x , y ) ≤ ε d(x,y)\leq\varepsilon d ( x , y ) ≤ ε Lemma 3.13 gives
∣ L ε ( x , y ) ∣ ≲ min { d ( x , y ) − n , ε − n , Ξ w ( x ) − n , Ξ w ( y ) − n } \lvert L_\varepsilon(x,y)\rvert\lesssim\min\{d(x,y)^{-n},\varepsilon^{-n},\Xi_w(x)^{-n},\Xi_w(y)^{-n}\} ∣ L ε ( x , y )∣ ≲ min { d ( x , y ) − n , ε − n , Ξ w ( x ) − n , Ξ w ( y ) − n } ≲ min { d ( x , y ) − n , Ψ ε ( x ) − n , Ψ ε ( y ) − n } . (159) \lesssim\min\{d(x,y)^{-n},\Psi_\varepsilon(x)^{-n},\Psi_\varepsilon(y)^{-n}\}. \tag*{(159)} ≲ min { d ( x , y ) − n , Ψ ε ( x ) − n , Ψ ε ( y ) − n } . (159) The function Ψ ε \Psi_\varepsilon Ψ ε m ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣ } \mathbf{m}\in\{\mu,\lvert\nu_1\rvert,\lvert\nu_2\rvert\} m ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣} x ∈ X x\in X x ∈ X t ≥ Ψ ε ( x ) t\geq\Psi_\varepsilon(x) t ≥ Ψ ε ( x ) t ≥ Ξ w ( x ) t\geq\Xi_w(x) t ≥ Ξ w ( x ) (155) applies, and (154) gives
m ( B ( x , t ) ) ≤ 2 n ( m + c 0 ) t n . \mathbf{m}(B(x,t))\leq2^n(\mathbf{m}+\mathfrak{c}_0)t^n. m ( B ( x , t )) ≤ 2 n ( m + c 0 ) t n . Therefore, by (158) and (159) , the kernel L ε L_\varepsilon L ε Ψ ε \Psi_\varepsilon Ψ ε m 1 , m 2 , m 3 ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣ } \mathbf{m}_1,\mathbf{m}_2,\mathbf{m}_3\in\{\mu,\lvert\nu_1\rvert,\lvert\nu_2\rvert\} m 1 , m 2 , m 3 ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣} Lemma 3.16 other than the L 2 L^2 L 2 δ 0 = ε \delta_0=\varepsilon δ 0 = ε ε \varepsilon ε w w w
Lemma 5.1. For every ε > 0 \varepsilon>0 ε > 0 m i ∈ { μ i , ∣ ν ^ i ∣ } \mathbf{m}_i\in\{\mu_i,\lvert\widehat{\nu}_i\rvert\} m i ∈ { μ i , ∣ ν i ∣} i = 1 , 2 i=1,2 i = 1 , 2
∥ T Ξ w , m 1 , ε − S m 1 , ε ∥ L 2 ( m 1 ) → L 2 ( m 2 ) ≲ 1. (160) \left\|T_{\Xi_w,\mathbf{m}_1,\varepsilon}-S_{\mathbf{m}_1,\varepsilon}\right\|_{L^2(\mathbf{m}_1)\to L^2(\mathbf{m}_2)}\lesssim1. \tag*{(160)} ∥ T Ξ w , m 1 , ε − S m 1 , ε ∥ L 2 ( m 1 ) → L 2 ( m 2 ) ≲ 1. (160) Proof. Since m i \mathbf{m}_i m i B i B_i B i (160) is
K Ξ w ( x , y ) χ { d ( x , y ) > ε } − L ε ( x , y ) = D 1 ( x , y ) + D 2 ( x , y ) , x ∈ B 2 , y ∈ B 1 , K_{\Xi_w}(x,y)\chi_{\{d(x,y)>\varepsilon\}}-L_\varepsilon(x,y)=\mathfrak{D}_1(x,y)+\mathfrak{D}_2(x,y),\qquad x\in B_2,\quad y\in B_1, K Ξ w ( x , y ) χ { d ( x , y ) > ε } − L ε ( x , y ) = D 1 ( x , y ) + D 2 ( x , y ) , x ∈ B 2 , y ∈ B 1 , where we split
D 1 ( x , y ) : = [ χ { d ( x , y ) > ε } − φ ( d ( x , y ) ε ) ] K Ξ w ( x , y ) , \mathcal{D}_1(x,y):=\left[\chi_{\{d(x,y)>\varepsilon\}}-\varphi\left(\frac{d(x,y)}{\varepsilon}\right)\right]K_{\Xi_w}(x,y), D 1 ( x , y ) := [ χ { d ( x , y ) > ε } − φ ( ε d ( x , y ) ) ] K Ξ w ( x , y ) , D 2 ( x , y ) : = [ 1 − χ 2 ( x ) χ 1 ( y ) ψ ( d ( x , y ) r 0 ) ] φ ( d ( x , y ) ε ) K Ξ w ( x , y ) . \mathcal{D}_2(x,y):=\left[1-\chi_2(x)\chi_1(y)\psi\left(\frac{d(x,y)}{r_0}\right)\right]\varphi\left(\frac{d(x,y)}{\varepsilon}\right)K_{\Xi_w}(x,y). D 2 ( x , y ) := [ 1 − χ 2 ( x ) χ 1 ( y ) ψ ( r 0 d ( x , y ) ) ] φ ( ε d ( x , y ) ) K Ξ w ( x , y ) . Since φ = 0 \varphi=0 φ = 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] φ = 1 \varphi=1 φ = 1 [ 2 , ∞ ) [2,\infty) [ 2 , ∞ ) D 1 ( x , y ) \mathcal{D}_1(x,y) D 1 ( x , y ) ε < d ( x , y ) < 2 ε \varepsilon<d(x,y)<2\varepsilon ε < d ( x , y ) < 2 ε ∣ D 1 ∣ ≤ ∣ K Ξ w ∣ |\mathcal{D}_1|\leq|K_{\Xi_w}| ∣ D 1 ∣ ≤ ∣ K Ξ w ∣ Lemma 3.13 and (154) , for m ∈ { m 1 , m 2 } \mathfrak{m}\in\{\mathfrak{m}_1,\mathfrak{m}_2\} m ∈ { m 1 , m 2 } z ∈ X z\in X z ∈ X
∫ ∣ D 1 ( z , y ) ∣ d m ( y ) + ∫ ∣ D 1 ( x , z ) ∣ d m ( x ) ≲ m ( B ( z , 2 ε ) ) max { ε , Ξ w ( z ) } n ≲ ( 2 ε + Ξ w ( z ) ) n max { ε , Ξ w ( z ) } n ≤ 3 n . \int|\mathcal{D}_1(z,y)|\,d\mathfrak{m}(y)+\int|\mathcal{D}_1(x,z)|\,d\mathfrak{m}(x)\lesssim\frac{\mathfrak{m}(B(z,2\varepsilon))}{\max\{\varepsilon,\Xi_w(z)\}^{n}}\lesssim\frac{(2\varepsilon+\Xi_w(z))^n}{\max\{\varepsilon,\Xi_w(z)\}^{n}}\leq3^n. ∫ ∣ D 1 ( z , y ) ∣ d m ( y ) + ∫ ∣ D 1 ( x , z ) ∣ d m ( x ) ≲ max { ε , Ξ w ( z ) } n m ( B ( z , 2 ε )) ≲ max { ε , Ξ w ( z ) } n ( 2 ε + Ξ w ( z ) ) n ≤ 3 n . By Schur’s test, the integral operator with kernel D 1 \mathcal{D}_1 D 1 L 2 ( m 1 ) L^2(\mathfrak{m}_1) L 2 ( m 1 ) L 2 ( m 2 ) L^2(\mathfrak{m}_2) L 2 ( m 2 ) ≲ 1 \lesssim1 ≲ 1
If D 2 ( x , y ) ≠ 0 \mathcal{D}_2(x,y)\neq0 D 2 ( x , y ) = 0 d ( x , y ) > r 0 d(x,y)>r_0 d ( x , y ) > r 0 χ 2 ( x ) ≠ 1 \chi_2(x)\neq1 χ 2 ( x ) = 1 χ 1 ( y ) ≠ 1 \chi_1(y)\neq1 χ 1 ( y ) = 1 (156) and r ( B i ) ≥ r 0 r(B_i)\geq r_0 r ( B i ) ≥ r 0
max { d ( x , y ) , Ξ w ( x ) , Ξ w ( y ) } ≥ θ r 0 2 . \max\{d(x,y),\Xi_w(x),\Xi_w(y)\}\geq\frac{\theta r_0}{2}. max { d ( x , y ) , Ξ w ( x ) , Ξ w ( y )} ≥ 2 θ r 0 . Since ∣ D 2 ∣ ≤ ∣ K Ξ w ∣ |\mathcal{D}_2|\leq|K_{\Xi_w}| ∣ D 2 ∣ ≤ ∣ K Ξ w ∣ Lemma 3.13 implies
∣ D 2 ( x , y ) ∣ ≲ 1 max { d ( x , y ) , Ξ w ( x ) , Ξ w ( y ) } n ≲ 1 ( r 0 + d ( x , y ) + Ξ w ( x ) + Ξ w ( y ) ) n = : H ( x , y ) . (161) |\mathcal{D}_2(x,y)|\lesssim\frac{1}{\max\{d(x,y),\Xi_w(x),\Xi_w(y)\}^{n}}\lesssim\frac{1}{(r_0+d(x,y)+\Xi_w(x)+\Xi_w(y))^n}=:H(x,y). \tag*{(161)} ∣ D 2 ( x , y ) ∣ ≲ max { d ( x , y ) , Ξ w ( x ) , Ξ w ( y ) } n 1 ≲ ( r 0 + d ( x , y ) + Ξ w ( x ) + Ξ w ( y ) ) n 1 =: H ( x , y ) . (161) Let z ∈ X z\in X z ∈ X m ∈ { m 1 , m 2 } \mathfrak{m}\in\{\mathfrak{m}_1,\mathfrak{m}_2\} m ∈ { m 1 , m 2 } r z : = r 0 + Ξ w ( z ) r_z:=r_0+\Xi_w(z) r z := r 0 + Ξ w ( z ) H ( x , z ) ≤ ( r z + d ( x , z ) ) − n H(x,z)\leq(r_z+d(x,z))^{-n} H ( x , z ) ≤ ( r z + d ( x , z ) ) − n H ( ⋅ , z ) ≤ 2 n ( 2 k r z ) − n H(\cdot,z)\leq2^n(2^kr_z)^{-n} H ( ⋅ , z ) ≤ 2 n ( 2 k r z ) − n B ( z , 2 k r z ) ∖ B ( z , 2 k − 1 r z ) B(z,2^kr_z)\setminus B(z,2^{k-1}r_z) B ( z , 2 k r z ) ∖ B ( z , 2 k − 1 r z ) k ≥ 1 k\geq1 k ≥ 1 B ( z , r z ) B(z,r_z) B ( z , r z ) (154) and Ξ w ( z ) ≤ r z \Xi_w(z)\leq r_z Ξ w ( z ) ≤ r z
∫ H ( x , z ) 2 d m ( x ) ≲ ∑ k ≥ 0 m ( B ( z , 2 k r z ) ) ( 2 k r z ) 2 n ≲ ∑ k ≥ 0 ( 2 k r z + Ξ w ( z ) ) n ( 2 k r z ) 2 n ≲ r z − n ≤ r 0 − n . \int H(x,z)^2\,d\mathfrak{m}(x)\lesssim\sum_{k\geq0}\frac{\mathfrak{m}(B(z,2^kr_z))}{(2^kr_z)^{2n}}\lesssim\sum_{k\geq0}\frac{(2^kr_z+\Xi_w(z))^n}{(2^kr_z)^{2n}}\lesssim r_z^{-n}\leq r_0^{-n}. ∫ H ( x , z ) 2 d m ( x ) ≲ k ≥ 0 ∑ ( 2 k r z ) 2 n m ( B ( z , 2 k r z )) ≲ k ≥ 0 ∑ ( 2 k r z ) 2 n ( 2 k r z + Ξ w ( z ) ) n ≲ r z − n ≤ r 0 − n . Since H H H y ∈ X y\in X y ∈ X
∫ H ( x , y ) 2 d m 2 ( x ) + ∫ H ( y , x ) 2 d m 1 ( x ) ≲ r 0 − n . (162) \int H(x,y)^2\,d\mathfrak{m}_2(x)+\int H(y,x)^2\,d\mathfrak{m}_1(x)\lesssim r_0^{-n}. \tag*{(162)} ∫ H ( x , y ) 2 d m 2 ( x ) + ∫ H ( y , x ) 2 d m 1 ( x ) ≲ r 0 − n . (162) By (41) and the n n n μ \mu μ i = 1 , 2 i=1,2 i = 1 , 2
m i ( X ) ≤ C 1 μ ( B i ) ≤ C 1 C 0 r ( B i ) n . \mathfrak{m}_i(X)\leq C_1\mu(B_i)\leq C_1C_0r(B_i)^n. m i ( X ) ≤ C 1 μ ( B i ) ≤ C 1 C 0 r ( B i ) n . Integrating (162) , we get
∬ H ( x , y ) 2 d m 1 ( y ) d m 2 ( x ) ≲ r 0 − n min { m 1 ( X ) , m 2 ( X ) } ≲ 1. \iint H(x,y)^2\,d\mathfrak{m}_1(y)d\mathfrak{m}_2(x)\lesssim r_0^{-n}\min\{\mathfrak{m}_1(X),\mathfrak{m}_2(X)\}\lesssim1. ∬ H ( x , y ) 2 d m 1 ( y ) d m 2 ( x ) ≲ r 0 − n min { m 1 ( X ) , m 2 ( X )} ≲ 1. By (161) and the Cauchy–Schwarz inequality, the integral operator with kernel D 2 \mathcal{D}_2 D 2 L 2 ( m 1 ) L^2(\mathfrak{m}_1) L 2 ( m 1 ) L 2 ( m 2 ) L^2(\mathfrak{m}_2) L 2 ( m 2 ) ≲ 1 \lesssim1 ≲ 1 D 1 \mathcal{D}_1 D 1 (160) . □
Comparison of the norms
As an application of Lemma 5.1 and the properties of the measures discussed at the beginning of the section, we obtain the following comparison result for the L 2 L^2 L 2
Lemma 5.2. There exists a constant c ^ 0 > 0 \widehat c_0>0 c 0 > 0 n n n D D D γ \gamma γ ε 0 \varepsilon_0 ε 0 C C C C 0 C_0 C 0 C 1 C_1 C 1 C 3 C_3 C 3 β \beta β ε > 0 \varepsilon>0 ε > 0
∥ T Ξ w , μ 1 , ε ∥ L 2 ( μ 1 ) → L 2 ( ∣ ν ^ 2 ∣ ) + ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( μ 2 ) ≤ c ^ 0 ( 1 + ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ) . (163) \left\|T_{\Xi_w,\mu_1,\varepsilon}\right\|_{L^2(\mu_1)\to L^2(|\widehat\nu_2|)}+\left\|T_{\Xi_w,|\widehat\nu_1|,\varepsilon}\right\|_{L^2(|\widehat\nu_1|)\to L^2(\mu_2)}
\leq\widehat c_0\left(1+\left\|T_{\Xi_w,|\widehat\nu_1|,\varepsilon}\right\|_{L^2(|\widehat\nu_1|)\to L^2(|\widehat\nu_2|)}\right). \tag*{(163)} ∥ T Ξ w , μ 1 , ε ∥ L 2 ( μ 1 ) → L 2 ( ∣ ν 2 ∣ ) + T Ξ w , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( μ 2 ) ≤ c 0 ( 1 + T Ξ w , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ) . (163) Moreover, there exists a constant c ^ > 0 \widehat{c}>0 c > 0
∥ T Ξ w , μ 1 , ε ∥ L 2 ( μ 1 ) → L 2 ( μ 2 ) ≤ c ^ ( 1 + ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ) . (164) \left\|T_{\Xi_w,\mu_1,\varepsilon}\right\|_{L^2(\mu_1)\to L^2(\mu_2)} \le\widehat{c}\left(1+\left\|T_{\Xi_w,\lvert\widehat{\nu}_1\rvert,\varepsilon}\right\|_{L^2(\lvert\widehat{\nu}_1\rvert)\to L^2(\lvert\widehat{\nu}_2\rvert)}\right). \tag*{(164)} ∥ T Ξ w , μ 1 , ε ∥ L 2 ( μ 1 ) → L 2 ( μ 2 ) ≤ c ( 1 + T Ξ w , ∣ ν 1 ∣ , ε L 2 (∣ ν 1 ∣) → L 2 (∣ ν 2 ∣) ) . (164) Proof. In this proof, we use the shorthand notation ∥ ⋅ ∥ m → m ′ : = ∥ ⋅ ∥ L 2 ( m ) → L 2 ( m ′ ) \left\|\cdot\right\|_{m\to m'}:=\left\|\cdot\right\|_{L^2(m)\to L^2(m')} ∥ ⋅ ∥ m → m ′ := ∥ ⋅ ∥ L 2 ( m ) → L 2 ( m ′ ) ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ ∣ ν ^ 1 ∣ → ∣ ν ^ 2 ∣ < ∞ \left\|T_{\Xi_w,\lvert\widehat{\nu}_1\rvert,\varepsilon}\right\|_{\lvert\widehat{\nu}_1\rvert\to\lvert\widehat{\nu}_2\rvert}<\infty T Ξ w , ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ < ∞ T Ξ w , ∣ ν ^ 1 ∣ , ε T_{\Xi_w,\lvert\widehat{\nu}_1\rvert,\varepsilon} T Ξ w , ∣ ν 1 ∣ , ε C ε − n C\varepsilon^{-n} C ε − n ∣ ν ^ 1 ∣ \lvert\widehat{\nu}_1\rvert ∣ ν 1 ∣ ∣ ν ^ 2 ∣ \lvert\widehat{\nu}_2\rvert ∣ ν 2 ∣
Let m 1 ∈ { μ , ∣ ν 1 ∣ } m_1\in\{\mu,\lvert\nu_1\rvert\} m 1 ∈ { μ , ∣ ν 1 ∣} m 2 ∈ { μ , ∣ ν 2 ∣ } m_2\in\{\mu,\lvert\nu_2\rvert\} m 2 ∈ { μ , ∣ ν 2 ∣} m i ∣ B i ∈ { μ i , ∣ ν ^ i ∣ } m_i|_{B_i}\in\{\mu_i,\lvert\widehat{\nu}_i\rvert\} m i ∣ B i ∈ { μ i , ∣ ν i ∣} L ε ( x , y ) = 0 L_\varepsilon(x,y)=0 L ε ( x , y ) = 0 ( x , y ) ∈ B 2 × B 1 (x,y)\in B_2\times B_1 ( x , y ) ∈ B 2 × B 1 ∥ S m 1 , ε ∥ m 1 → m 2 = ∥ S m 1 ∣ B 1 , ε ∥ m 1 ∣ B 1 → m 2 ∣ B 2 \left\|S_{m_1,\varepsilon}\right\|_{m_1\to m_2}=\left\|S_{m_1|_{B_1},\varepsilon}\right\|_{m_1|_{B_1}\to m_2|_{B_2}} ∥ S m 1 , ε ∥ m 1 → m 2 = S m 1 ∣ B 1 , ε m 1 ∣ B 1 → m 2 ∣ B 2 (160) gives
∣ ∥ T Ξ w , m 1 ∣ B 1 , ε ∥ m 1 ∣ B 1 → m 2 ∣ B 2 − ∥ S m 1 , ε ∥ m 1 → m 2 ∣ ≤ C . (165) \left|\left\|T_{\Xi_w,m_1|_{B_1},\varepsilon}\right\|_{m_1|_{B_1}\to m_2|_{B_2}}-\left\|S_{m_1,\varepsilon}\right\|_{m_1\to m_2}\right|\le C. \tag*{(165)} T Ξ w , m 1 ∣ B 1 , ε m 1 ∣ B 1 → m 2 ∣ B 2 − ∥ S m 1 , ε ∥ m 1 → m 2 ≤ C . (165) In particular,
∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → ∣ ν 2 ∣ ≤ ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ ∣ ν ^ 1 ∣ → ∣ ν ^ 2 ∣ + C < ∞ . (166) \left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\lvert\nu_2\rvert}\le\left\|T_{\Xi_w,\lvert\widehat{\nu}_1\rvert,\varepsilon}\right\|_{\lvert\widehat{\nu}_1\rvert\to\lvert\widehat{\nu}_2\rvert}+C<\infty. \tag*{(166)} S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ ≤ T Ξ w , ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ + C < ∞. (166) Thus, by (165) and (166) with ( m 1 , m 2 ) (m_1,m_2) ( m 1 , m 2 ) ( ∣ ν 1 ∣ , μ ) (\lvert\nu_1\rvert,\mu) (∣ ν 1 ∣ , μ ) ( μ , ∣ ν 2 ∣ ) (\mu,\lvert\nu_2\rvert) ( μ , ∣ ν 2 ∣) ( μ , μ ) (\mu,\mu) ( μ , μ ) (163) and (164) follow from
∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → μ + ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ + ∥ S μ , ε ∥ μ → μ ≲ 1 + ∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → ∣ ν 2 ∣ . (167) \left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\mu}+\left\|S_{\mu,\varepsilon}\right\|_{\mu\to\lvert\nu_2\rvert}+\left\|S_{\mu,\varepsilon}\right\|_{\mu\to\mu}\lesssim1+\left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\lvert\nu_2\rvert}. \tag*{(167)} S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → μ + ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ + ∥ S μ , ε ∥ μ → μ ≲ 1 + S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ . (167) As observed after (159) , Lemma 3.16 applies to L ε L_\varepsilon L ε L ε t ( x , y ) : = L ε ( y , x ) L_\varepsilon^t(x,y):=L_\varepsilon(y,x) L ε t ( x , y ) := L ε ( y , x ) Ξ = Ψ ε \Xi=\Psi_\varepsilon Ξ = Ψ ε δ 0 = ε \delta_0=\varepsilon δ 0 = ε m , m ′ , m ′ ′ ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣ } m,m',m''\in\{\mu,\lvert\nu_1\rvert,\lvert\nu_2\rvert\} m , m ′ , m ′′ ∈ { μ , ∣ ν 1 ∣ , ∣ ν 2 ∣} S m , ε t S^t_{m,\varepsilon} S m , ε t L ε t L_\varepsilon^t L ε t m m m ∥ S m ′ , ε t ∥ m ′ → m = ∥ S m , ε ∥ m → m ′ \left\|S^t_{m',\varepsilon}\right\|_{m'\to m}=\left\|S_{m,\varepsilon}\right\|_{m\to m'} S m ′ , ε t m ′ → m = ∥ S m , ε ∥ m → m ′ L ε ( x , y ) = 0 L_\varepsilon(x,y)=0 L ε ( x , y ) = 0 d ( x , y ) ≤ ε d(x,y)\le\varepsilon d ( x , y ) ≤ ε S m , ε S_{m,\varepsilon} S m , ε δ < ε \delta<\varepsilon δ < ε S m , ε S_{m,\varepsilon} S m , ε (78) gives
∥ S m , ε ∥ m → m ′ ′ ≲ 1 + ∥ S m , ε ∥ m → m ′ (168) \left\|S_{m,\varepsilon}\right\|_{m\to m''}\lesssim1+\left\|S_{m,\varepsilon}\right\|_{m\to m'} \tag*{(168)} ∥ S m , ε ∥ m → m ′′ ≲ 1 + ∥ S m , ε ∥ m → m ′ (168) whenever the right-hand side is finite, and the same holds for S m , ε t S^t_{m,\varepsilon} S m , ε t (168) three times, we obtain
∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → μ ≲ 1 + ∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → ∣ ν 2 ∣ , \left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\mu}\lesssim1+\left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\lvert\nu_2\rvert}, S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → μ ≲ 1 + S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ , ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ = ∥ S ∣ ν 2 ∣ , ε t ∥ ∣ ν 2 ∣ → μ ≲ 1 + ∥ S ∣ ν 2 ∣ , ε t ∥ ∣ ν 2 ∣ → ∣ ν 1 ∣ = 1 + ∥ S ∣ ν 1 ∣ , ε ∥ ∣ ν 1 ∣ → ∣ ν 2 ∣ , \left\|S_{\mu,\varepsilon}\right\|_{\mu\to\lvert\nu_2\rvert}=\left\|S^t_{\lvert\nu_2\rvert,\varepsilon}\right\|_{\lvert\nu_2\rvert\to\mu}\lesssim1+\left\|S^t_{\lvert\nu_2\rvert,\varepsilon}\right\|_{\lvert\nu_2\rvert\to\lvert\nu_1\rvert}=1+\left\|S_{\lvert\nu_1\rvert,\varepsilon}\right\|_{\lvert\nu_1\rvert\to\lvert\nu_2\rvert}, ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ = S ∣ ν 2 ∣ , ε t ∣ ν 2 ∣ → μ ≲ 1 + S ∣ ν 2 ∣ , ε t ∣ ν 2 ∣ → ∣ ν 1 ∣ = 1 + S ∣ ν 1 ∣ , ε ∣ ν 1 ∣ → ∣ ν 2 ∣ , and
∥ S μ , ε ∥ μ → μ ≲ 1 + ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ . \left\|S_{\mu,\varepsilon}\right\|_{\mu\to\mu}\lesssim1+\left\|S_{\mu,\varepsilon}\right\|_{\mu\to\lvert\nu_2\rvert}. ∥ S μ , ε ∥ μ → μ ≲ 1 + ∥ S μ , ε ∥ μ → ∣ ν 2 ∣ . By (166) and the second inequality, all the right-hand sides are finite, and the inequalities above prove (167) . □
Let R R R R 1 R_1 R 1 R 2 R_2 R 2 M : = 2 M ∗ M:=2M_* M := 2 M ∗ R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 R 2 ⊂ R 1 R_2\subset R_1 R 2 ⊂ R 1
84 B ( R ) ⊂ M B ~ ( R ) ⊂ M B 1 ∩ M B 2 . (169) 84B(R)\subset M\widetilde{B}(R)\subset MB_1\cap MB_2. \tag*{(169)} 84 B ( R ) ⊂ M B ( R ) ⊂ M B 1 ∩ M B 2 . (169) Lemma 5.3. For every ε > 0 \varepsilon>0 ε > 0
∥ T Ξ w , μ ∣ 84 B ( R ) , ε ∥ L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≲ 1 + ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) . (170) \left\|T_{\Xi_w,\mu|_{84B(R)},\varepsilon}\right\|_{L^2(\mu|_{84B(R)})\to L^2(\mu|_{84B(R)})}\lesssim1+\left\|T_{\Xi_w,\lvert\widehat{\nu}_1\rvert,\varepsilon}\right\|_{L^2(\lvert\widehat{\nu}_1\rvert)\to L^2(\lvert\widehat{\nu}_2\rvert)}. \tag*{(170)} T Ξ w , μ ∣ 84 B ( R ) , ε L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≲ 1 + T Ξ w , ∣ ν 1 ∣ , ε L 2 (∣ ν 1 ∣) → L 2 (∣ ν 2 ∣) . (170) The same estimate holds with Ξ w \Xi_w Ξ w Φ ~ R \widetilde{\Phi}_R Φ R (138) .
Proof. We use the notation of the proof of Lemma 5.2, and we set μ ^ i : = μ ∣ M B i \widehat{\mu}_i := \mu|_{MB_i} μ i := μ ∣ M B i i = 1 , 2 i = 1,2 i = 1 , 2 μ ^ : = μ ∣ M B ~ ( R ) \widehat{\mu} := \mu|_{M\widetilde{B}(R)} μ := μ ∣ M B ( R ) μ ∣ 84 B ( R ) ≤ μ ^ \mu|_{84B(R)} \leq\widehat{\mu} μ ∣ 84 B ( R ) ≤ μ (169) , it suffices to prove (170) with μ ^ \widehat{\mu} μ μ ∣ 84 B ( R ) \mu|_{84B(R)} μ ∣ 84 B ( R )
Let i = 1 , 2 i = 1,2 i = 1 , 2 χ i = 0 \chi_i = 0 χ i = 0 X ∖ B i X \setminus B_i X ∖ B i (156) gives Ξ w ≥ θ r ( B i ) / 2 \Xi_{\mathbf w} \geq\theta r(B_i)/2 Ξ w ≥ θ r ( B i ) /2 X ∖ B i X \setminus B_i X ∖ B i Lemma 3.13 , ∣ K Ξ w ( x , y ) ∣ ≲ r ( B i ) − n |K_{\Xi_{\mathbf w}}(x,y)| \lesssim r(B_i)^{-n} ∣ K Ξ w ( x , y ) ∣ ≲ r ( B i ) − n x ∉ B i x \notin B_i x ∈ / B i y ∉ B i y \notin B_i y ∈ / B i μ ( M B i ) ≤ C 0 M n r ( B i ) n \mu(MB_i) \leq C_0M^n r(B_i)^n μ ( M B i ) ≤ C 0 M n r ( B i ) n
∬ M B i × M B i ( χ X ∖ B i ( x ) + χ X ∖ B i ( y ) ) ∣ K Ξ w ( x , y ) ∣ 2 d μ ( y ) d μ ( x ) ≲ r ( B i ) − 2 n μ ( M B i ) 2 ≲ 1. \iint_{MB_i \times MB_i} \bigl(\chi_{X\setminus B_i}(x) + \chi_{X\setminus B_i}(y)\bigr)|K_{\Xi_{\mathbf w}}(x,y)|^2\,d\mu(y)\,d\mu(x) \lesssim r(B_i)^{-2n}\mu(MB_i)^2 \lesssim1. ∬ M B i × M B i ( χ X ∖ B i ( x ) + χ X ∖ B i ( y ) ) ∣ K Ξ w ( x , y ) ∣ 2 d μ ( y ) d μ ( x ) ≲ r ( B i ) − 2 n μ ( M B i ) 2 ≲ 1. Hence, by the Cauchy-Schwarz inequality, for every ε > 0 \varepsilon> 0 ε > 0
∥ χ X ∖ B i T Ξ w , μ ^ i , ε ∥ μ ^ i → μ ^ i + ∥ T Ξ w , μ ^ i , ε ( χ X ∖ B i ⋅ ) ∥ μ ^ i → μ ^ i ≲ 1. (171) \left\|\chi_{X\setminus B_i}T_{\Xi_{\mathbf w},\widehat{\mu}_i,\varepsilon}\right\|_{\widehat{\mu}_i\to\widehat{\mu}_i}+\left\|T_{\Xi_{\mathbf w},\widehat{\mu}_i,\varepsilon}(\chi_{X\setminus B_i}\mathbin{\cdot})\right\|_{\widehat{\mu}_i\to\widehat{\mu}_i}\lesssim1. \tag*{(171)} χ X ∖ B i T Ξ w , μ i , ε μ i → μ i + T Ξ w , μ i , ε ( χ X ∖ B i ⋅ ) μ i → μ i ≲ 1. (171) Let f ∈ L 2 ( μ ^ ) f \in L^2(\widehat{\mu}) f ∈ L 2 ( μ ) M B ~ ( R ) M\widetilde{B}(R) M B ( R ) M B ~ ( R ) ⊂ M B 1 ∩ M B 2 M\widetilde{B}(R) \subset MB_1 \cap MB_2 M B ( R ) ⊂ M B 1 ∩ M B 2
T Ξ w , μ ^ , ε f = χ B 2 T Ξ w , μ 1 , ε f + χ B 2 T Ξ w , μ ^ 1 , ε ( χ X ∖ B 1 f ) + χ X ∖ B 2 T Ξ w , μ ^ 2 , ε f = : I 1 + I 2 + I 3 . T_{\Xi_{\mathbf w},\widehat{\mu},\varepsilon}f=\chi_{B_2}T_{\Xi_{\mathbf w},\mu_1,\varepsilon}f+\chi_{B_2}T_{\Xi_{\mathbf w},\widehat{\mu}_1,\varepsilon}(\chi_{X\setminus B_1}f)+\chi_{X\setminus B_2}T_{\Xi_{\mathbf w},\widehat{\mu}_2,\varepsilon}f=: \mathcal I_1+\mathcal I_2+\mathcal I_3. T Ξ w , μ , ε f = χ B 2 T Ξ w , μ 1 , ε f + χ B 2 T Ξ w , μ 1 , ε ( χ X ∖ B 1 f ) + χ X ∖ B 2 T Ξ w , μ 2 , ε f =: I 1 + I 2 + I 3 . Since μ ^ ∣ B 2 ≤ μ 2 \widehat{\mu}|_{B_2}\leq\mu_2 μ ∣ B 2 ≤ μ 2 ∥ f ∥ L 2 ( μ 1 ) ≤ ∥ f ∥ L 2 ( μ ^ ) \|f\|_{L^2(\mu_1)}\leq\|f\|_{L^2(\widehat{\mu})} ∥ f ∥ L 2 ( μ 1 ) ≤ ∥ f ∥ L 2 ( μ ) (164) gives
∥ I 1 ∥ L 2 ( μ ^ ) ≤ ∥ T Ξ w , μ 1 , ε ∥ μ 1 → μ 2 ∥ f ∥ L 2 ( μ 1 ) ≲ ( 1 + ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ) ∥ f ∥ L 2 ( μ ^ ) . \|\mathcal I_1\|_{L^2(\widehat{\mu})}\leq\|T_{\Xi_{\mathbf w},\mu_1,\varepsilon}\|_{\mu_1\to\mu_2}\|f\|_{L^2(\mu_1)}\lesssim\left(1+\|T_{\Xi_{\mathbf w},|\widehat{\nu}_1|,\varepsilon}\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}\right)\|f\|_{L^2(\widehat{\mu})}. ∥ I 1 ∥ L 2 ( μ ) ≤ ∥ T Ξ w , μ 1 , ε ∥ μ 1 → μ 2 ∥ f ∥ L 2 ( μ 1 ) ≲ ( 1 + ∥ T Ξ w , ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ) ∥ f ∥ L 2 ( μ ) . Moreover μ ^ ≤ μ ^ i \widehat{\mu}\leq\widehat{\mu}_i μ ≤ μ i ∥ f ∥ L 2 ( μ ^ i ) = ∥ f ∥ L 2 ( μ ^ ) \|f\|_{L^2(\widehat{\mu}_i)}=\|f\|_{L^2(\widehat{\mu})} ∥ f ∥ L 2 ( μ i ) = ∥ f ∥ L 2 ( μ ) i = 1 , 2 i=1,2 i = 1 , 2 (171) gives
∥ I 2 ∥ L 2 ( μ ^ ) + ∥ I 3 ∥ L 2 ( μ ^ ) ≤ ∥ T Ξ w , μ ^ 1 , ε ( χ X ∖ B 1 f ) ∥ L 2 ( μ ^ 1 ) + ∥ χ X ∖ B 2 T Ξ w , μ ^ 2 , ε f ∥ L 2 ( μ ^ 2 ) ≲ ∥ f ∥ L 2 ( μ ^ ) , \|\mathcal I_2\|_{L^2(\widehat{\mu})}+\|\mathcal I_3\|_{L^2(\widehat{\mu})}\leq\|T_{\Xi_{\mathbf w},\widehat{\mu}_1,\varepsilon}(\chi_{X\setminus B_1}f)\|_{L^2(\widehat{\mu}_1)}+\|\chi_{X\setminus B_2}T_{\Xi_{\mathbf w},\widehat{\mu}_2,\varepsilon}f\|_{L^2(\widehat{\mu}_2)}\lesssim\|f\|_{L^2(\widehat{\mu})}, ∥ I 2 ∥ L 2 ( μ ) + ∥ I 3 ∥ L 2 ( μ ) ≤ ∥ T Ξ w , μ 1 , ε ( χ X ∖ B 1 f ) ∥ L 2 ( μ 1 ) + ∥ χ X ∖ B 2 T Ξ w , μ 2 , ε f ∥ L 2 ( μ 2 ) ≲ ∥ f ∥ L 2 ( μ ) , which proves (170) for the SIO associated with the suppression Ξ w \Xi_{\mathbf w} Ξ w
Finally, let i = 1 , 2 i=1,2 i = 1 , 2 w i ∈ Ω w_i\in\Omega w i ∈ Ω F i ( w i ) F_i(w_i) F i ( w i ) ( 1 − θ ) B i (1-\theta)B_i ( 1 − θ ) B i B ( x , r j , i ( x ) ) B(x,r_{j,i}(x)) B ( x , r j , i ( x )) x ∈ B i x\in B_i x ∈ B i j = 1 , 2 j=1,2 j = 1 , 2 B ( x , R i ( x ) ) B(x,R_i(x)) B ( x , R i ( x )) x ∈ X x\in X x ∈ X Φ i , R i , w i \Phi_{i,R_i,w_i} Φ i , R i , w i r 1 , i r_{1,i} r 1 , i r 2 , i r_{2,i} r 2 , i B i B_i B i R i R_i R i dist ( ⋅ , ( 1 − θ ) B i ) \operatorname{dist}(\cdot,(1-\theta)B_i) dist ( ⋅ , ( 1 − θ ) B i ) X X X w i w_i w i (99) , they also hold for Φ i , R i \Phi_{i,R_i} Φ i , R i Φ ~ R = max { Φ 1 , R 1 , Φ 2 , R 2 } \widetilde{\Phi}_R=\max\{\Phi_{1,R_1},\Phi_{2,R_2}\} Φ R = max { Φ 1 , R 1 , Φ 2 , R 2 } Φ ~ R \widetilde{\Phi}_R Φ R 1 1 1 Ξ w \Xi_{\mathbf w} Ξ w Φ ~ R \widetilde{\Phi}_R Φ R Ξ w \Xi_{\mathbf w} Ξ w
The Key Lemma for the suppressed operator
The setup. We keep the setting and the notation of § 5 . In particular, R 1 ∈ Top ′ ′ R_1\in\operatorname{Top}^{\prime\prime} R 1 ∈ Top ′′ R 2 ∈ Top ′ ′ R_2\in\operatorname{Top}^{\prime\prime} R 2 ∈ Top ′′ Tree ( R 1 , R 2 ) ≠ ∅ \operatorname{Tree}(R_1,R_2)\neq\varnothing Tree ( R 1 , R 2 ) = ∅ B i = B ~ ( R i ) B_i=\widetilde{B}(R_i) B i = B ( R i ) ν ^ 1 \widehat{\nu}_1 ν 1 ν ^ 2 \widehat{\nu}_2 ν 2 T T T B 1 B_1 B 1 T ∗ T^* T ∗ B 2 B_2 B 2 § 4 , for i = 1 , 2 i=1,2 i = 1 , 2 μ \mu μ ν ^ i \widehat{\nu}_i ν i B = B i B=B_i B = B i C 2 = 1 C_2=1 C 2 = 1 T T T T ∗ T^* T ∗ i = 2 i=2 i = 2 (43) for T T T B 1 B_1 B 1 T ∗ T^* T ∗
We use Ω \Omega Ω Ω 2 = Ω × Ω \Omega^2=\Omega\times\Omega Ω 2 = Ω × Ω w = ( w 1 , w 2 ) ∈ Ω 2 \mathbf w=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2 w i ∈ Ω w_i\in\Omega w i ∈ Ω D ( w i ) \mathcal D(w_i) D ( w i ) W i ( w i ) W_i(w_i) W i ( w i )
Let i = 1 , 2 i=1,2 i = 1 , 2 (2) , we have ν ^ i ( B i ) = μ ( B i ) \widehat{\nu}_i(B_i)=\mu(B_i) ν i ( B i ) = μ ( B i ) ν ^ i = b ^ i ∣ ν ^ i ∣ \widehat{\nu}_i=\widehat{b}_i|\widehat{\nu}_i| ν i = b i ∣ ν i ∣ B i B_i B i b ^ i \widehat{b}_i b i 1 1 1 X ∖ B i X\setminus B_i X ∖ B i ν i = ν ^ i + μ ∣ X ∖ B i \nu_i=\widehat{\nu}_i+\mu|_{X\setminus B_i} ν i = ν i + μ ∣ X ∖ B i ν i = b ^ i ∣ ν i ∣ \nu_i=\widehat{b}_i|\nu_i| ν i = b i ∣ ν i ∣ ∣ b ^ i ∣ = 1 |\widehat{b}_i|=1 ∣ b i ∣ = 1 ∣ ν i ∣ |\nu_i| ∣ ν i ∣
By symmetry, we may assume that R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 Otherwise, it suffices to interchange the indices and to replace T T T T ∗ T^* T ∗ r ( B 1 ) ≤ r ( B 2 ) r(B_1)\leq r(B_2) r ( B 1 ) ≤ r ( B 2 ) 2 M ∗ B 1 ⊂ 2 M ∗ B 2 2M_*B_1\subset2M_*B_2 2 M ∗ B 1 ⊂ 2 M ∗ B 2
Let κ ∈ ( 0 , 1 ) \kappa\in(0,1) κ ∈ ( 0 , 1 ) § 7.1 . We fix the generations k 1 k_1 k 1 k 2 k_2 k 2 Lemma A.4 , applied to B 1 B_1 B 1 B 2 B_2 B 2 κ \kappa κ i , j = 1 , 2 i,j=1,2 i , j = 1 , 2 w j ∈ Ω w_j\in\Omega w j ∈ Ω Q i ( w j ) Q_i(w_j) Q i ( w j ) D k i ( w j ) \mathcal D_{k_i}(w_j) D k i ( w j ) B i B_i B i Q i ( w j ) Q_i(w_j) Q i ( w j ) w j w_j w j
As is customary in T b Tb T b [72 ] , we adopt the notion of good and bad cubes, depending on the exponent α : = γ / ( 2 ( n + γ ) ) ∈ ( 0 , 1 ) \alpha:=\gamma/(2(n+\gamma))\in(0,1) α := γ / ( 2 ( n + γ )) ∈ ( 0 , 1 ) η > 0 \eta>0 η > 0 d = log 2 ( D ) \mathfrak{d} = \log_{2}(D) d = log 2 ( D ) D D D X X X ℓ ∈ ( 0 , 1 ) \ell\in(0,1) ℓ ∈ ( 0 , 1 ) Theorem 2.9 .
Definition 6.1 (Good and bad cubes). Let w = ( w 1 , w 2 ) \mathbf{w}=(w_{1},w_{2}) w = ( w 1 , w 2 ) m ∈ N m\in\mathbb{N} m ∈ N Q ∈ D ( w 1 ) Q\in\mathcal{D}(w_{1}) Q ∈ D ( w 1 ) R ∈ D ( w 2 ) R\in\mathcal{D}(w_{2}) R ∈ D ( w 2 ) ℓ ( R ) ≥ ℓ − m ℓ ( Q ) \ell(R)\geq\ell^{-m}\ell(Q) ℓ ( R ) ≥ ℓ − m ℓ ( Q )
dist ( Q , R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α and dist ( Q , X ∖ R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α . \operatorname{dist}(Q,R)\leq\ell(Q)^{\alpha}\ell(R)^{1-\alpha}
\qquad\text{and}\qquad
\operatorname{dist}(Q,X\setminus R)\leq\ell(Q)^{\alpha}\ell(R)^{1-\alpha}. dist ( Q , R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α and dist ( Q , X ∖ R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α . Otherwise Q Q Q D 1 b ( w , m ) \mathcal{D}^{\mathrm{b}}_{1}(\mathbf{w},m) D 1 b ( w , m ) D 1 g ( w , m ) \mathcal{D}^{\mathrm{g}}_{1}(\mathbf{w},m) D 1 g ( w , m ) m m m D 2 b ( w , m ) \mathcal{D}^{\mathrm{b}}_{2}(\mathbf{w},m) D 2 b ( w , m ) D 2 g ( w , m ) \mathcal{D}^{\mathrm{g}}_{2}(\mathbf{w},m) D 2 g ( w , m )
By Lemma A.4 , for 0 < κ ≪ 1 0<\kappa\ll1 0 < κ ≪ 1 m m m § 7.1 , there exists L = L ( κ ) > 0 L=L(\kappa)>0 L = L ( κ ) > 0 Lemma A.4 , such that, denoting P Ω 2 : = P ⊗ P \mathbb{P}^{\Omega^{2}}:=\mathbb{P}\otimes\mathbb{P} P Ω 2 := P ⊗ P
P Ω 2 ( { w = ( w 1 , w 2 ) ∈ Ω 2 : for every i = 1 , 2 , j = 1 , 2 there exists Q i ( w j ) ∈ D ( w j ) , \mathbb{P}^{\Omega^{2}}\left(\left\{\mathbf{w}=(w_{1},w_{2})\in\Omega^{2}: \text{for every }i=1,2,\ j=1,2\text{ there exists }Q_{i}(w_{j})\in\mathcal{D}(w_{j}),
\right.\right. P Ω 2 ( { w = ( w 1 , w 2 ) ∈ Ω 2 : for every i = 1 , 2 , j = 1 , 2 there exists Q i ( w j ) ∈ D ( w j ) , such that 6 M ∗ B i ⊂ Q i ( w j ) , Q 1 ( w j ) ⊂ Q 2 ( w j ) for j = 1 , 2 , \left.\left.
\text{such that }6M_{*}B_{i}\subset Q_{i}(w_{j}),\ Q_{1}(w_{j})\subset Q_{2}(w_{j})\text{ for }j=1,2,
\right.\right. such that 6 M ∗ B i ⊂ Q i ( w j ) , Q 1 ( w j ) ⊂ Q 2 ( w j ) for j = 1 , 2 , diam Q i ( w j ) ≤ L r ( B i ) , and Q i ( w j ) ∈ D j g ( w , m ) } ) > 1 − κ . (172) \left.\left.
\operatorname{diam}Q_{i}(w_{j})\leq Lr(B_{i}),\text{ and }Q_{i}(w_{j})\in\mathcal{D}^{\mathrm{g}}_{j}(\mathbf{w},m)\right\}\right)>1-\kappa. \tag*{(172)} diam Q i ( w j ) ≤ L r ( B i ) , and Q i ( w j ) ∈ D j g ( w , m ) } ) > 1 − κ . (172) Hence, we denote by G ⊂ Ω 2 \mathcal{G}\subset\Omega^{2} G ⊂ Ω 2 (172) . For the rest of the argument, we hence fix ( w 1 , w 2 ) ∈ G (w_{1},w_{2})\in\mathcal{G} ( w 1 , w 2 ) ∈ G Q i ( w j ) Q_{i}(w_{j}) Q i ( w j ) i , j = 1 , 2 i,j=1,2 i , j = 1 , 2 G \mathcal{G} G
P G Ω 2 ( A ) : = P Ω 2 ( A ∩ G ) P Ω 2 ( G ) , for A ⊂ Ω 2 , \mathbb{P}_{\mathcal{G}}^{\Omega^{2}}(A):=\frac{\mathbb{P}^{\Omega^{2}}(A\cap\mathcal{G})}{\mathbb{P}^{\Omega^{2}}(\mathcal{G})},\qquad\text{for }A\subset\Omega^{2}, P G Ω 2 ( A ) := P Ω 2 ( G ) P Ω 2 ( A ∩ G ) , for A ⊂ Ω 2 , and its associated expectations.
Adapted martingale differences. Let w ∈ G \mathbf{w}\in\mathcal{G} w ∈ G i = 1 , 2 i=1,2 i = 1 , 2 Q 2 ( w i ) Q_{2}(w_{i}) Q 2 ( w i ) D ( w i ) \mathcal{D}(w_{i}) D ( w i ) G \mathcal{G} G B 1 ⊂ Q 1 ( w i ) ⊂ Q 2 ( w i ) B_{1}\subset Q_{1}(w_{i})\subset Q_{2}(w_{i}) B 1 ⊂ Q 1 ( w i ) ⊂ Q 2 ( w i ) B 2 ⊂ Q 2 ( w i ) B_{2}\subset Q_{2}(w_{i}) B 2 ⊂ Q 2 ( w i ) L 2 ( ∣ ν i ∣ ) L^{2}(|\nu_{i}|) L 2 ( ∣ ν i ∣ ) X ∖ B i X\setminus B_{i} X ∖ B i L 2 ( ∣ ν ^ i ∣ ) L^{2}(|\widehat{\nu}_{i}|) L 2 ( ∣ ν i ∣ ) L 2 ( ∣ ν i ∣ ) L^{2}(|\nu_{i}|) L 2 ( ∣ ν i ∣ ) D ( w i ) \mathcal{D}(w_{i}) D ( w i ) Q 2 ( w i ) Q_{2}(w_{i}) Q 2 ( w i )
Definition 6.2 (“Terminal” and “transit” cubes). Let i = 1 , 2 i=1,2 i = 1 , 2 Q ∈ D ( w i ) Q\in\mathcal{D}(w_{i}) Q ∈ D ( w i ) Q ⊂ W i ( w i ) Q\subset\mathcal{W}_{i}(w_{i}) Q ⊂ W i ( w i ) D i t e r m \mathcal{D}_{i}^{\mathrm{term}} D i term D i t r \mathcal{D}_{i}^{\mathrm{tr}} D i tr
Since B i ⊂ Q 2 ( w i ) B_{i}\subset Q_{2}(w_{i}) B i ⊂ Q 2 ( w i ) Q 0 = Q 2 ( w i ) Q_{0}=Q_{2}(w_{i}) Q 0 = Q 2 ( w i ) B = B i B=B_{i} B = B i ν = ν i \nu=\nu_{i} ν = ν i St ( Q 2 ( w i ) , w i ) \operatorname{St}(Q_{2}(w_{i}),w_{i}) St ( Q 2 ( w i ) , w i ) W i ( w i ) \mathcal{W}_{i}(w_{i}) W i ( w i ) Q ⊂ Q 2 ( w i ) Q\subset Q_{2}(w_{i}) Q ⊂ Q 2 ( w i ) Q 2 ( w i ) Q_{2}(w_{i}) Q 2 ( w i ) ∅ ≠ F i ( w i ) ⊂ B i ∖ W i ( w i ) ⊂ Q 2 ( w i ) ∖ W i ( w i ) \varnothing\neq F_{i}(w_{i})\subset B_{i}\setminus\mathcal{W}_{i}(w_{i})\subset Q_{2}(w_{i})\setminus\mathcal{W}_{i}(w_{i}) ∅ = F i ( w i ) ⊂ B i ∖ W i ( w i ) ⊂ Q 2 ( w i ) ∖ W i ( w i )
m − 1 μ ( Q ) ≤ ∣ ν i ∣ ( Q ) ≤ m μ ( Q ) and ∣ ν i ( Q ) ∣ ≥ ∣ ν i ∣ ( Q ) 2 ε 1 , Q ∈ D i t r , Q ⊂ Q 2 ( w i ) . \mathfrak{m}^{-1}\mu(Q)\leq|\nu_{i}|(Q)\leq\mathfrak{m}\mu(Q)
\qquad\text{and}\qquad
|\nu_{i}(Q)|\geq\frac{|\nu_{i}|(Q)}{2\varepsilon_{1}},
\qquad Q\in\mathcal{D}_{i}^{\mathrm{tr}},\quad Q\subset Q_{2}(w_{i}). m − 1 μ ( Q ) ≤ ∣ ν i ∣ ( Q ) ≤ m μ ( Q ) and ∣ ν i ( Q ) ∣ ≥ 2 ε 1 ∣ ν i ∣ ( Q ) , Q ∈ D i tr , Q ⊂ Q 2 ( w i ) . In particular, since ν i = b ^ i ∣ ν i ∣ \nu_{i}=\widehat{b}_{i}|\nu_{i}| ν i = b i ∣ ν i ∣ ∣ ⟨ b ^ i ⟩ ∣ ν i ∣ , Q ∣ ≥ ( 2 ε 1 ) − 1 |\langle\widehat{b}_{i}\rangle_{|\nu_{i}|,Q}|\geq(2\varepsilon_{1})^{-1} ∣ ⟨ b i ⟩ ∣ ν i ∣ , Q ∣ ≥ ( 2 ε 1 ) − 1 ∣ ν i ∣ ( Q ) > 0 |\nu_{i}|(Q)>0 ∣ ν i ∣ ( Q ) > 0
Remark 6.3. Let Q ∈ D i t r Q\in\mathcal{D}_{i}^{\mathrm{tr}} Q ∈ D i tr P P P Q Q Q C : = 2 C 1 ℓ θ − 1 C:=2C_{1}\ell^{\theta-1} C := 2 C 1 ℓ θ − 1
∣ ν i ∣ ( B ( z , t ) ) ≲ t n , for z ∈ P , t ≥ ℓ ( P ) . |\nu_{i}|(B(z,t))\lesssim t^{n},\qquad\text{for }z\in P,\quad t\geq\ell(P). ∣ ν i ∣ ( B ( z , t )) ≲ t n , for z ∈ P , t ≥ ℓ ( P ) . Indeed, let x ∈ Q ∖ W i ( w i ) x \in Q \setminus\mathcal{W}_i(w_i) x ∈ Q ∖ W i ( w i ) d ( x , z ) ≤ diam Q ≤ 2 C 1 ℓ ( Q ) = C ℓ ( P ) ≤ C t d(x,z) \le\operatorname{diam} Q \le2C_1\ell(Q) = C\ell(P) \le Ct d ( x , z ) ≤ diam Q ≤ 2 C 1 ℓ ( Q ) = C ℓ ( P ) ≤ C t B ( z , t ) ⊂ B ( x , ( C + 1 ) t ) B(z,t) \subset B(x,(C+1)t) B ( z , t ) ⊂ B ( x , ( C + 1 ) t ) x ∉ H i , R i x \notin\mathcal{H}_{i,R_i} x ∈ / H i , R i H i , R i ⊂ W i ( w i ) \mathcal{H}_{i,R_i} \subset\mathcal{W}_i(w_i) H i , R i ⊂ W i ( w i ) (52) and (53) give ∣ ν ^ i ∣ ( B ( x , r ) ) ≤ m r n |\widehat{\nu}_i|(B(x,r)) \le\mathbf{m}r^n ∣ ν i ∣ ( B ( x , r )) ≤ m r n r > 0 r > 0 r > 0
∣ ν i ∣ ( B ( z , t ) ) ≤ ∣ ν ^ i ∣ ( B ( x , ( C + 1 ) t ) ) + μ ( B ( z , t ) ) ≤ ( m ( C + 1 ) n + ε 0 ) t n . |\nu_i|(B(z,t)) \le|\widehat{\nu}_i|(B(x,(C+1)t))+\mu(B(z,t)) \le\bigl(\mathbf{m}(C+1)^n+\varepsilon_0\bigr)t^n. ∣ ν i ∣ ( B ( z , t )) ≤ ∣ ν i ∣ ( B ( x , ( C + 1 ) t )) + μ ( B ( z , t )) ≤ ( m ( C + 1 ) n + ε 0 ) t n . Let m ∈ M + ( X ) \mathbf{m} \in M_+(X) m ∈ M + ( X ) f ∈ L l o c 1 ( m ) f \in L^1_{\mathrm{loc}}(\mathbf{m}) f ∈ L loc 1 ( m ) Q Q Q X X X m ( Q ) ≠ 0 \mathbf{m}(Q)\ne0 m ( Q ) = 0
⟨ f ⟩ m , Q : = 1 m ( Q ) ∫ Q f d m . (173) \langle f\rangle_{\mathbf{m},Q}:=\frac{1}{\mathbf{m}(Q)}\int_Q f\,d\mathbf{m}. \tag*{(173)} ⟨ f ⟩ m , Q := m ( Q ) 1 ∫ Q f d m . (173) We also note that
∫ Q 2 ( w i ) b ^ i d ∣ ν i ∣ = μ ( Q 2 ( w i ) ) , μ ( Q 2 ( w i ) ) ≤ ∣ ν i ∣ ( Q 2 ( w i ) ) ≤ E 1 μ ( Q 2 ( w i ) ) , \int_{Q_2(w_i)}\widehat{b}_i\,d|\nu_i|=\mu(Q_2(w_i)),\qquad\mu(Q_2(w_i))\le|\nu_i|(Q_2(w_i))\le\mathcal{E}_1\mu(Q_2(w_i)), ∫ Q 2 ( w i ) b i d ∣ ν i ∣ = μ ( Q 2 ( w i )) , μ ( Q 2 ( w i )) ≤ ∣ ν i ∣ ( Q 2 ( w i )) ≤ E 1 μ ( Q 2 ( w i )) , and define the top averages by
E i f : = χ Q 2 ( w i ) b ^ i ∫ Q 2 ( w i ) f d ∣ ν i ∣ μ ( Q 2 ( w i ) ) and E i ∗ f : = χ Q 2 ( w i ) ∫ Q 2 ( w i ) b ^ i f d ∣ ν i ∣ μ ( Q 2 ( w i ) ) . (174) E_i f:=\chi_{Q_2(w_i)}\widehat{b}_i\frac{\int_{Q_2(w_i)}f\,d|\nu_i|}{\mu(Q_2(w_i))}
\qquad\text{and}\qquad
E_i^*f:=\chi_{Q_2(w_i)}\frac{\int_{Q_2(w_i)}\widehat{b}_i f\,d|\nu_i|}{\mu(Q_2(w_i))}. \tag*{(174)} E i f := χ Q 2 ( w i ) b i μ ( Q 2 ( w i )) ∫ Q 2 ( w i ) f d ∣ ν i ∣ and E i ∗ f := χ Q 2 ( w i ) μ ( Q 2 ( w i )) ∫ Q 2 ( w i ) b i f d ∣ ν i ∣ . (174) For i = 1 , 2 i=1,2 i = 1 , 2 A ⊂ X A\subset X A ⊂ X ∣ ν i ∣ ( A ) > 0 |\nu_i|(A)>0 ∣ ν i ∣ ( A ) > 0 ⟨ f ⟩ i , A : = ⟨ f ⟩ ∣ ν i ∣ , A \langle f\rangle_{i,A}:=\langle f\rangle_{|\nu_i|,A} ⟨ f ⟩ i , A := ⟨ f ⟩ ∣ ν i ∣ , A
For Q ∈ D i Q\in\mathcal{D}_i Q ∈ D i Ch ( Q ) = ∅ \operatorname{Ch}(Q)=\varnothing Ch ( Q ) = ∅ ∣ ν i ∣ ( Q ) = 0 |\nu_i|(Q)=0 ∣ ν i ∣ ( Q ) = 0 ∣ ν i ∣ ( Q ) > 0 |\nu_i|(Q)>0 ∣ ν i ∣ ( Q ) > 0 Ch ( Q ) \operatorname{Ch}(Q) Ch ( Q ) P P P Q Q Q ∣ ν i ∣ ( P ) > 0 |\nu_i|(P)>0 ∣ ν i ∣ ( P ) > 0 Q ∈ D i t r Q\in\mathcal{D}_i^{\mathrm{tr}} Q ∈ D i tr f ∈ L l o c 1 ( ∣ ν i ∣ ) f\in L^1_{\mathrm{loc}}(|\nu_i|) f ∈ L loc 1 ( ∣ ν i ∣ )
Δ i , Q f ( x ) : = { 0 , for x ∈ X ∖ ⋃ P ∈ Ch ( Q ) P , ( ⟨ f ⟩ i , P ⟨ b ^ i ⟩ i , P − ⟨ f ⟩ i , Q ⟨ b ^ i ⟩ i , Q ) b ^ i ( x ) , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i t r , f ( x ) − ⟨ f ⟩ i , Q ⟨ b ^ i ⟩ i , Q b ^ i ( x ) , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i t e r m . (175) \Delta_{i,Q}f(x):=
\begin{cases}
0, & \text{for }x\in X\setminus\bigcup_{P\in\operatorname{Ch}(Q)}P,\\
\left(\dfrac{\langle f\rangle_{i,P}}{\langle\widehat{b}_i\rangle_{i,P}}-\dfrac{\langle f\rangle_{i,Q}}{\langle\widehat{b}_i\rangle_{i,Q}}\right)\widehat{b}_i(x), & \text{for }x\in P,\ \text{for some }P\in\operatorname{Ch}(Q)\cap\mathcal{D}_i^{\mathrm{tr}},\\
f(x)-\dfrac{\langle f\rangle_{i,Q}}{\langle\widehat{b}_i\rangle_{i,Q}}\widehat{b}_i(x), & \text{for }x\in P,\ \text{for some }P\in\operatorname{Ch}(Q)\cap\mathcal{D}_i^{\mathrm{term}}.
\end{cases} \tag*{(175)} Δ i , Q f ( x ) := ⎩ ⎨ ⎧ 0 , ( ⟨ b i ⟩ i , P ⟨ f ⟩ i , P − ⟨ b i ⟩ i , Q ⟨ f ⟩ i , Q ) b i ( x ) , f ( x ) − ⟨ b i ⟩ i , Q ⟨ f ⟩ i , Q b i ( x ) , for x ∈ X ∖ ⋃ P ∈ Ch ( Q ) P , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i tr , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i term . (175) We also set Δ i , Q : = 0 \Delta_{i,Q}:=0 Δ i , Q := 0 Q Q Q Q ⊄ Q 2 ( w i ) Q\not\subset Q_2(w_i) Q ⊂ Q 2 ( w i )
A standard computation, see [77 Lemma 5.10] for a close analogue, shows that
Δ i , Q Δ i , R = { Δ i , Q , if R = Q , 0 , if R ∈ D i t r , R ≠ Q , (176) \Delta_{i,Q}\Delta_{i,R}=
\begin{cases}
\Delta_{i,Q}, & \text{if }R=Q,\\
0, & \text{if }R\in\mathcal{D}_i^{\mathrm{tr}},\ R\ne Q,
\end{cases} \tag*{(176)} Δ i , Q Δ i , R = { Δ i , Q , 0 , if R = Q , if R ∈ D i tr , R = Q , (176) and every f ∈ L 2 ( ∣ ν i ∣ ) f\in L^2(|\nu_i|) f ∈ L 2 ( ∣ ν i ∣ ) Q 2 ( w i ) Q_2(w_i) Q 2 ( w i )
f = E i f + ∑ Q ∈ D i t r Δ i , Q f , (177) f=E_i f+\sum_{Q\in\mathcal{D}_i^{\mathrm{tr}}}\Delta_{i,Q}f, \tag*{(177)} f = E i f + Q ∈ D i tr ∑ Δ i , Q f , (177) where the sum is unconditionally convergent in L 2 ( ∣ ν i ∣ ) L^2(|\nu_i|) L 2 ( ∣ ν i ∣ ) [77 Lemma 5.11] , there exists a constant c ~ = c ~ ( E 1 ) > 0 \widetilde{c}=\widetilde{c}(\mathcal{E}_1)>0 c = c ( E 1 ) > 0
c ~ − 1 ∥ f ∥ L 2 ( ∣ ν i ∣ ) 2 ≤ ∥ E i f ∥ L 2 ( ∣ ν i ∣ ) 2 + ∑ Q ∈ D i t r ∥ Δ i , Q f ∥ L 2 ( ∣ ν i ∣ ) 2 ≤ c ~ ∥ f ∥ L 2 ( ∣ ν i ∣ ) 2 . (178) \widetilde{c}^{-1}\|f\|_{L^2(|\nu_i|)}^2
\le\|E_i f\|_{L^2(|\nu_i|)}^2+
\sum_{Q\in\mathcal{D}_i^{\mathrm{tr}}}\|\Delta_{i,Q}f\|_{L^2(|\nu_i|)}^2
\le\widetilde{c}\|f\|_{L^2(|\nu_i|)}^2. \tag*{(178)} c − 1 ∥ f ∥ L 2 ( ∣ ν i ∣ ) 2 ≤ ∥ E i f ∥ L 2 ( ∣ ν i ∣ ) 2 + Q ∈ D i tr ∑ ∥ Δ i , Q f ∥ L 2 ( ∣ ν i ∣ ) 2 ≤ c ∥ f ∥ L 2 ( ∣ ν i ∣ ) 2 . (178) Here we use ∣ b ^ i ∣ = 1 |\widehat{b}_i|=1 ∣ b i ∣ = 1 ∣ ν i ∣ |\nu_i| ∣ ν i ∣ ∣ ⟨ b ^ i ⟩ i , Q ∣ |\langle\widehat{b}_i\rangle_{i,Q}| ∣ ⟨ b i ⟩ i , Q ∣ c ~ \widetilde{c} c ( 2 E 1 ) − 1 (2\mathcal{E}_1)^{-1} ( 2 E 1 ) − 1 Δ i , Q \Delta_{i,Q} Δ i , Q
Δ i , Q ∗ f ( x ) : = { 0 , for x ∈ X ∖ ⋃ P ∈ Ch ( Q ) P , ⟨ f b ^ i ⟩ i , P ⟨ b ^ i ⟩ i , P − ⟨ f b ^ i ⟩ i , Q ⟨ b ^ i ⟩ i , Q , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i t r , f ( x ) − ⟨ f b ^ i ⟩ i , Q ⟨ b ^ i ⟩ i , Q , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i t e r m . \Delta_{i,Q}^{*}f(x):=
\begin{cases}
0, & \text{for }x\in X\setminus\bigcup_{P\in\operatorname{Ch}(Q)}P,\\
\dfrac{\langle f\widehat b_i\rangle_{i,P}}{\langle\widehat b_i\rangle_{i,P}}-\dfrac{\langle f\widehat b_i\rangle_{i,Q}}{\langle\widehat b_i\rangle_{i,Q}}, & \text{for }x\in P,\ \text{for some }P\in\operatorname{Ch}(Q)\cap\mathbb{D}_i^{\mathrm{tr}},\\
f(x)-\dfrac{\langle f\widehat b_i\rangle_{i,Q}}{\langle\widehat b_i\rangle_{i,Q}}, & \text{for }x\in P,\ \text{for some }P\in\operatorname{Ch}(Q)\cap\mathbb{D}_i^{\mathrm{term}}.
\end{cases} Δ i , Q ∗ f ( x ) := ⎩ ⎨ ⎧ 0 , ⟨ b i ⟩ i , P ⟨ f b i ⟩ i , P − ⟨ b i ⟩ i , Q ⟨ f b i ⟩ i , Q , f ( x ) − ⟨ b i ⟩ i , Q ⟨ f b i ⟩ i , Q , for x ∈ X ∖ ⋃ P ∈ Ch ( Q ) P , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i tr , for x ∈ P , for some P ∈ Ch ( Q ) ∩ D i term . which is such that
Δ i , Q ∗ Δ i , R ∗ = { Δ i , Q ∗ , if R = Q , 0 , if R ∈ D i t r , R ≠ Q . \Delta_{i,Q}^{*}\Delta_{i,R}^{*}=
\begin{cases}
\Delta_{i,Q}^{*}, & \text{if }R=Q,\\
0, & \text{if }R\in\mathbb{D}_i^{\mathrm{tr}},\ R\ne Q.
\end{cases} Δ i , Q ∗ Δ i , R ∗ = { Δ i , Q ∗ , 0 , if R = Q , if R ∈ D i tr , R = Q . Similarly to (177) , a function g ∈ L 2 ( ∣ ν i ∣ ) g\in L^{2}(|\nu_i|) g ∈ L 2 ( ∣ ν i ∣ ) Q 2 ( w i ) Q_2(w_i) Q 2 ( w i )
g = E i ∗ g + ∑ Q ∈ D i t r Δ i , Q ∗ g (179) g=E_i^{*}g+\sum_{Q\in\mathbb{D}_i^{\mathrm{tr}}}\Delta_{i,Q}^{*}g \tag*{(179)} g = E i ∗ g + Q ∈ D i tr ∑ Δ i , Q ∗ g (179) where the sum is unconditionally convergent in L 2 ( ∣ ν i ∣ ) L^{2}(|\nu_i|) L 2 ( ∣ ν i ∣ ) (178) holds.
Projection operators. Next, we define the projection operators associated with good and bad cubes.
Definition 6.4. Let w = ( w 1 , w 2 ) ∈ Ω 2 w=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2 f ∈ L 2 ( ∣ ν 1 ∣ ) f\in L^{2}(|\nu_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν 2 ∣ ) g\in L^{2}(|\nu_2|) g ∈ L 2 ( ∣ ν 2 ∣ )
P G 1 ( w ) f : = ∑ Q ∈ D 1 G ( w ) ∩ D 1 t r Δ 1 , Q f , P B 1 ( w ) f : = ∑ Q ∈ D 1 B ( w ) ∩ D 1 t r Δ 1 , Q f , \mathcal{P}_{\mathcal{G}}^{1}(w)f:=\sum_{Q\in\mathbb{D}_{1}^{\mathcal{G}}(w)\cap\mathbb{D}_{1}^{\mathrm{tr}}}\Delta_{1,Q}f,
\qquad
\mathcal{P}_{\mathcal{B}}^{1}(w)f:=\sum_{Q\in\mathbb{D}_{1}^{\mathcal{B}}(w)\cap\mathbb{D}_{1}^{\mathrm{tr}}}\Delta_{1,Q}f, P G 1 ( w ) f := Q ∈ D 1 G ( w ) ∩ D 1 tr ∑ Δ 1 , Q f , P B 1 ( w ) f := Q ∈ D 1 B ( w ) ∩ D 1 tr ∑ Δ 1 , Q f , and
P G 2 ( w ) g : = ∑ R ∈ D 2 G ( w ) ∩ D 2 t r Δ 2 , R g , P B 2 ( w ) g : = ∑ R ∈ D 2 B ( w ) ∩ D 2 t r Δ 2 , R g . \mathcal{P}_{\mathcal{G}}^{2}(w)g:=\sum_{R\in\mathbb{D}_{2}^{\mathcal{G}}(w)\cap\mathbb{D}_{2}^{\mathrm{tr}}}\Delta_{2,R}g,
\qquad
\mathcal{P}_{\mathcal{B}}^{2}(w)g:=\sum_{R\in\mathbb{D}_{2}^{\mathcal{B}}(w)\cap\mathbb{D}_{2}^{\mathrm{tr}}}\Delta_{2,R}g. P G 2 ( w ) g := R ∈ D 2 G ( w ) ∩ D 2 tr ∑ Δ 2 , R g , P B 2 ( w ) g := R ∈ D 2 B ( w ) ∩ D 2 tr ∑ Δ 2 , R g . Given the probability space ( Ω , P Ω ) (\Omega,\mathbb{P}^{\Omega}) ( Ω , P Ω ) E w : = E w Ω \mathbb{E}_{w}:=\mathbb{E}_{w}^{\Omega} E w := E w Ω w ∈ Ω w\in\Omega w ∈ Ω E ~ w \widetilde{\mathbb{E}}_{w} E w ( w 1 , w 2 ) ∈ Ω × Ω (w_1,w_2)\in\Omega\times\Omega ( w 1 , w 2 ) ∈ Ω × Ω P G Ω 2 \mathbb{P}_{\mathcal{G}}^{\Omega^{2}} P G Ω 2
Lemma 6.5. There is a quantity ε B ( m ) → 0 \varepsilon_{\mathcal{B}}(m)\to0 ε B ( m ) → 0 m → ∞ m\to\infty m → ∞ w 1 w_1 w 1
E w 2 ∥ ∑ Q ∈ D 1 B ( w 1 , w 2 ) ∩ D 1 t r Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) ≤ ε B ( m ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) . \mathbb{E}_{w_2}\left\|
\sum_{Q\in\mathbb{D}_{1}^{\mathcal{B}}(w_1,w_2)\cap\mathbb{D}_{1}^{\mathrm{tr}}}
\Delta_{1,Q}f
\right\|_{L^{2}(|\nu_1|)}
\leq\varepsilon_{\mathcal{B}}(m)\|f\|_{L^{2}(|\nu_1|)}. E w 2 Q ∈ D 1 B ( w 1 , w 2 ) ∩ D 1 tr ∑ Δ 1 , Q f L 2 ( ∣ ν 1 ∣ ) ≤ ε B ( m ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) . The analogous estimate holds with the indices interchanged.
Proof. Fix w 1 ∈ Ω w_1\in\Omega w 1 ∈ Ω Q 2 ( w 1 ) Q_2(w_1) Q 2 ( w 1 ) D 1 t r \mathbb{D}_1^{\mathrm{tr}} D 1 tr D 1 t e r m \mathbb{D}_1^{\mathrm{term}} D 1 term Δ 1 , Q \Delta_{1,Q} Δ 1 , Q w 1 w_1 w 1 B 1 ∪ B 2 ⊄ Q 2 ( w 1 ) B_1\cup B_2\not\subset Q_2(w_1) B 1 ∪ B 2 ⊂ Q 2 ( w 1 ) Δ 1 , Q : = 0 \Delta_{1,Q}:=0 Δ 1 , Q := 0 Q Q Q Δ 1 , Q \Delta_{1,Q} Δ 1 , Q w 1 ∈ Ω w_1\in\Omega w 1 ∈ Ω (175) on G \mathcal{G} G (178) uniformly in w 1 w_1 w 1 w 2 w_2 w 2
The argument leading to (A.11) applies to every Q ∈ D ( w 1 ) Q\in\mathbb{D}(w_1) Q ∈ D ( w 1 )
P w 2 ( { Q is bad } ) ≤ C 2 ( 1 + 2 C 1 ) η 1 − ℓ α η ℓ α η m = : p B ( m ) . \mathbb{P}_{w_2}(\{Q\text{ is bad}\})
\leq
\frac{C_2(1+2C_1)^{\eta}}{1-\ell^{\alpha\eta}}\ell^{\alpha\eta m}
=:p_{\mathcal{B}}(m). P w 2 ({ Q is bad }) ≤ 1 − ℓ α η C 2 ( 1 + 2 C 1 ) η ℓ α η m =: p B ( m ) . Moreover, ∫ Δ 1 , Q f d ∣ ν 1 ∣ = 0 \int\Delta_{1,Q}f\,d|\nu_1|=0 ∫ Δ 1 , Q f d ∣ ν 1 ∣ = 0 E 1 Δ 1 , Q f = 0 E_1\Delta_{1,Q}f=0 E 1 Δ 1 , Q f = 0 (176) and the lower bound in (178) , we have
∥ ∑ Q ∈ Q Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≤ c ~ ∑ Q ∈ Q ∥ Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 , for every Q ⊂ D 1 t r . \left\|\sum_{Q\in\mathcal{Q}}\Delta_{1,Q}f\right\|_{L^{2}(|\nu_1|)}^{2}
\leq\widetilde c\sum_{Q\in\mathcal{Q}}\|\Delta_{1,Q}f\|_{L^{2}(|\nu_1|)}^{2},
\qquad\text{for every }\mathcal{Q}\subset\mathbb{D}_{1}^{\mathrm{tr}}. Q ∈ Q ∑ Δ 1 , Q f L 2 ( ∣ ν 1 ∣ ) 2 ≤ c Q ∈ Q ∑ ∥ Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 , for every Q ⊂ D 1 tr . If B 1 ∪ B 2 ⊄ Q 2 ( w 1 ) B_1 \cup B_2 \not\subset Q_2(w_1) B 1 ∪ B 2 ⊂ Q 2 ( w 1 ) Q = D 1 B ( w 1 , w 2 ) ∩ D 1 t r Q = \mathbb{D}_1^{\mathcal{B}}(w_1,w_2) \cap\mathbb{D}_1^{\mathrm{tr}} Q = D 1 B ( w 1 , w 2 ) ∩ D 1 tr (178) give
E w 2 ∥ ∑ Q ∈ D 1 B ( w 1 , w 2 ) ∩ D 1 t r Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≤ c ~ ∑ Q ∈ D 1 t r P w 2 ( { Q is bad } ) ∥ Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≤ c ~ 2 p B ( m ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) 2 . \mathbb{E}_{w_2}\left\|\sum_{Q\in\mathbb{D}_1^{\mathcal{B}}(w_1,w_2)\cap\mathbb{D}_1^{\mathrm{tr}}}\Delta_{1,Q}f\right\|_{L^2(|\nu_1|)}^2
\leq\widetilde{c}\sum_{Q\in\mathbb{D}_1^{\mathrm{tr}}}\mathbb{P}_{w_2}(\{Q\text{ is bad}\})\|\Delta_{1,Q}f\|_{L^2(|\nu_1|)}^2
\leq\widetilde{c}^{\,2}p_B(m)\|f\|_{L^2(|\nu_1|)}^2. E w 2 Q ∈ D 1 B ( w 1 , w 2 ) ∩ D 1 tr ∑ Δ 1 , Q f L 2 ( ∣ ν 1 ∣ ) 2 ≤ c Q ∈ D 1 tr ∑ P w 2 ({ Q is bad }) ∥ Δ 1 , Q f ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≤ c 2 p B ( m ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) 2 . The lemma follows from the Cauchy-Schwarz inequality, with ε B ( m ) : = c ~ p B ( m ) 1 / 2 \varepsilon_B(m):=\widetilde{c}\,p_B(m)^{1/2} ε B ( m ) := c p B ( m ) 1/2
As an immediate corollary, we get the following bounds on projections and their associated conditional expectations.
Corollary 6.6. Let w = ( w 1 , w 2 ) ∈ Ω 2 \mathbf{w}=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2 f ∈ L 2 ( ∣ ν 1 ∣ ) f\in L^2(|\nu_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν 2 ∣ ) g\in L^2(|\nu_2|) g ∈ L 2 ( ∣ ν 2 ∣ )
E ~ w ∥ P B 1 ( w ) f ∥ L 2 ( ∣ ν 1 ∣ ) ≤ ε B ( m ) P Ω 2 ( G ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) and E ~ w ∥ P B 2 ( w ) g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ ε B ( m ) P Ω 2 ( G ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (180) \widetilde{\mathbb{E}}_{\mathbf{w}}\|P_B^1(\mathbf{w})f\|_{L^2(|\nu_1|)}
\leq\frac{\varepsilon_B(m)}{\mathbb{P}_{\Omega^2}(\mathcal{G})}\|f\|_{L^2(|\nu_1|)}
\quad\text{and}\quad
\widetilde{\mathbb{E}}_{\mathbf{w}}\|P_B^2(\mathbf{w})g\|_{L^2(|\nu_2|)}
\leq\frac{\varepsilon_B(m)}{\mathbb{P}_{\Omega^2}(\mathcal{G})}\|g\|_{L^2(|\nu_2|)}. \tag*{(180)} E w ∥ P B 1 ( w ) f ∥ L 2 ( ∣ ν 1 ∣ ) ≤ P Ω 2 ( G ) ε B ( m ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) and E w ∥ P B 2 ( w ) g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ P Ω 2 ( G ) ε B ( m ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (180) We also have
∥ P G 1 f ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) and ∥ P G 2 g ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) , (181) \|P_{\mathcal{G}}^1f\|_{L^2(|\nu_1|)}\lesssim\|f\|_{L^2(|\nu_1|)}
\quad\text{and}\quad
\|P_{\mathcal{G}}^2g\|_{L^2(|\nu_2|)}\lesssim\|g\|_{L^2(|\nu_2|)}, \tag*{(181)} ∥ P G 1 f ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) and ∥ P G 2 g ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) , (181) where the implicit constants do not depend on ( w 1 , w 2 ) (w_1,w_2) ( w 1 , w 2 )
Proof. The bounds (180) are direct consequences of Lemma 6.5 and the definition of P Ω 2 ( G ) \mathbb{P}_{\Omega^2}(\mathcal{G}) P Ω 2 ( G ) w 1 w_1 w 1 w 2 w_2 w 2 (181) follow from (178) . □
The Key Lemma
Our goal is to bound the norm of T Ξ w , ∣ ν ^ 1 ∣ , ε T_{\Xi_{\mathbf{w}},|\widehat{\nu}_1|,\varepsilon} T Ξ w , ∣ ν 1 ∣ , ε L 2 ( ∣ ν ^ 1 ∣ ) L^2(|\widehat{\nu}_1|) L 2 ( ∣ ν 1 ∣ ) L 2 ( ∣ ν ^ 2 ∣ ) L^2(|\widehat{\nu}_2|) L 2 ( ∣ ν 2 ∣ ) ε \varepsilon ε w ∈ G \mathbf{w}\in\mathcal{G} w ∈ G [72 ] , in Section 7 we split f f f g g g w \mathbf{w} w
We write
∥ T Ξ w , ε ∥ o p : = ∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) . \|T_{\Xi_{\mathbf{w}},\varepsilon}\|_{\mathrm{op}}:=\|T_{\Xi_{\mathbf{w}},|\widehat{\nu}_1|,\varepsilon}\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}. ∥ T Ξ w , ε ∥ op := ∥ T Ξ w , ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) . Since ∣ K Ξ w ( x , y ) ∣ ≲ d ( x , y ) − n |K_{\Xi_{\mathbf{w}}}(x,y)|\lesssim d(x,y)^{-n} ∣ K Ξ w ( x , y ) ∣ ≲ d ( x , y ) − n
∥ T Ξ w , ε ∥ o p ≲ ε − n ∣ ν ^ 1 ∣ ( B 1 ) 1 / 2 ∣ ν ^ 2 ∣ ( B 2 ) 1 / 2 , \|T_{\Xi_{\mathbf{w}},\varepsilon}\|_{\mathrm{op}}\lesssim\varepsilon^{-n}|\widehat{\nu}_1|(B_1)^{1/2}|\widehat{\nu}_2|(B_2)^{1/2}, ∥ T Ξ w , ε ∥ op ≲ ε − n ∣ ν 1 ∣ ( B 1 ) 1/2 ∣ ν 2 ∣ ( B 2 ) 1/2 , uniformly in w \mathbf{w} w
N ε : = sup w ∈ G ∥ T Ξ w , ε ∥ o p < ∞ . (182) N_\varepsilon:=\sup_{\mathbf{w}\in\mathcal{G}}\|T_{\Xi_{\mathbf{w}},\varepsilon}\|_{\mathrm{op}}<\infty. \tag*{(182)} N ε := w ∈ G sup ∥ T Ξ w , ε ∥ op < ∞. (182) The main result of this section is the following.
Lemma 6.7 (Key Lemma). Let 0 < a < 1 / 4 0<a<1/4 0 < a < 1/4 0 < λ , υ < 1 0<\lambda,\upsilon<1 0 < λ , υ < 1 m m m N ε N_\varepsilon N ε (182) . For f ∈ L 2 ( ∣ ν ^ 1 ∣ ) f\in L^2(|\widehat{\nu}_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν ^ 2 ∣ ) g\in L^2(|\widehat{\nu}_2|) g ∈ L 2 ( ∣ ν 2 ∣ )
E ~ w ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( P G 1 f ) , P G 2 g ⟩ ∣ ν ^ 2 ∣ ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + C m , υ ( M Λ a ) 1 / 2 ( 1 + N ε ) + C m ( ( 2 m + 1 ) λ + υ P Ω 2 ( G ) ) 1 / 2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν ^ 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν ^ 2 ∣ ) . (183) \widetilde{\mathbb{E}}_{\mathbf{w}}\left|\left\langle T_{\Xi_{\mathbf{w}},|\widehat{\nu}_1|,\varepsilon}(P_{\mathcal{G}}^1f),P_{\mathcal{G}}^2g\right\rangle_{|\widehat{\nu}_2|}\right|
\leq\left[
C_{m,\lambda,\upsilon}(1+a^{-n})+C_{m,\upsilon}(M\Lambda a)^{1/2}(1+N_\varepsilon)
+C_m\left(\frac{(2m+1)^\lambda+\upsilon}{\mathbb{P}_{\Omega^2}(\mathcal{G})}\right)^{1/2}(1+N_\varepsilon)
\right]\|f\|_{L^2(|\widehat{\nu}_1|)}\|g\|_{L^2(|\widehat{\nu}_2|)}. \tag*{(183)} E w ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( P G 1 f ) , P G 2 g ⟩ ∣ ν 2 ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + C m , υ ( M Λ a ) 1/2 ( 1 + N ε ) + C m ( P Ω 2 ( G ) ( 2 m + 1 ) λ + υ ) 1/2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (183) Here, M Λ ≥ 1 M_{\Lambda} \ge1 M Λ ≥ 1 small-boundary constant in (212) , which is fixed once m m m υ \upsilon υ C m , λ , υ C_{m,\lambda,\upsilon} C m , λ , υ C m , υ C_{m,\upsilon} C m , υ C m C_m C m a a a ε \varepsilon ε
For fixed m m m N ε N_{\varepsilon} N ε (183) can be made arbitrarily small by choosing λ , υ \lambda,\upsilon λ , υ a a a S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε
We remark that the value of the constants in the bound (183) does not depend on the truncation ε \varepsilon ε T T T S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε
Reduction to the localized operator. We recall that the kernel L ε L_{\varepsilon} L ε S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε § 5 , defined in (157) , vanishes unless ( x , y ) ∈ B 2 × B 1 (x,y) \in B_2 \times B_1 ( x , y ) ∈ B 2 × B 1 d ( x , y ) > ε d(x,y) > \varepsilon d ( x , y ) > ε L ε L_{\varepsilon} L ε (158) and (159) , uniformly in ε \varepsilon ε w w w m \mathfrak m m S m , ε ∗ S_{\mathfrak m,\varepsilon}^{*} S m , ε ∗ L ε ( y , x ) L_{\varepsilon}(y,x) L ε ( y , x ) m \mathfrak m m ⟨ S ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ = ⟨ f , S ∣ ν 2 ∣ , ε ∗ g ⟩ ∣ ν 1 ∣ \langle S_{|\nu_1|,\varepsilon}f,g\rangle_{|\nu_2|}=\langle f,S_{|\nu_2|,\varepsilon}^{*}g\rangle_{|\nu_1|} ⟨ S ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ = ⟨ f , S ∣ ν 2 ∣ , ε ∗ g ⟩ ∣ ν 1 ∣
⟨ ⋅ , ⋅ ⟩ : = ⟨ ⋅ , ⋅ ⟩ ∣ ν 1 ∣ , ⟨ ⋅ , ⋅ ⟩ ∗ : = ⟨ ⋅ , ⋅ ⟩ ∣ ν 2 ∣ , ∥ S ε ∥ o p : = ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) . \langle\cdot,\cdot\rangle:=\langle\cdot,\cdot\rangle_{|\nu_1|},\qquad\langle\cdot,\cdot\rangle_{*}:=\langle\cdot,\cdot\rangle_{|\nu_2|},\qquad\|S_{\varepsilon}\|_{\mathrm{op}}:=\|S_{|\nu_1|,\varepsilon}\|_{L^2(|\nu_1|)\to L^2(|\nu_2|)}. ⟨ ⋅ , ⋅ ⟩ := ⟨ ⋅ , ⋅ ⟩ ∣ ν 1 ∣ , ⟨ ⋅ , ⋅ ⟩ ∗ := ⟨ ⋅ , ⋅ ⟩ ∣ ν 2 ∣ , ∥ S ε ∥ op := ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) . By (165) with m i = ∣ ν i ∣ \mathfrak m_i=|\nu_i| m i = ∣ ν i ∣
∥ S ε ∥ o p − ∥ T Ξ w , ε ∥ o p ≲ 1. (184) \|S_{\varepsilon}\|_{\mathrm{op}}-\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op}}\lesssim1. \tag*{(184)} ∥ S ε ∥ op − ∥ T Ξ w , ε ∥ op ≲ 1. (184) and, in particular, ∥ S ε ∥ o p ≲ 1 + N ε \|S_{\varepsilon}\|_{\mathrm{op}}\lesssim1+N_{\varepsilon} ∥ S ε ∥ op ≲ 1 + N ε w ∈ G w\in\mathcal G w ∈ G ∣ ν i ∣ = ∣ ν ^ i ∣ |\nu_i|=|\widehat{\nu}_i| ∣ ν i ∣ = ∣ ν i ∣ B i B_i B i m i = ∣ ν ^ i ∣ \mathfrak m_i=|\widehat{\nu}_i| m i = ∣ ν i ∣ f ∈ L 2 ( ∣ ν 1 ∣ ) f\in L^2(|\nu_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν 2 ∣ ) g\in L^2(|\nu_2|) g ∈ L 2 ( ∣ ν 2 ∣ )
∣ ⟨ T Ξ w , ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ − ⟨ S ∣ ν 1 ∣ , ε f , g ⟩ ∣ ≲ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (185) \left|\langle T_{\Xi_w,|\nu_1|,\varepsilon}f,g\rangle_{|\nu_2|}-\langle S_{|\nu_1|,\varepsilon}f,g\rangle\right|\lesssim\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \tag*{(185)} ⟨ T Ξ w , ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ − ⟨ S ∣ ν 1 ∣ , ε f , g ⟩ ≲ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (185) By (181) , it therefore suffices to prove (183) with ⟨ S ∣ ν 1 ∣ , ε ( P G 1 f ) , P G 2 g ⟩ ∗ \langle S_{|\nu_1|,\varepsilon}(P_{\mathcal G}^{1}f),P_{\mathcal G}^{2}g\rangle_{*} ⟨ S ∣ ν 1 ∣ , ε ( P G 1 f ) , P G 2 g ⟩ ∗
Testing for the localized operator. In the proof of the Key Lemma, b ^ 1 \widehat b_1 b 1 b ^ 2 \widehat b_2 b 2 T b Tb T b (187) below for the truncations of S ∣ ν 1 ∣ , ε b ^ 1 S_{|\nu_1|,\varepsilon}\widehat b_1 S ∣ ν 1 ∣ , ε b 1 S ∣ ν 2 ∣ , ε ∗ b ^ 2 S_{|\nu_2|,\varepsilon}^{*}\widehat b_2 S ∣ ν 2 ∣ , ε ∗ b 2
Let i = 1 , 2 i=1,2 i = 1 , 2 F i ( w i ) ⊂ G i ⊂ B i ∖ ( H i , R i ∪ E i , R i ) F_i(w_i)\subset G_i\subset B_i\setminus(\mathcal H_{i,R_i}\cup\mathcal E_{i,R_i}) F i ( w i ) ⊂ G i ⊂ B i ∖ ( H i , R i ∪ E i , R i )
Ξ w ( x ) ≥ dist ( x , F i ( w i ) ) ≥ dist ( x , B i ∖ ( H i , R i ∪ E i , R i ) ) , x ∈ X , \Xi_w(x)\ge\operatorname{dist}(x,F_i(w_i))\ge\operatorname{dist}(x,B_i\setminus(\mathcal H_{i,R_i}\cup\mathcal E_{i,R_i})),\qquad x\in X, Ξ w ( x ) ≥ dist ( x , F i ( w i )) ≥ dist ( x , B i ∖ ( H i , R i ∪ E i , R i )) , x ∈ X , that is, Ξ w \Xi_w Ξ w (74) with B = B i B=B_i B = B i
T Ξ w , ♯ ν ^ 1 ≤ c ♯ on B 1 and ( T ∗ ) Ξ w , ♯ ν ^ 2 ≤ c ♯ on B 2 . T_{\Xi_w,\sharp}\widehat{\nu}_1\le c_{\sharp}\quad\text{on }B_1\qquad\text{and}\qquad(T^{*})_{\Xi_w,\sharp}\widehat{\nu}_2\le c_{\sharp}\quad\text{on }B_2. T Ξ w , ♯ ν 1 ≤ c ♯ on B 1 and ( T ∗ ) Ξ w , ♯ ν 2 ≤ c ♯ on B 2 . Moreover, since χ i = 0 \chi_i=0 χ i = 0 X ∖ B i X\setminus B_i X ∖ B i (156) gives Ξ w ≥ θ r ( B i ) / 2 \Xi_w\ge\theta r(B_i)/2 Ξ w ≥ θ r ( B i ) /2 X ∖ B i X\setminus B_i X ∖ B i (70) , (41) , and the n n n μ \mu μ
T Ξ w , ♯ ν ^ 1 ( x ) ≤ ∫ ∣ K Ξ w ( x , y ) ∣ d ∣ ν ^ 1 ∣ ( y ) ≲ ∣ ν ^ 1 ∣ ( B 1 ) ( θ r ( B 1 ) ) n ≲ 1 , x ∈ X ∖ B 1 , T_{\Xi_w,\sharp}\widehat{\nu}_1(x)\le\int|K_{\Xi_w}(x,y)|\,d|\widehat{\nu}_1|(y)\lesssim\frac{|\widehat{\nu}_1|(B_1)}{(\theta r(B_1))^n}\lesssim1,\qquad x\in X\setminus B_1, T Ξ w , ♯ ν 1 ( x ) ≤ ∫ ∣ K Ξ w ( x , y ) ∣ d ∣ ν 1 ∣ ( y ) ≲ ( θ r ( B 1 ) ) n ∣ ν 1 ∣ ( B 1 ) ≲ 1 , x ∈ X ∖ B 1 , and, in the same way, ( T ∗ ) Ξ w , ♯ ν ^ 2 ≲ 1 (T^{*})_{\Xi_w,\sharp}\widehat{\nu}_2\lesssim1 ( T ∗ ) Ξ w , ♯ ν 2 ≲ 1 X ∖ B 2 X\setminus B_2 X ∖ B 2
∥ T Ξ w , ♯ ν ^ 1 ∥ L ∞ ( X ) + ∥ ( T ∗ ) Ξ w , ♯ ν ^ 2 ∥ L ∞ ( X ) ≲ 1. (186) \|T_{\Xi_w,\sharp}\widehat{\nu}_1\|_{L^\infty(X)}+\|(T^{*})_{\Xi_w,\sharp}\widehat{\nu}_2\|_{L^\infty(X)}\lesssim1. \tag*{(186)} ∥ T Ξ w , ♯ ν 1 ∥ L ∞ ( X ) + ∥ ( T ∗ ) Ξ w , ♯ ν 2 ∥ L ∞ ( X ) ≲ 1. (186) For ρ > 0 \rho>0 ρ > 0 S ∣ ν 1 ∣ , ε , ρ S_{|\nu_1|,\varepsilon,\rho} S ∣ ν 1 ∣ , ε , ρ S ∣ ν 2 ∣ , ε , ρ ∗ S^{*}_{|\nu_2|,\varepsilon,\rho} S ∣ ν 2 ∣ , ε , ρ ∗ S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε S ∣ ν 2 ∣ , ε ∗ S^{*}_{|\nu_2|,\varepsilon} S ∣ ν 2 ∣ , ε ∗ ρ \rho ρ
Lemma 6.8. For every x ∈ X x \in X x ∈ X
sup ρ > 0 ∣ S ∣ ν 1 ∣ , ε , ρ b ^ 1 ( x ) ∣ + sup ρ > 0 ∣ S ∣ ν 2 ∣ , ε , ρ ∗ b ^ 2 ( x ) ∣ ≲ 1 , (187) \sup_{\rho>0}|S_{|\nu_1|,\varepsilon,\rho}\widehat{b}_1(x)|+\sup_{\rho>0}|S^*_{|\nu_2|,\varepsilon,\rho}\widehat{b}_2(x)|\lesssim1, \tag*{(187)} ρ > 0 sup ∣ S ∣ ν 1 ∣ , ε , ρ b 1 ( x ) ∣ + ρ > 0 sup ∣ S ∣ ν 2 ∣ , ε , ρ ∗ b 2 ( x ) ∣ ≲ 1 , (187) where the implicit constant depends only on the structural parameters.
Proof. Let x ∈ X x\in X x ∈ X ρ > 0 \rho>0 ρ > 0 a ε ( t ) = φ ( t / ε ) ψ ( t / r 0 ) a_\varepsilon(t)=\varphi(t/\varepsilon)\psi(t/r_0) a ε ( t ) = φ ( t / ε ) ψ ( t / r 0 ) § 5 . Since b ^ 1 ∣ ν 1 ∣ = ν ^ 1 \widehat{b}_1|\nu_1|=\widehat{\nu}_1 b 1 ∣ ν 1 ∣ = ν 1 B 1 B_1 B 1 χ 1 = 0 \chi_1=0 χ 1 = 0 X ∖ B 1 X\setminus B_1 X ∖ B 1
S ∣ ν 1 ∣ , ε , ρ b ^ 1 ( x ) = χ 2 ( x ) ∫ d ( x , y ) > ρ a ε ( d ( x , y ) ) K Ξ w ( x , y ) d ν ^ 1 ( y ) S_{|\nu_1|,\varepsilon,\rho}\widehat{b}_1(x)=\chi_2(x)\int_{d(x,y)>\rho}a_\varepsilon(d(x,y))K_{\Xi_w}(x,y)\,d\widehat{\nu}_1(y) S ∣ ν 1 ∣ , ε , ρ b 1 ( x ) = χ 2 ( x ) ∫ d ( x , y ) > ρ a ε ( d ( x , y )) K Ξ w ( x , y ) d ν 1 ( y ) − χ 2 ( x ) ∫ d ( x , y ) > ρ ( 1 − χ 1 ( y ) ) a ε ( d ( x , y ) ) K Ξ w ( x , y ) d ν ^ 1 ( y ) = : I 1 − I 2 . -\chi_2(x)\int_{d(x,y)>\rho}(1-\chi_1(y))a_\varepsilon(d(x,y))K_{\Xi_w}(x,y)\,d\widehat{\nu}_1(y)=:\mathcal{I}_1-\mathcal{I}_2. − χ 2 ( x ) ∫ d ( x , y ) > ρ ( 1 − χ 1 ( y )) a ε ( d ( x , y )) K Ξ w ( x , y ) d ν 1 ( y ) =: I 1 − I 2 . Since 0 ≤ a ε ≤ 1 0\leq a_\varepsilon\leq1 0 ≤ a ε ≤ 1 ∫ 0 ∞ ∣ a ε ′ ∣ ≤ 2 \int_0^\infty|a_\varepsilon'|\leq2 ∫ 0 ∞ ∣ a ε ′ ∣ ≤ 2 Lemma 3.2 ,
I 1 = χ 2 ( x ) ( a ε ( ρ ) T Ξ w , ρ ν ^ 1 ( x ) + ∫ ρ ∞ a ε ′ ( t ) T Ξ w , t ν ^ 1 ( x ) d t ) , \mathcal{I}_1=\chi_2(x)\left(a_\varepsilon(\rho)T_{\Xi_w,\rho}\widehat{\nu}_1(x)+\int_\rho^\infty a_\varepsilon'(t)T_{\Xi_w,t}\widehat{\nu}_1(x)\,dt\right), I 1 = χ 2 ( x ) ( a ε ( ρ ) T Ξ w , ρ ν 1 ( x ) + ∫ ρ ∞ a ε ′ ( t ) T Ξ w , t ν 1 ( x ) d t ) , so that ∣ I 1 ∣ ≤ 3 T Ξ w , ♯ ν ^ 1 ( x ) ≲ 1 |\mathcal{I}_1|\leq3T_{\Xi_w,\sharp}\widehat{\nu}_1(x)\lesssim1 ∣ I 1 ∣ ≤ 3 T Ξ w , ♯ ν 1 ( x ) ≲ 1 (186) . If χ 1 ( y ) ≠ 1 \chi_1(y)\ne1 χ 1 ( y ) = 1 Ξ w ( y ) ≥ θ r ( B 1 ) / 2 \Xi_w(y)\geq\theta r(B_1)/2 Ξ w ( y ) ≥ θ r ( B 1 ) /2 (156) . Hence, as above,
∣ I 2 ∣ ≤ ∫ B 1 ( 1 − χ 1 ( y ) ) ∣ K Ξ w ( x , y ) ∣ d ∣ ν ^ 1 ∣ ( y ) ≲ ∣ ν ^ 1 ∣ ( B 1 ) ( θ r ( B 1 ) ) n ≲ 1. |\mathcal{I}_2|\leq\int_{B_1}(1-\chi_1(y))|K_{\Xi_w}(x,y)|\,d|\widehat{\nu}_1|(y)\lesssim\frac{|\widehat{\nu}_1|(B_1)}{(\theta r(B_1))^n}\lesssim1. ∣ I 2 ∣ ≤ ∫ B 1 ( 1 − χ 1 ( y )) ∣ K Ξ w ( x , y ) ∣ d ∣ ν 1 ∣ ( y ) ≲ ( θ r ( B 1 ) ) n ∣ ν 1 ∣ ( B 1 ) ≲ 1. This proves the bound for the first term in (187) . The second one follows in the same way, by interchanging the variables and the roles of χ 1 \chi_1 χ 1 χ 2 \chi_2 χ 2 ( T ∗ ) Ξ w , ♯ ν ^ 2 (T^*)_{\Xi_w,\sharp}\widehat{\nu}_2 ( T ∗ ) Ξ w , ♯ ν 2 T Ξ w , ♯ ν ^ 1 T_{\Xi_w,\sharp}\widehat{\nu}_1 T Ξ w , ♯ ν 1
Increasing c ♯ c_\sharp c ♯ (186) and (187) are bounded by c ♯ c_\sharp c ♯
Before proving the Key Lemma for the operator S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε f f f g g g Lemma 6.7 and we define
f K 1 = E 1 f + ∑ Q ∈ D 1 t r ℓ ( Q ) > ℓ K 1 Δ 1 , Q f and g K 2 = E 2 g + ∑ P ∈ D 2 t r ℓ ( P ) > ℓ K 2 Δ 2 , P g , (188) f_{K_1}=E_1f+\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1\\ \ell(Q)>\ell K_1}}\Delta_{1,Q}f
\qquad\text{and}\qquad
g_{K_2}=E_2g+\sum_{\substack{P\in\mathcal{D}^{\mathrm{tr}}_2\\ \ell(P)>\ell K_2}}\Delta_{2,P}g, \tag*{(188)} f K 1 = E 1 f + Q ∈ D 1 tr ℓ ( Q ) > ℓ K 1 ∑ Δ 1 , Q f and g K 2 = E 2 g + P ∈ D 2 tr ℓ ( P ) > ℓ K 2 ∑ Δ 2 , P g , (188) then, by (177) ,
f K 1 → f in L 2 ( ∣ ν 1 ∣ ) and g K 2 → g in L 2 ( ∣ ν 2 ∣ ) unconditionally. (189) f_{K_1}\to f\text{ in }L^2(|\nu_1|)\qquad\text{and}\qquad g_{K_2}\to g\text{ in }L^2(|\nu_2|)\text{ unconditionally.} \tag*{(189)} f K 1 → f in L 2 ( ∣ ν 1 ∣ ) and g K 2 → g in L 2 ( ∣ ν 2 ∣ ) unconditionally. (189) Hence, we provide a truncated analogue of Definition 6.4.
Definition 6.9. Let K 1 , K 2 > 0 K_1,K_2>0 K 1 , K 2 > 0 w = ( w 1 , w 2 ) ∈ Ω 2 \mathbf{w}=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2 f ∈ L 2 ( ∣ ν 1 ∣ ) f\in L^2(|\nu_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν 2 ∣ ) g\in L^2(|\nu_2|) g ∈ L 2 ( ∣ ν 2 ∣ )
P G , 1 K 1 ( w ) f : = ∑ Q ∈ D 1 G ( w ) ∩ D 1 t r ℓ ( Q ) ≥ ℓ K 1 Δ 1 , Q f , P B , 1 K 1 ( w ) f : = ∑ Q ∈ D 1 B ( w ) ∩ D 1 t r ℓ ( Q ) ≥ ℓ K 1 Δ 1 , Q f , \mathcal{P}^{K_1}_{\mathcal{G},1}(\mathbf{w})f:=\sum_{\substack{Q\in\mathcal{D}^{\mathcal{G}}_1(\mathbf{w})\cap\mathcal{D}^{\mathrm{tr}}_1\\ \ell(Q)\geq\ell K_1}}\Delta_{1,Q}f,
\qquad
\mathcal{P}^{K_1}_{\mathcal{B},1}(\mathbf{w})f:=\sum_{\substack{Q\in\mathcal{D}^{\mathcal{B}}_1(\mathbf{w})\cap\mathcal{D}^{\mathrm{tr}}_1\\ \ell(Q)\geq\ell K_1}}\Delta_{1,Q}f, P G , 1 K 1 ( w ) f := Q ∈ D 1 G ( w ) ∩ D 1 tr ℓ ( Q ) ≥ ℓ K 1 ∑ Δ 1 , Q f , P B , 1 K 1 ( w ) f := Q ∈ D 1 B ( w ) ∩ D 1 tr ℓ ( Q ) ≥ ℓ K 1 ∑ Δ 1 , Q f , and
P G , 2 K 2 ( w ) g : = ∑ R ∈ D 2 G ( w ) ∩ D 2 t r ℓ ( R ) ≥ ℓ K 2 Δ 2 , R g , P B , 2 K 2 ( w ) g : = ∑ R ∈ D 2 B ( w ) ∩ D 2 t r ℓ ( R ) ≥ ℓ K 2 Δ 2 , R g . \mathcal{P}^{K_2}_{\mathcal{G},2}(\mathbf{w})g:=\sum_{\substack{R\in\mathcal{D}^{\mathcal{G}}_2(\mathbf{w})\cap\mathcal{D}^{\mathrm{tr}}_2\\ \ell(R)\geq\ell K_2}}\Delta_{2,R}g,
\qquad
\mathcal{P}^{K_2}_{\mathcal{B},2}(\mathbf{w})g:=\sum_{\substack{R\in\mathcal{D}^{\mathcal{B}}_2(\mathbf{w})\cap\mathcal{D}^{\mathrm{tr}}_2\\ \ell(R)\geq\ell K_2}}\Delta_{2,R}g. P G , 2 K 2 ( w ) g := R ∈ D 2 G ( w ) ∩ D 2 tr ℓ ( R ) ≥ ℓ K 2 ∑ Δ 2 , R g , P B , 2 K 2 ( w ) g := R ∈ D 2 B ( w ) ∩ D 2 tr ℓ ( R ) ≥ ℓ K 2 ∑ Δ 2 , R g . To simplify notation, for w = ( w 1 , w 2 ) ∈ Ω 2 \mathbf{w}=(w_1,w_2)\in\Omega^2 w = ( w 1 , w 2 ) ∈ Ω 2 K 1 , K 2 > 1 K_1,K_2>1 K 1 , K 2 > 1 f w : = f w K 1 : = P G , 1 K 1 ( w ) f f_{\mathbf{w}}:=f_{\mathbf{w}}^{K_1}:=P_{\mathcal{G},1}^{K_1}(\mathbf{w})f f w := f w K 1 := P G , 1 K 1 ( w ) f g w : = g w K 2 : = P G , 2 K 2 ( w ) g g_{\mathbf{w}}:=g_{\mathbf{w}}^{K_2}:=P_{\mathcal{G},2}^{K_2}(\mathbf{w})g g w := g w K 2 := P G , 2 K 2 ( w ) g
∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) < ∞ and ∥ S ∣ ν 2 ∣ , ε ∗ ∥ L 2 ( ∣ ν 2 ∣ ) → L 2 ( ∣ ν 1 ∣ ) < ∞ . \left\|S_{|\nu_1|,\varepsilon}\right\|_{L^2(|\nu_1|)\to L^2(|\nu_2|)}<\infty
\qquad\text{and}\qquad
\left\|S^{*}_{|\nu_2|,\varepsilon}\right\|_{L^2(|\nu_2|)\to L^2(|\nu_1|)}<\infty. S ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) < ∞ and S ∣ ν 2 ∣ , ε ∗ L 2 ( ∣ ν 2 ∣ ) → L 2 ( ∣ ν 1 ∣ ) < ∞. So, (189) yields
S ∣ ν 1 ∣ , ε f w K 1 = S ∣ ν 1 ∣ , ε ( E 1 f w K 1 ) + ∑ Q ∈ D 1 t r S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w K 1 ) → K 1 → ∞ S ∣ ν 1 ∣ , ε ( P G , 1 1 ( w ) f ) S_{|\nu_1|,\varepsilon}f_{\mathbf{w}}^{K_1}
=
S_{|\nu_1|,\varepsilon}(E_1f_{\mathbf{w}}^{K_1})
+
\sum_{Q\in\mathcal{D}^{\mathrm{tr}}_1}
S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_{\mathbf{w}}^{K_1})
\xrightarrow[K_1\to\infty]{}
S_{|\nu_1|,\varepsilon}(P_{\mathcal{G},1}^{1}(\mathbf{w})f) S ∣ ν 1 ∣ , ε f w K 1 = S ∣ ν 1 ∣ , ε ( E 1 f w K 1 ) + Q ∈ D 1 tr ∑ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w K 1 ) K 1 → ∞ S ∣ ν 1 ∣ , ε ( P G , 1 1 ( w ) f ) and
S ∣ ν 2 ∣ , ε ∗ g w K 2 = S ∣ ν 2 ∣ , ε ∗ ( E 2 g w K 2 ) + ∑ P ∈ D 2 t r S ∣ ν 2 ∣ , ε ∗ ( Δ 2 , P g w K 2 ) → K 2 → ∞ S ∣ ν 2 ∣ , ε ∗ ( P G , 2 2 ( w ) g ) , S^{*}_{|\nu_2|,\varepsilon}g_{\mathbf{w}}^{K_2}
=
S^{*}_{|\nu_2|,\varepsilon}(E_2g_{\mathbf{w}}^{K_2})
+
\sum_{P\in\mathcal{D}^{\mathrm{tr}}_2}
S^{*}_{|\nu_2|,\varepsilon}(\Delta_{2,P}g_{\mathbf{w}}^{K_2})
\xrightarrow[K_2\to\infty]{}
S^{*}_{|\nu_2|,\varepsilon}(P_{\mathcal{G},2}^{2}(\mathbf{w})g), S ∣ ν 2 ∣ , ε ∗ g w K 2 = S ∣ ν 2 ∣ , ε ∗ ( E 2 g w K 2 ) + P ∈ D 2 tr ∑ S ∣ ν 2 ∣ , ε ∗ ( Δ 2 , P g w K 2 ) K 2 → ∞ S ∣ ν 2 ∣ , ε ∗ ( P G , 2 2 ( w ) g ) , where the sums converge in L 2 ( ∣ ν 2 ∣ ) L^2(|\nu_2|) L 2 ( ∣ ν 2 ∣ ) L 2 ( ∣ ν 1 ∣ ) L^2(|\nu_1|) L 2 ( ∣ ν 1 ∣ )
Remark 6.10. The sums appearing in the expressions of S ∣ ν 1 ∣ , ε f w K 1 S_{|\nu_1|,\varepsilon}f_{\mathbf{w}}^{K_1} S ∣ ν 1 ∣ , ε f w K 1 S ∣ ν 2 ∣ , ε ∗ g w K 2 S^{*}_{|\nu_2|,\varepsilon}g_{\mathbf{w}}^{K_2} S ∣ ν 2 ∣ , ε ∗ g w K 2 K 1 K_1 K 1 K 2 K_2 K 2 f w f_{\mathbf{w}} f w g w g_{\mathbf{w}} g w [77 Chapter 5] , studying the contributions of S ∣ ν 1 ∣ , ε f w K 1 S_{|\nu_1|,\varepsilon}f_{\mathbf{w}}^{K_1} S ∣ ν 1 ∣ , ε f w K 1 S ∣ ν 2 ∣ , ε ∗ g w K 2 S^{*}_{|\nu_2|,\varepsilon}g_{\mathbf{w}}^{K_2} S ∣ ν 2 ∣ , ε ∗ g w K 2 very good functions .
Proof of Key Lemma 6.7
By (185) , we estimate the bilinear form of S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε f w f_{\mathbf{w}} f w g w g_{\mathbf{w}} g w Q 2 ( w 1 ) Q_2(w_1) Q 2 ( w 1 ) Q 2 ( w 2 ) Q_2(w_2) Q 2 ( w 2 ) B 1 ∪ B 2 B_1\cup B_2 B 1 ∪ B 2
⟨ S ∣ ν 1 ∣ , ε f w , g w ⟩ ∗ = − ⟨ S ∣ ν 1 ∣ , ε ( E 1 f w ) , E 2 g w ⟩ ∗ + ⟨ S ∣ ν 1 ∣ , ε ( E 1 f w ) , g w ⟩ ∗ + ⟨ S ∣ ν 1 ∣ , ε f w , E 2 g w ⟩ ∗ + ∑ Q ∈ D 1 t r , R ∈ D 2 t r ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ = : I 1 ( w ) + I 2 ( w ) + I 3 ( w ) + I 4 ( w ) . (190) \begin{aligned}
\langle S_{|\nu_1|,\varepsilon}f_{\mathbf{w}},g_{\mathbf{w}}\rangle_*
={}&-\langle S_{|\nu_1|,\varepsilon}(E_1f_{\mathbf{w}}),E_2g_{\mathbf{w}}\rangle_*+\langle S_{|\nu_1|,\varepsilon}(E_1f_{\mathbf{w}}),g_{\mathbf{w}}\rangle_*\\
&+\langle S_{|\nu_1|,\varepsilon}f_{\mathbf{w}},E_2g_{\mathbf{w}}\rangle_*+
\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1,\ R\in\mathcal{D}^{\mathrm{tr}}_2}}
\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_{\mathbf{w}}),\Delta_{2,R}g_{\mathbf{w}}\rangle_*\\
=:{}&I_1(\mathbf{w})+I_2(\mathbf{w})+I_3(\mathbf{w})+I_4(\mathbf{w}). \tag*{(190)}
\end{aligned} ⟨ S ∣ ν 1 ∣ , ε f w , g w ⟩ ∗ = =: − ⟨ S ∣ ν 1 ∣ , ε ( E 1 f w ) , E 2 g w ⟩ ∗ + ⟨ S ∣ ν 1 ∣ , ε ( E 1 f w ) , g w ⟩ ∗ + ⟨ S ∣ ν 1 ∣ , ε f w , E 2 g w ⟩ ∗ + Q ∈ D 1 tr , R ∈ D 2 tr ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ I 1 ( w ) + I 2 ( w ) + I 3 ( w ) + I 4 ( w ) . (190) Since ∫ Δ i , Q h d ∣ ν i ∣ = 0 \int\Delta_{i,Q}h\,d|\nu_i|=0 ∫ Δ i , Q h d ∣ ν i ∣ = 0 Q ∈ D i t r Q\in\mathcal{D}^{\mathrm{tr}}_i Q ∈ D i tr h ∈ L 2 ( ∣ ν i ∣ ) h\in L^2(|\nu_i|) h ∈ L 2 ( ∣ ν i ∣ ) E 1 f w = 0 E_1f_{\mathbf{w}}=0 E 1 f w = 0 E 2 g w = 0 E_2g_{\mathbf{w}}=0 E 2 g w = 0 I 1 ( w ) I_1(\mathbf{w}) I 1 ( w ) I 2 ( w ) I_2(\mathbf{w}) I 2 ( w ) I 3 ( w ) I_3(\mathbf{w}) I 3 ( w ) f ∈ L 2 ( ∣ ν 1 ∣ ) f\in L^2(|\nu_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν 2 ∣ ) g\in L^2(|\nu_2|) g ∈ L 2 ( ∣ ν 2 ∣ ) f w f_{\mathbf{w}} f w g w g_{\mathbf{w}} g w
∣ ν i ^ ∣ ( B i ) ≤ ϵ 1 μ ( B i ) and μ ( Q 2 ( w i ) ) ≤ ∣ ν i ∣ ( Q 2 ( w i ) ) ≤ ϵ 1 μ ( Q 2 ( w i ) ) . (191) |\widehat{\nu_i}|(B_i)\le\epsilon_1\mu(B_i)
\qquad\text{and}\qquad
\mu(Q_2(w_i))\le|\nu_i|(Q_2(w_i))\le\epsilon_1\mu(Q_2(w_i)). \tag*{(191)} ∣ ν i ∣ ( B i ) ≤ ϵ 1 μ ( B i ) and μ ( Q 2 ( w i )) ≤ ∣ ν i ∣ ( Q 2 ( w i )) ≤ ϵ 1 μ ( Q 2 ( w i )) . (191) The Cauchy–Schwarz inequality and (191) yield
∣ ∫ Q 2 ( w i ) u d ∣ ν i ∣ μ ( Q 2 ( w i ) ) ∣ ≲ μ ( Q 2 ( w i ) ) − 1 / 2 ∥ u ∥ L 2 ( ∣ ν i ∣ ) , \left|\frac{\int_{Q_2(w_i)}u\,d|\nu_i|}{\mu(Q_2(w_i))}\right|
\lesssim\mu(Q_2(w_i))^{-1/2}\|u\|_{L^2(|\nu_i|)}, μ ( Q 2 ( w i )) ∫ Q 2 ( w i ) u d ∣ ν i ∣ ≲ μ ( Q 2 ( w i ) ) − 1/2 ∥ u ∥ L 2 ( ∣ ν i ∣ ) , so ∥ E i u ∥ L 2 ( ∣ ν i ∣ ) ≲ ∥ u ∥ L 2 ( ∣ ν i ∣ ) \|E_iu\|_{L^2(|\nu_i|)}\lesssim\|u\|_{L^2(|\nu_i|)} ∥ E i u ∥ L 2 ( ∣ ν i ∣ ) ≲ ∥ u ∥ L 2 ( ∣ ν i ∣ ) L ε L_\varepsilon L ε
S ∣ ν 1 ∣ , ε ( χ Q 2 ( w 1 ) b ^ 1 ) = S ∣ ν 1 ∣ , ε b ^ 1 and S ∣ ν 2 ∣ , ε ∗ ( χ Q 2 ( w 2 ) b ^ 2 ) = S ∣ ν 2 ∣ , ε ∗ b ^ 2 . S_{|\nu_1|,\varepsilon}(\chi_{Q_2(w_1)}\widehat{b}_1)=S_{|\nu_1|,\varepsilon}\widehat{b}_1
\qquad\text{and}\qquad
S^{*}_{|\nu_2|,\varepsilon}(\chi_{Q_2(w_2)}\widehat{b}_2)=S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2. S ∣ ν 1 ∣ , ε ( χ Q 2 ( w 1 ) b 1 ) = S ∣ ν 1 ∣ , ε b 1 and S ∣ ν 2 ∣ , ε ∗ ( χ Q 2 ( w 2 ) b 2 ) = S ∣ ν 2 ∣ , ε ∗ b 2 . Moreover, the two outputs are supported in B 2 B_2 B 2 B 1 B_1 B 1 (187) and (191) imply
∥ S ∣ ν 1 ∣ , ε b ^ 1 ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≲ μ ( B 2 ) ≤ μ ( Q 2 ( w 1 ) ) , ∥ S ∣ ν 2 ∣ , ε ∗ b ^ 2 ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≲ μ ( B 1 ) ≤ μ ( Q 2 ( w 2 ) ) , \left\|S_{|\nu_1|,\varepsilon}\widehat{b}_1\right\|_{L^2(|\nu_2|)}^2\lesssim\mu(B_2)\le\mu(Q_2(w_1)),
\qquad
\left\|S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\right\|_{L^2(|\nu_1|)}^2\lesssim\mu(B_1)\le\mu(Q_2(w_2)), S ∣ ν 1 ∣ , ε b 1 L 2 ( ∣ ν 2 ∣ ) 2 ≲ μ ( B 2 ) ≤ μ ( Q 2 ( w 1 )) , S ∣ ν 2 ∣ , ε ∗ b 2 L 2 ( ∣ ν 1 ∣ ) 2 ≲ μ ( B 1 ) ≤ μ ( Q 2 ( w 2 )) , and combining these bounds with the controls on the top averages, we obtain
∣ I 1 ( w ) ∣ ≤ ∥ S ∣ ν 1 ∣ , ε ( E 1 f w ) ∥ L 2 ( ∣ ν 2 ∣ ) ∥ E 2 g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (192) |\mathcal{I}_1(\mathbf{w})| \le\|S_{|\nu_1|,\varepsilon}(E_1f_{\mathbf{w}})\|_{L^2(|\nu_2|)}\|E_2g_{\mathbf{w}}\|_{L^2(|\nu_2|)} \lesssim\|f_{\mathbf{w}}\|_{L^2(|\nu_1|)}\|g_{\mathbf{w}}\|_{L^2(|\nu_2|)}. \tag*{(192)} ∣ I 1 ( w ) ∣ ≤ ∥ S ∣ ν 1 ∣ , ε ( E 1 f w ) ∥ L 2 ( ∣ ν 2 ∣ ) ∥ E 2 g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (192) We also have the following estimates.
Lemma 6.11. We have
∥ S ∣ ν 1 ∣ , ε ^ b ^ 1 ∥ L 2 ( ∣ ν 2 ∣ ) ≲ μ ( Q 2 ( w 1 ) ) 1 / 2 (193) \|S_{|\nu_1|,\widehat{\varepsilon}\widehat{b}_1\|_{L^2(|\nu_2|)} \lesssim\mu(Q_2(w_1))^{1/2}} \tag*{(193)} ∥ S ∣ ν 1 ∣ , ε b 1 ∥ L 2 ( ∣ ν 2 ∣ ) ≲ μ ( Q 2 ( w 1 ) ) 1/2 (193) and
∣ I 2 ( w ) ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (194) |\mathcal{I}_2(\mathbf{w})| \lesssim\|f_{\mathbf{w}}\|_{L^2(|\nu_1|)}\|g_{\mathbf{w}}\|_{L^2(|\nu_2|)}. \tag*{(194)} ∣ I 2 ( w ) ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (194) Proof. The first estimate was proved above. For the second, by the definition of E 1 E_1 E 1
∣ I 2 ( w ) ∣ ≤ ∣ ∫ Q 2 ( w 1 ) f w d ∣ ν 1 ∣ ∣ μ ( Q 2 ( w 1 ) ) ∥ S ∣ ν 1 ∣ , ε ^ b ^ 1 ∥ L 2 ( ∣ ν 2 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . |\mathcal{I}_2(\mathbf{w})| \le\frac{\left|\int_{Q_2(w_1)} f_{\mathbf{w}}\,d|\nu_1|\right|}{\mu(Q_2(w_1))}\|S_{|\nu_1|,\widehat{\varepsilon}\widehat{b}_1\|_{L^2(|\nu_2|)}\|g_{\mathbf{w}}\|_{L^2(|\nu_2|)} \lesssim\|f_{\mathbf{w}}\|_{L^2(|\nu_1|)}\|g_{\mathbf{w}}\|_{L^2(|\nu_2|)}.} ∣ I 2 ( w ) ∣ ≤ μ ( Q 2 ( w 1 )) ∫ Q 2 ( w 1 ) f w d ∣ ν 1 ∣ ∥ S ∣ ν 1 ∣ , ε b 1 ∥ L 2 ( ∣ ν 2 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . Finally, by (187) and (191) , we obtain
∣ I 3 ( w ) ∣ ≤ ∣ ∫ Q 2 ( w 2 ) g w d ∣ ν 2 ∣ ∣ μ ( Q 2 ( w 2 ) ) ∥ S ∣ ν 2 ∣ , ε b ^ 2 ∗ ∥ L 2 ( ∣ ν 1 ∣ ) ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) |\mathcal{I}_3(\mathbf{w})| \le\frac{\left|\int_{Q_2(w_2)} g_{\mathbf{w}}\,d|\nu_2|\right|}{\mu(Q_2(w_2))}\|S^*_{|\nu_2|,\varepsilon\widehat{b}_2}\|_{L^2(|\nu_1|)}\|f_{\mathbf{w}}\|_{L^2(|\nu_1|)} ∣ I 3 ( w ) ∣ ≤ μ ( Q 2 ( w 2 )) ∫ Q 2 ( w 2 ) g w d ∣ ν 2 ∣ ∥ S ∣ ν 2 ∣ , ε b 2 ∗ ∥ L 2 ( ∣ ν 1 ∣ ) ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (195) \lesssim\|f_{\mathbf{w}}\|_{L^2(|\nu_1|)}\|g_{\mathbf{w}}\|_{L^2(|\nu_2|)}. \tag*{(195)} ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (195) It remains to estimate I 4 ( w ) \mathcal{I}_4(\mathbf{w}) I 4 ( w ) α = γ / ( 2 ( n + γ ) ) \alpha=\gamma/(2(n+\gamma)) α = γ / ( 2 ( n + γ )) ( 0 , 1 ) (0,1) ( 0 , 1 ) Q ∈ D 1 t r Q\in\mathcal{D}_1^{\mathrm{tr}} Q ∈ D 1 tr R ∈ D 2 t r R\in\mathcal{D}_2^{\mathrm{tr}} R ∈ D 2 tr
dist ( Q , R ) ≥ min ( ℓ ( Q ) , ℓ ( R ) ) α ⋅ max ( ℓ ( Q ) , ℓ ( R ) ) 1 − α . (196) \operatorname{dist}(Q,R)\ge\min(\ell(Q),\ell(R))^\alpha\cdot\max(\ell(Q),\ell(R))^{1-\alpha}. \tag*{(196)} dist ( Q , R ) ≥ min ( ℓ ( Q ) , ℓ ( R ) ) α ⋅ max ( ℓ ( Q ) , ℓ ( R ) ) 1 − α . (196) Note that, if Q Q Q R R R Definition 6.1 it holds that ℓ m ℓ ( R ) < ℓ ( Q ) < ℓ − m ℓ ( R ) \ell^m\ell(R)<\ell(Q)<\ell^{-m}\ell(R) ℓ m ℓ ( R ) < ℓ ( Q ) < ℓ − m ℓ ( R )
Hence, we split I 4 ( w ) \mathcal{I}_4(\mathbf{w}) I 4 ( w )
I 4 ( w ) = ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q , R d i s t a n t ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ = ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q , R d i s t a n t + ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q ∩ R = ∅ , Q , R n o t d i s t a n t + ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q ∩ R ≠ ∅ ℓ m ℓ ( R ) ≤ ℓ ( Q ) ≤ ℓ − m ℓ ( R ) + ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q ∩ R ≠ ∅ ℓ ( Q ) < ℓ m ℓ ( R ) o r ℓ ( Q ) > ℓ − m ℓ ( R ) = : S 1 ( w ) + S 2 ( w ) + S 3 ( w ) + S 4 ( w ) . (197) \begin{aligned}
\mathcal{I}_4(\mathbf{w})=\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q,R\ \mathrm{distant}}}\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_{\mathbf{w}}),\Delta_{2,R}g_{\mathbf{w}}\rangle_* \\
&=\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q,R\ \mathrm{distant}}}
+\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q\cap R=\varnothing,\ Q,R\ \mathrm{not\ distant}}}
+\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q\cap R\ne\varnothing\\\ell^m\ell(R)\le\ell(Q)\le\ell^{-m}\ell(R)}}
+\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q\cap R\ne\varnothing\\\ell(Q)<\ell^m\ell(R)\ \mathrm{or}\ \ell(Q)>\ell^{-m}\ell(R)}} \\
&=:S_1(\mathbf{w})+S_2(\mathbf{w})+S_3(\mathbf{w})+S_4(\mathbf{w}). \tag*{(197)}
\end{aligned} I 4 ( w ) = Q ∈ D 1 tr , R ∈ D 2 tr Q , R distant ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ = Q ∈ D 1 tr , R ∈ D 2 tr Q , R distant ∑ + Q ∈ D 1 tr , R ∈ D 2 tr Q ∩ R = ∅ , Q , R not distant ∑ + Q ∈ D 1 tr , R ∈ D 2 tr Q ∩ R = ∅ ℓ m ℓ ( R ) ≤ ℓ ( Q ) ≤ ℓ − m ℓ ( R ) ∑ + Q ∈ D 1 tr , R ∈ D 2 tr Q ∩ R = ∅ ℓ ( Q ) < ℓ m ℓ ( R ) or ℓ ( Q ) > ℓ − m ℓ ( R ) ∑ =: S 1 ( w ) + S 2 ( w ) + S 3 ( w ) + S 4 ( w ) . (197) We estimate these four terms separately.
Estimate of S1
We first write
∣ S 1 ( w ) ∣ ≤ ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q , R d i s t a n t ∣ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ ∣ = ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q , R d i s t a n t ℓ ( Q ) ≤ ℓ ( R ) + ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q , R d i s t a n t ℓ ( Q ) > ℓ ( R ) = : S 1 ′ + S 1 ′ ′ . \begin{aligned}
|S_1(\mathbf{w})|\le\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q,R\ \mathrm{distant}}}\left|\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_{\mathbf{w}}),\Delta_{2,R}g_{\mathbf{w}}\rangle_*\right| \\
&=\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q,R\ \mathrm{distant}\\\ell(Q)\le\ell(R)}}+\sum_{\substack{Q\in\mathcal{D}_1^{\mathrm{tr}},\,R\in\mathcal{D}_2^{\mathrm{tr}}\\Q,R\ \mathrm{distant}\\\ell(Q)>\ell(R)}}
=:S'_1+S''_1.
\end{aligned} ∣ S 1 ( w ) ∣ ≤ Q ∈ D 1 tr , R ∈ D 2 tr Q , R distant ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ = Q ∈ D 1 tr , R ∈ D 2 tr Q , R distant ℓ ( Q ) ≤ ℓ ( R ) ∑ + Q ∈ D 1 tr , R ∈ D 2 tr Q , R distant ℓ ( Q ) > ℓ ( R ) ∑ =: S 1 ′ + S 1 ′′ . The long distance is defined as
D ( Q , R ) : = ℓ ( Q ) + ℓ ( R ) + dist ( Q , R ) , for Q ∈ D 1 , R ∈ D 2 , D(Q,R):=\ell(Q)+\ell(R)+\operatorname{dist}(Q,R),\qquad\text{for }Q\in\mathcal{D}_1,\ R\in\mathcal{D}_2, D ( Q , R ) := ℓ ( Q ) + ℓ ( R ) + dist ( Q , R ) , for Q ∈ D 1 , R ∈ D 2 , and it is used in the bound of S 1 ′ S'_1 S 1 ′
Lemma 6.12. Let Q 1 Q_1 Q 1 Q 2 Q_2 Q 2 X X X i = 1 , 2 i=1,2 i = 1 , 2 φ i ∈ L 2 ( ∣ ν i ∣ ) \varphi_i \in L^2(|\nu_i|) φ i ∈ L 2 ( ∣ ν i ∣ ) supp ( φ i ) ⊂ Q i \operatorname{supp}(\varphi_i) \subset Q_i supp ( φ i ) ⊂ Q i dist ( Q 1 , supp ( φ 2 ) ) ≥ ℓ ( Q 1 ) \operatorname{dist}(Q_1,\operatorname{supp}(\varphi_2)) \geq\ell(Q_1) dist ( Q 1 , supp ( φ 2 )) ≥ ℓ ( Q 1 ) ∫ φ 1 d ∣ ν 1 ∣ = 0 \int\varphi_1\,d|\nu_1|=0 ∫ φ 1 d ∣ ν 1 ∣ = 0
∣ ⟨ S ∣ ν 1 ∣ , ε φ 1 , φ 2 ⟩ ∗ ∣ ≤ C ℓ ( Q 1 ) γ dist ( Q 1 , supp ( φ 2 ) ) n + γ ∥ φ 1 ∥ L 1 ( ∣ ν 1 ∣ ) ∥ φ 2 ∥ L 1 ( ∣ ν 2 ∣ ) . (198) \left|\langle S_{|\nu_1|,\varepsilon}\varphi_1,\varphi_2\rangle_*\right| \leq C\frac{\ell(Q_1)^\gamma}{\operatorname{dist}(Q_1,\operatorname{supp}(\varphi_2))^{n+\gamma}}\|\varphi_1\|_{L^1(|\nu_1|)}\|\varphi_2\|_{L^1(|\nu_2|)}. \tag*{(198)} ⟨ S ∣ ν 1 ∣ , ε φ 1 , φ 2 ⟩ ∗ ≤ C dist ( Q 1 , supp ( φ 2 ) ) n + γ ℓ ( Q 1 ) γ ∥ φ 1 ∥ L 1 ( ∣ ν 1 ∣ ) ∥ φ 2 ∥ L 1 ( ∣ ν 2 ∣ ) . (198) Moreover, if ℓ ( Q 1 ) ≤ ℓ ( Q 2 ) \ell(Q_1)\leq\ell(Q_2) ℓ ( Q 1 ) ≤ ℓ ( Q 2 ) c ′ > 0 c'>0 c ′ > 0
dist ( Q 1 , supp ( φ 2 ) ) ≥ c ′ ℓ ( Q 1 ) α ℓ ( Q 2 ) 1 − α , \operatorname{dist}(Q_1,\operatorname{supp}(\varphi_2))\geq c'\ell(Q_1)^\alpha\ell(Q_2)^{1-\alpha}, dist ( Q 1 , supp ( φ 2 )) ≥ c ′ ℓ ( Q 1 ) α ℓ ( Q 2 ) 1 − α , then, denoting by C 1 C_1 C 1
∣ ⟨ S ∣ ν 1 ∣ , ε φ 1 , φ 2 ⟩ ∗ ∣ ≲ C , c ′ , C 1 ℓ ( Q 1 ) γ / 2 ℓ ( Q 2 ) γ / 2 D ( Q 1 , Q 2 ) n + γ ∥ φ 1 ∥ L 1 ( ∣ ν 1 ∣ ) ∥ φ 2 ∥ L 1 ( ∣ ν 2 ∣ ) . (199) \left|\langle S_{|\nu_1|,\varepsilon}\varphi_1,\varphi_2\rangle_*\right|\lesssim_{C,c',C_1}\frac{\ell(Q_1)^{\gamma/2}\ell(Q_2)^{\gamma/2}}{D(Q_1,Q_2)^{n+\gamma}}\|\varphi_1\|_{L^1(|\nu_1|)}\|\varphi_2\|_{L^1(|\nu_2|)}. \tag*{(199)} ⟨ S ∣ ν 1 ∣ , ε φ 1 , φ 2 ⟩ ∗ ≲ C , c ′ , C 1 D ( Q 1 , Q 2 ) n + γ ℓ ( Q 1 ) γ /2 ℓ ( Q 2 ) γ /2 ∥ φ 1 ∥ L 1 ( ∣ ν 1 ∣ ) ∥ φ 2 ∥ L 1 ( ∣ ν 2 ∣ ) . (199) Proof. The proof is a minor variation of the proof of [77 Lemma 5.14] , using Calderón-Zygmund kernel estimates and the choice α = γ / ( 2 ( n + γ ) ) \alpha=\gamma/(2(n+\gamma)) α = γ / ( 2 ( n + γ ))
Since the cubes Q Q Q R R R S 1 ′ S'_1 S 1 ′ ℓ ( Q ) ≤ ℓ ( R ) \ell(Q)\leq\ell(R) ℓ ( Q ) ≤ ℓ ( R )
S 1 ′ ≲ C ∑ Q ∈ D 1 t r , R ∈ D 2 t r ℓ ( Q ) ≤ ℓ ( R ) ℓ ( Q ) γ / 2 ℓ ( R ) γ / 2 D ( Q , R ) n + γ ∣ ν 1 ∣ ( Q ) 1 / 2 ∣ ν 2 ∣ ( R ) 1 / 2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) . (200) S'_1\lesssim_C\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1,R\in\mathcal{D}^{\mathrm{tr}}_2\\ \ell(Q)\leq\ell(R)}}\frac{\ell(Q)^{\gamma/2}\ell(R)^{\gamma/2}}{D(Q,R)^{n+\gamma}}|\nu_1|(Q)^{1/2}|\nu_2|(R)^{1/2}\|\Delta_{1,Q}fw\|_{L^2(|\nu_1|)}\|\Delta_{2,R}gw\|_{L^2(|\nu_2|)}. \tag*{(200)} S 1 ′ ≲ C Q ∈ D 1 tr , R ∈ D 2 tr ℓ ( Q ) ≤ ℓ ( R ) ∑ D ( Q , R ) n + γ ℓ ( Q ) γ /2 ℓ ( R ) γ /2 ∣ ν 1 ∣ ( Q ) 1/2 ∣ ν 2 ∣ ( R ) 1/2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) . (200) For transit cubes Q ∈ D 1 t r Q\in\mathcal{D}^{\mathrm{tr}}_1 Q ∈ D 1 tr R ∈ D 2 t r R\in\mathcal{D}^{\mathrm{tr}}_2 R ∈ D 2 tr
T Q , R : = ℓ ( Q ) γ / 2 ℓ ( R ) γ / 2 D ( Q , R ) n + γ ∣ ν 1 ∣ ( Q ) 1 / 2 ∣ ν 2 ∣ ( R ) 1 / 2 , if ℓ ( Q ) ≤ ℓ ( R ) T_{Q,R}:=\frac{\ell(Q)^{\gamma/2}\ell(R)^{\gamma/2}}{D(Q,R)^{n+\gamma}}|\nu_1|(Q)^{1/2}|\nu_2|(R)^{1/2},\qquad\text{if }\ell(Q)\leq\ell(R) T Q , R := D ( Q , R ) n + γ ℓ ( Q ) γ /2 ℓ ( R ) γ /2 ∣ ν 1 ∣ ( Q ) 1/2 ∣ ν 2 ∣ ( R ) 1/2 , if ℓ ( Q ) ≤ ℓ ( R ) and set T Q , R : = 0 T_{Q,R}:=0 T Q , R := 0
∑ ℓ ( Q ) ≤ ℓ ( R ) T Q , R ∣ ν 1 ∣ ( Q ) 1 / 2 ≲ ∣ ν 2 ∣ ( R ) 1 / 2 , \sum_{\ell(Q)\leq\ell(R)}T_{Q,R}|\nu_1|(Q)^{1/2}\lesssim|\nu_2|(R)^{1/2}, ℓ ( Q ) ≤ ℓ ( R ) ∑ T Q , R ∣ ν 1 ∣ ( Q ) 1/2 ≲ ∣ ν 2 ∣ ( R ) 1/2 , ∑ ℓ ( R ) ≥ ℓ ( Q ) T Q , R ∣ ν 2 ∣ ( R ) 1 / 2 ≲ ∣ ν 1 ∣ ( Q ) 1 / 2 . (201) \sum_{\ell(R)\geq\ell(Q)}T_{Q,R}|\nu_2|(R)^{1/2}\lesssim|\nu_1|(Q)^{1/2}. \tag*{(201)} ℓ ( R ) ≥ ℓ ( Q ) ∑ T Q , R ∣ ν 2 ∣ ( R ) 1/2 ≲ ∣ ν 1 ∣ ( Q ) 1/2 . (201) Note that, by Schur’s test, (201) implies that the matrix ( T Q , R ) (T_{Q,R}) ( T Q , R ) ℓ 2 \ell^2 ℓ 2
We now prove (201) . Let i = 1 , 2 i=1,2 i = 1 , 2 A ⊂ D i t r A\subset\mathcal{D}^{\mathrm{tr}}_i A ⊂ D i tr B ( z , t ) B(z,t) B ( z , t ) A A A x ∈ ⋃ P ∈ A P ∖ W i ( w i ) x\in\bigcup_{P\in A}P\setminus\mathcal{W}_i(w_i) x ∈ ⋃ P ∈ A P ∖ W i ( w i ) x ∉ H i , R i x\notin\mathcal{H}_{i,R_i} x ∈ / H i , R i n n n μ \mu μ ∣ ν i ∣ ( B ( x , ρ ) ) ≤ ( m + C 0 ) ρ n |\nu_i|(B(x,\rho))\leq(\mathfrak m+C_0)\rho^n ∣ ν i ∣ ( B ( x , ρ )) ≤ ( m + C 0 ) ρ n ρ > 0 \rho>0 ρ > 0 A A A B ( z , t ) ⊂ B ( x , 2 t ) B(z,t)\subset B(x,2t) B ( z , t ) ⊂ B ( x , 2 t )
∑ P ∈ A ∣ ν i ∣ ( P ) = ∣ ν i ∣ ( ⋃ P ∈ A P ) ≤ ∣ ν i ∣ ( B ( x , 2 t ) ) ≤ 2 n ( m + C 0 ) t n . (202) \sum_{P\in A}|\nu_i|(P)=|\nu_i|\left(\bigcup_{P\in A}P\right)\leq|\nu_i|(B(x,2t))\leq2^n(\mathfrak m+C_0)t^n. \tag*{(202)} P ∈ A ∑ ∣ ν i ∣ ( P ) = ∣ ν i ∣ ( P ∈ A ⋃ P ) ≤ ∣ ν i ∣ ( B ( x , 2 t )) ≤ 2 n ( m + C 0 ) t n . (202) Let R ∈ D 2 t r R\in\mathcal{D}^{\mathrm{tr}}_2 R ∈ D 2 tr r : = ℓ ( R ) r:=\ell(R) r := ℓ ( R ) D ( Q , R ) ≥ r D(Q,R)\geq r D ( Q , R ) ≥ r Q ∈ D 1 t r Q\in\mathcal{D}^{\mathrm{tr}}_1 Q ∈ D 1 tr ℓ ( Q ) ≤ r \ell(Q)\leq r ℓ ( Q ) ≤ r k , j ≥ 0 k,j\geq0 k , j ≥ 0 ℓ ( Q ) = ℓ k r \ell(Q)=\ell^k r ℓ ( Q ) = ℓ k r 2 j r ≤ D ( Q , R ) < 2 j + 1 r 2^j r\leq D(Q,R)<2^{j+1}r 2 j r ≤ D ( Q , R ) < 2 j + 1 r (33) , each group is contained in a ball of radius ( 2 C 1 + 1 ) 2 j + 1 r (2C_1+1)2^{j+1}r ( 2 C 1 + 1 ) 2 j + 1 r (202) yields
∑ Q ∈ D 1 t r ℓ ( Q ) ≤ r ℓ ( Q ) γ / 2 r γ / 2 D ( Q , R ) n + γ ∣ ν 1 ∣ ( Q ) ≲ ∑ k , j ≥ 0 ( ℓ k r ) γ / 2 r γ / 2 ( 2 j r ) n + γ ( 2 j r ) n = ∑ k , j ≥ 0 2 k γ / 2 2 − j γ ≲ 1. \sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1\\ \ell(Q)\le r}}\frac{\ell(Q)^{\gamma/2}r^{\gamma/2}}{D(Q,R)^{n+\gamma}}|\nu_1|(Q)\lesssim\sum_{k,j\ge0}\frac{(\ell^k r)^{\gamma/2}r^{\gamma/2}}{(2^j r)^{n+\gamma}}(2^j r)^n=\sum_{k,j\ge0}2^{k\gamma/2}2^{-j\gamma}\lesssim1. Q ∈ D 1 tr ℓ ( Q ) ≤ r ∑ D ( Q , R ) n + γ ℓ ( Q ) γ /2 r γ /2 ∣ ν 1 ∣ ( Q ) ≲ k , j ≥ 0 ∑ ( 2 j r ) n + γ ( ℓ k r ) γ /2 r γ /2 ( 2 j r ) n = k , j ≥ 0 ∑ 2 k γ /2 2 − j γ ≲ 1. which implies the first estimate in (201) . The second estimate follows analogously.
Thus, by (200) , (201) , (178) , and Schur’s test we obtain
S 1 ′ ( w ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . S_1'(w)\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. S 1 ′ ( w ) ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . We observe that, for Q ∈ D 1 t r Q\in\mathcal{D}^{\mathrm{tr}}_1 Q ∈ D 1 tr R ∈ D 2 t r R\in\mathcal{D}^{\mathrm{tr}}_2 R ∈ D 2 tr
⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ = ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ Δ 2 , R g w ⟩ . (203) \left\langle S_{|\nu_1|,\varepsilon}\Delta_{1,Q}f_w,\Delta_{2,R}g_w\right\rangle_*=\left\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}\Delta_{2,R}g_w\right\rangle. \tag*{(203)} ⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ = ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ Δ 2 , R g w ⟩ . (203) so the same argument can be applied to estimate S 1 ∗ S_1^* S 1 ∗
∣ S 1 ( w ) ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ xref‘(6.7)‘ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) , |S_1(w)|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}\overset{\text{\scriptsize{xref}`(6.7)`}}{\lesssim}\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}, ∣ S 1 ( w ) ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ xref ‘(6.7)‘ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) , which implies that there exists c S 1 > 0 c_{S_1}>0 c S 1 > 0
E ~ w ∣ S 1 ( w ) ∣ ≤ c S 1 ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (204) \widetilde{\mathbb{E}}_w|S_1(w)|\le c_{S_1}\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \tag*{(204)} E w ∣ S 1 ( w ) ∣ ≤ c S 1 ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (204) Estimates of S 2 S_2 S 2 S 3 S_3 S 3
We recall that Δ 1 , Q f w = 0 \Delta_{1,Q}f_w=0 Δ 1 , Q f w = 0 Q Q Q Δ 2 , R g w = 0 \Delta_{2,R}g_w=0 Δ 2 , R g w = 0 R R R Q ∈ D 1 t r Q\in\mathcal{D}^{\mathrm{tr}}_1 Q ∈ D 1 tr R ∈ D 2 t r R\in\mathcal{D}^{\mathrm{tr}}_2 R ∈ D 2 tr ℓ m ℓ ( R ) < ℓ ( Q ) < ℓ − m ℓ ( R ) \ell^m\ell(R)<\ell(Q)<\ell^{-m}\ell(R) ℓ m ℓ ( R ) < ℓ ( Q ) < ℓ − m ℓ ( R ) Q , R ⊂ X Q,R\subset X Q , R ⊂ X ℓ ( Q ) ∼ m ℓ ( R ) \ell(Q)\sim_m\ell(R) ℓ ( Q ) ∼ m ℓ ( R ) ℓ − m ℓ ( Q ) ≤ ℓ ( R ) ≤ ℓ − m ℓ ( Q ) \ell^{-m}\ell(Q)\le\ell(R)\le\ell^{-m}\ell(Q) ℓ − m ℓ ( Q ) ≤ ℓ ( R ) ≤ ℓ − m ℓ ( Q )
Moreover, for α = γ / ( 2 ( n + γ ) ) \alpha=\gamma/(2(n+\gamma)) α = γ / ( 2 ( n + γ )) (196) ,
dist ( Q , R ) ≤ min ( ℓ ( Q ) , ℓ ( R ) ) α max ( ℓ ( Q ) , ℓ ( R ) ) 1 − α ≤ max ( ℓ ( Q ) , ℓ ( R ) ) . \operatorname{dist}(Q,R)\le\min(\ell(Q),\ell(R))^\alpha\max(\ell(Q),\ell(R))^{1-\alpha}\le\max(\ell(Q),\ell(R)). dist ( Q , R ) ≤ min ( ℓ ( Q ) , ℓ ( R ) ) α max ( ℓ ( Q ) , ℓ ( R ) ) 1 − α ≤ max ( ℓ ( Q ) , ℓ ( R )) . Thus we have
∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ ∑ Q ∈ D 1 t r , R ∈ D 2 t r dist ( Q , R ) ≤ max ( ℓ ( Q ) , ℓ ( R ) ) ℓ ( Q ) ∼ m ℓ ( R ) ∣ ⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ ∣ . (205) |S_2(w)+S_3(w)|\le\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1,\ R\in\mathcal{D}^{\mathrm{tr}}_2\\ \operatorname{dist}(Q,R)\le\max(\ell(Q),\ell(R))\\ \ell(Q)\sim_m\ell(R)}}\left|\left\langle S_{|\nu_1|,\varepsilon}\Delta_{1,Q}f_w,\Delta_{2,R}g_w\right\rangle_*\right|. \tag*{(205)} ∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ Q ∈ D 1 tr , R ∈ D 2 tr dist ( Q , R ) ≤ m a x ( ℓ ( Q ) , ℓ ( R )) ℓ ( Q ) ∼ m ℓ ( R ) ∑ ⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ . (205) Let D ∗ ( w ) ⊂ D 1 t r × D 2 t r \mathcal{D}^*(w)\subset\mathcal{D}^{\mathrm{tr}}_1\times\mathcal{D}^{\mathrm{tr}}_2 D ∗ ( w ) ⊂ D 1 tr × D 2 tr ( Q , R ) (Q,R) ( Q , R ) dist ( Q , R ) ≤ max ( ℓ ( Q ) , ℓ ( R ) ) \operatorname{dist}(Q,R)\le\max(\ell(Q),\ell(R)) dist ( Q , R ) ≤ max ( ℓ ( Q ) , ℓ ( R )) ℓ ( Q ) ∼ m ℓ ( R ) \ell(Q)\sim_m\ell(R) ℓ ( Q ) ∼ m ℓ ( R ) D ( w ) \mathcal{D}(w) D ( w ) ( I , J ) ∈ D ( w 1 ) × D ( w 2 ) (I,J)\in\mathcal{D}(w_1)\times\mathcal{D}(w_2) ( I , J ) ∈ D ( w 1 ) × D ( w 2 ) I ∈ Ch ( Q ) I\in\operatorname{Ch}(Q) I ∈ Ch ( Q ) J ∈ Ch ( R ) J\in\operatorname{Ch}(R) J ∈ Ch ( R ) ( Q , R ) ∈ D ∗ ( w ) (Q,R)\in\mathcal{D}^*(w) ( Q , R ) ∈ D ∗ ( w )
Thus, the right-hand side of (205) reads
∑ ( Q , R ) ∈ D ∗ ( w ) ∣ ⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ ∣ ≤ ∑ ( I , J ) ∈ D ( w ) ∣ ⟨ S ∣ ν 1 ∣ , ε χ I Δ 1 , I ∗ f w , χ J Δ 2 , J ∗ g w ⟩ ∗ ∣ , (206) \sum_{(Q,R)\in\mathcal{D}^*(w)}\left|\left\langle S_{|\nu_1|,\varepsilon}\Delta_{1,Q}f_w,\Delta_{2,R}g_w\right\rangle_*\right|
\le\sum_{(I,J)\in\mathcal{D}(w)}\left|\left\langle S_{|\nu_1|,\varepsilon}\chi_I\Delta_{1,I^*}f_w,\chi_J\Delta_{2,J^*}g_w\right\rangle_*\right|, \tag*{(206)} ( Q , R ) ∈ D ∗ ( w ) ∑ ⟨ S ∣ ν 1 ∣ , ε Δ 1 , Q f w , Δ 2 , R g w ⟩ ∗ ≤ ( I , J ) ∈ D ( w ) ∑ ⟨ S ∣ ν 1 ∣ , ε χ I Δ 1 , I ∗ f w , χ J Δ 2 , J ∗ g w ⟩ ∗ , (206) where I ∗ ∈ D 1 t r I^*\in\mathcal{D}^{\mathrm{tr}}_1 I ∗ ∈ D 1 tr J ∗ ∈ D 2 t r J^*\in\mathcal{D}^{\mathrm{tr}}_2 J ∗ ∈ D 2 tr I I I J J J ( I , J ) ∈ D ( w ) (I,J)\in\mathcal{D}(w) ( I , J ) ∈ D ( w ) ℓ ( J ) ∼ m ℓ ( I ) \ell(J)\sim_m\ell(I) ℓ ( J ) ∼ m ℓ ( I ) dist ( I , J ∗ ) ≤ max ( ℓ ( I ∗ ) , ℓ ( J ∗ ) ) ≤ ℓ − m − 1 ℓ ( I ) \operatorname{dist}(I,J^*)\le\max(\ell(I^*),\ell(J^*))\le\ell^{-m-1}\ell(I) dist ( I , J ∗ ) ≤ max ( ℓ ( I ∗ ) , ℓ ( J ∗ )) ≤ ℓ − m − 1 ℓ ( I ) (33) , for fixed I I I J J J ( I , J ) ∈ D ( w ) (I,J)\in\mathcal{D}(w) ( I , J ) ∈ D ( w ) 2 m + 1 2m+1 2 m + 1 C m ℓ ( I ) C_m\ell(I) C m ℓ ( I ) X X X J J J c 1 ℓ ( J ) c_1\ell(J) c 1 ℓ ( J )
# { J : ( I , J ) ∈ D ( w ) } ≤ C m and # { I : ( I , J ) ∈ D ( w ) } ≤ C m , (207) \#\{J:(I,J)\in\mathcal{D}(w)\}\le C_m
\qquad\text{and}\qquad
\#\{I:(I,J)\in\mathcal{D}(w)\}\le C_m, \tag*{(207)} # { J : ( I , J ) ∈ D ( w )} ≤ C m and # { I : ( I , J ) ∈ D ( w )} ≤ C m , (207) and the second one, for fixed J J J
We define D t r t r ( w ) : = { ( I , J ) ∈ D ( w ) : both I and J are transit } \mathcal{D}^{\mathrm{tr}}_{\mathrm{tr}}(\mathrm{w}) := \{(I,J) \in\mathcal{D}(\mathrm{w}) : \text{both } I \text{ and } J \text{ are transit}\} D tr tr ( w ) := {( I , J ) ∈ D ( w ) : both I and J are transit } ( I 1 , I 2 ) ∈ D t r t r ( w ) (I_1,I_2) \in\mathcal{D}^{\mathrm{tr}}_{\mathrm{tr}}(\mathrm{w}) ( I 1 , I 2 ) ∈ D tr tr ( w )
D I j j ϕ : = ⟨ ϕ ⟩ j , I j ⟨ b ^ j ⟩ j , I j − ⟨ ϕ ⟩ j , I j ∗ ⟨ b ^ j ⟩ j , I j ∗ , for j = 1 , 2 and ϕ ∈ L 2 ( ∣ ν j ∣ ) , D^{j}_{I_j}\phi:= \frac{\langle\phi\rangle_{j,I_j}}{\langle\widehat b_j\rangle_{j,I_j}}-\frac{\langle\phi\rangle_{j,I_j^*}}{\langle\widehat b_j\rangle_{j,I_j^*}},
\qquad\text{for } j=1,2 \text{ and } \phi\in L^2(|\nu_j|), D I j j ϕ := ⟨ b j ⟩ j , I j ⟨ ϕ ⟩ j , I j − ⟨ b j ⟩ j , I j ∗ ⟨ ϕ ⟩ j , I j ∗ , for j = 1 , 2 and ϕ ∈ L 2 ( ∣ ν j ∣ ) , where I j ∗ ∈ D ( w j ) I_j^* \in\mathcal{D}(w_j) I j ∗ ∈ D ( w j ) I j I_j I j (175) , if ( I , J ) ∈ D t r t r ( w ) (I,J) \in\mathcal{D}^{\mathrm{tr}}_{\mathrm{tr}}(\mathrm{w}) ( I , J ) ∈ D tr tr ( w )
χ I D I 1 f w b ^ 1 = χ I Δ 1 , I ∗ f w and χ J D J 2 g w b ^ 2 = χ J Δ 2 , J ∗ g w , (208) \chi_I D^1_I f_{\mathrm{w}}\widehat b_1=\chi_I\Delta_{1,I^*}f_{\mathrm{w}}
\qquad\text{and}\qquad
\chi_J D^2_J g_{\mathrm{w}}\widehat b_2=\chi_J\Delta_{2,J^*}g_{\mathrm{w}}, \tag*{(208)} χ I D I 1 f w b 1 = χ I Δ 1 , I ∗ f w and χ J D J 2 g w b 2 = χ J Δ 2 , J ∗ g w , (208) so
∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ 1 , I ∗ f w ) , χ J Δ 2 , J ∗ g w ⟩ ∗ ∣ = ∣ ⟨ S ∣ ν 1 ∣ , ε ( D I 1 f w χ I b ^ 1 ) , D J 2 g w χ J b ^ 2 ⟩ ∗ ∣ . (209) \left|\left\langle S_{|\nu_1|,\varepsilon}\left(\chi_I\Delta_{1,I^*}f_{\mathrm{w}}\right),\chi_J\Delta_{2,J^*}g_{\mathrm{w}}\right\rangle_*\right|
=
\left|\left\langle S_{|\nu_1|,\varepsilon}\left(D^1_I f_{\mathrm{w}}\chi_I\widehat b_1\right),D^2_Jg_{\mathrm{w}}\chi_J\widehat b_2\right\rangle_*\right|. \tag*{(209)} ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ 1 , I ∗ f w ) , χ J Δ 2 , J ∗ g w ⟩ ∗ = ⟨ S ∣ ν 1 ∣ , ε ( D I 1 f w χ I b 1 ) , D J 2 g w χ J b 2 ⟩ ∗ . (209) The random almost-covering balls. In the next proposition, we state the properties of a “random” family of balls constructed by Hytönen and Martikainen in the quasi-metric setting. The balls have radii comparable to ϑ k \vartheta^k ϑ k k ∈ Z k\in\mathbb{Z} k ∈ Z 0 < ϑ < 1 / 32 0<\vartheta<1/32 0 < ϑ < 1/32 1 / 2 1/2 1/2 ξ \xi ξ ( Ω ^ , P ^ ) (\widehat{\Omega},\widehat{\mathbb{P}}) ( Ω , P ) w \mathrm{w} w E ^ \widehat{\mathbb{E}} E k ∈ Z k\in\mathbb{Z} k ∈ Z ξ \xi ξ
Proposition 6.13 (see Proposition 8.2 in [41 ] ). Let 0 < ϑ < 1 / 32 0<\vartheta<1/32 0 < ϑ < 1/32 k ∈ Z k\in\mathbb{Z} k ∈ Z v ∈ ( 0 , 1 ) v\in(0,1) v ∈ ( 0 , 1 ) Λ ~ ∈ ( 0 , 1 ) \widetilde{\Lambda}\in(0,1) Λ ∈ ( 0 , 1 ) s , M ≥ 1 s,M\ge1 s , M ≥ 1 D D D ϑ \vartheta ϑ v v v ξ ∈ Ω ^ \xi\in\widehat{\Omega} ξ ∈ Ω B ξ \mathcal{B}_{\xi} B ξ X X X
If B , B ′ ∈ B ξ B,B'\in\mathcal{B}_{\xi} B , B ′ ∈ B ξ
1 32 ϑ k + n 1 s ≤ r ( B ) ≤ 1 16 ϑ k + n 1 s and 1 32 ϑ k + n 2 s ≤ r ( B ′ ) ≤ 1 16 ϑ k + n 2 s , (210) \frac{1}{32}\vartheta^{k+n_1s}\le r(B)\le\frac{1}{16}\vartheta^{k+n_1s}
\qquad\text{and}\qquad
\frac{1}{32}\vartheta^{k+n_2s}\le r(B')\le\frac{1}{16}\vartheta^{k+n_2s}, \tag*{(210)} 32 1 ϑ k + n 1 s ≤ r ( B ) ≤ 16 1 ϑ k + n 1 s and 32 1 ϑ k + n 2 s ≤ r ( B ′ ) ≤ 16 1 ϑ k + n 2 s , (210) for some n 1 , n 2 = 0 , 1 , … , M − 1 n_1,n_2=0,1,\ldots,M-1 n 1 , n 2 = 0 , 1 , … , M − 1 B ≠ B ′ B\ne B' B = B ′
d ( x B , x B ′ ) ≥ r ( B ) + r ( B ′ ) + Λ ~ max ( r ( B ) , r ( B ′ ) ) , d(x_B,x_{B'})\ge r(B)+r(B')+\widetilde{\Lambda}\max(r(B),r(B')), d ( x B , x B ′ ) ≥ r ( B ) + r ( B ′ ) + Λ max ( r ( B ) , r ( B ′ )) , where x B x_B x B x B ′ x_{B'} x B ′ B B B B ′ B' B ′
P ^ ( x ∈ ⋃ B ∈ B ξ B ) > 1 − v , for every x ∈ X . (211) \widehat{\mathbb{P}}\left(x\in\bigcup_{B\in\mathcal{B}_{\xi}}B\right)>1-v,
\qquad\text{for every }x\in X. \tag*{(211)} P x ∈ B ∈ B ξ ⋃ B > 1 − v , for every x ∈ X . (211) Moreover, the set { ( ξ , x ) : x ∈ ⋃ B ∈ B ξ B } \{(\xi,x):x\in\bigcup_{B\in\mathcal{B}_{\xi}}B\} {( ξ , x ) : x ∈ ⋃ B ∈ B ξ B }
In [41 ] , the separation is stated for the balls themselves, but the separation of their centers follows from the construction. Indeed, if n 1 = n 2 n_1=n_2 n 1 = n 2 (210) , then
d ( x B , x B ′ ) ≥ ϑ k + n 1 s / 4 ≥ 2 ( r ( B ) + r ( B ′ ) ) . d(x_B,x_{B'})\ge\vartheta^{k+n_1s}/4\ge2(r(B)+r(B')). d ( x B , x B ′ ) ≥ ϑ k + n 1 s /4 ≥ 2 ( r ( B ) + r ( B ′ )) . If n 1 < n 2 n_1<n_2 n 1 < n 2 x B ′ ∉ ( 1 + ω ) B x_{B'}\notin(1+\omega)B x B ′ ∈ / ( 1 + ω ) B r ( B ′ ) ≤ 2 ϑ s r ( B ) < ω r ( B ) / 3 r(B')\le2\vartheta^s r(B)<\omega r(B)/3 r ( B ′ ) ≤ 2 ϑ s r ( B ) < ω r ( B ) /3 ω = ω ( v ) ∈ ( 0 , 1 ) \omega=\omega(v)\in(0,1) ω = ω ( v ) ∈ ( 0 , 1 ) [41 Section 8] , and s s s 2 ϑ s < ω / 3 2\vartheta^s<\omega/3 2 ϑ s < ω /3 Λ ~ = 2 ω / 3 \widetilde{\Lambda}=2\omega/3 Λ = 2 ω /3
Note that the balls in B ξ \mathcal{B}_{\xi} B ξ (210) implies
1 2 ϑ M s ≤ r ( B ) r ( B ′ ) ≤ 2 ϑ − M s . \frac{1}{2}\vartheta^{Ms}\le\frac{r(B)}{r(B')}\le2\vartheta^{-Ms}. 2 1 ϑ M s ≤ r ( B ′ ) r ( B ) ≤ 2 ϑ − M s . For ϑ \vartheta ϑ k k k υ \upsilon υ Proposition 6.13 and ξ ∈ Ω ^ \xi\in\widehat{\Omega} ξ ∈ Ω B ξ \mathcal{B}_{\xi} B ξ M Λ M_{\Lambda} M Λ Λ ~ \widetilde{\Lambda} Λ B ∈ B ξ B \in\mathcal{B}_{\xi} B ∈ B ξ B ^ \widehat{B} B x B x_B x B M Λ − ∣ ν 1 ∣ M_{\Lambda}-|\nu_1| M Λ − ∣ ν 1 ∣ M Λ − ∣ ν 2 ∣ M_{\Lambda}-|\nu_2| M Λ − ∣ ν 2 ∣ x B x_B x B r ( B ) r(B) r ( B )
B ⊂ B ^ ⊂ ( 1 + Λ ~ 8 ) B . (212) B \subset\widehat{B} \subset\left(1+\frac{\widetilde{\Lambda}}{8}\right)B. \tag*{(212)} B ⊂ B ⊂ ( 1 + 8 Λ ) B . (212) Let λ ∈ ( 0 , 1 ) \lambda\in(0,1) λ ∈ ( 0 , 1 ) ( I , J ) ∈ D ( w ) (I,J) \in\mathcal{D}(w) ( I , J ) ∈ D ( w ) k ∈ Z k \in\mathbb{Z} k ∈ Z
ϑ k ≤ λ min ( ℓ ( I ) , ℓ ( J ) ) < ϑ k − 1 . (213) \vartheta^k \leq\lambda\min(\ell(I),\ell(J)) < \vartheta^{k-1}. \tag*{(213)} ϑ k ≤ λ min ( ℓ ( I ) , ℓ ( J )) < ϑ k − 1 . (213) Thus k k k λ \lambda λ min ( ℓ ( I ) , ℓ ( J ) ) \min(\ell(I),\ell(J)) min ( ℓ ( I ) , ℓ ( J ))
We denote B ^ ξ : = { B ^ : B ∈ B ξ } \widehat{\mathcal{B}}_{\xi}:=\{\widehat{B}:B\in\mathcal{B}_{\xi}\} B ξ := { B : B ∈ B ξ } B ^ , B ^ ′ ∈ B ^ ξ \widehat{B},\widehat{B}'\in\widehat{\mathcal{B}}_{\xi} B , B ′ ∈ B ξ (212) , the triangle inequality, and Proposition 6.13 give
dist ( B ^ , B ^ ′ ) ≥ d ( x B , x B ′ ) − ( 1 + Λ ~ 8 ) ( r ( B ) + r ( B ′ ) ) ≥ 3 4 Λ ~ max ( r ( B ) , r ( B ′ ) ) . \operatorname{dist}(\widehat{B},\widehat{B}')\geq d(x_B,x_{B'})-\left(1+\frac{\widetilde{\Lambda}}{8}\right)(r(B)+r(B'))\geq\frac{3}{4}\widetilde{\Lambda}\max(r(B),r(B')). dist ( B , B ′ ) ≥ d ( x B , x B ′ ) − ( 1 + 8 Λ ) ( r ( B ) + r ( B ′ )) ≥ 4 3 Λ max ( r ( B ) , r ( B ′ )) . Since r ( B ^ ) ≤ 2 r ( B ) r(\widehat{B})\leq2r(B) r ( B ) ≤ 2 r ( B ) r ( B ^ ′ ) ≤ 2 r ( B ′ ) r(\widehat{B}')\leq2r(B') r ( B ′ ) ≤ 2 r ( B ′ )
dist ( B ^ , B ^ ′ ) ≥ Λ ~ 8 max ( r ( B ^ ) , r ( B ^ ′ ) ) . (214) \operatorname{dist}(\widehat{B},\widehat{B}')\geq\frac{\widetilde{\Lambda}}{8}\max(r(\widehat{B}),r(\widehat{B}')). \tag*{(214)} dist ( B , B ′ ) ≥ 8 Λ max ( r ( B ) , r ( B ′ )) . (214) The bound (211) readily implies
P ^ ( x ∈ ⋃ B ^ ∈ B ^ ξ B ^ ) > 1 − υ , for every x ∈ X , \widehat{\mathbb{P}}\left(x\in\bigcup_{\widehat{B}\in\widehat{\mathcal{B}}_{\xi}}\widehat{B}\right)>1-\upsilon,\qquad\text{for every }x\in X, P x ∈ B ∈ B ξ ⋃ B > 1 − υ , for every x ∈ X , so for every non-negative Radon measure m m m X X X A ⊂ X A\subset X A ⊂ X
E ^ ( m ( A ∖ ⋃ B ^ ∈ B ^ ξ B ^ ) ) ≤ υ m ( A ) . (215) \widehat{\mathbb{E}}\left(m\left(A\setminus\bigcup_{\widehat{B}\in\widehat{\mathcal{B}}_{\xi}}\widehat{B}\right)\right)\leq\upsilon m(A). \tag*{(215)} E m A ∖ B ∈ B ξ ⋃ B ≤ υ m ( A ) . (215) The surgery argument. Let ( I , J ) ∈ D ( w ) (I,J)\in\mathcal{D}(w) ( I , J ) ∈ D ( w ) h : = λ min ( ℓ ( I ) , ℓ ( J ) ) h:=\lambda\min(\ell(I),\ell(J)) h := λ min ( ℓ ( I ) , ℓ ( J )) B ^ ξ \widehat{\mathcal{B}}_{\xi} B ξ (213) to decompose both I I I J J J B ^ ξ \widehat{\mathcal{B}}_{\xi} B ξ 6 r ( B ) ≤ h 6r(B)\leq h 6 r ( B ) ≤ h
C J : = { B ∈ B ^ ξ : 5 B ⊂ J } , I Δ = I ∩ ⋃ B ∈ C J B , \mathcal{C}_J:=\{B\in\widehat{\mathcal{B}}_{\xi}:5B\subset J\},\qquad I_{\Delta}=I\cap\bigcup_{B\in\mathcal{C}_J}B, C J := { B ∈ B ξ : 5 B ⊂ J } , I Δ = I ∩ B ∈ C J ⋃ B , I s e p = { x ∈ I : dist ( x , J ) > h } , I ∂ = I ∖ ( I s e p ∪ I Δ ) . I_{\mathrm{sep}}=\{x\in I:\operatorname{dist}(x,J)>h\},\qquad I_{\partial}=I\setminus(I_{\mathrm{sep}}\cup I_{\Delta}). I sep = { x ∈ I : dist ( x , J ) > h } , I ∂ = I ∖ ( I sep ∪ I Δ ) . The sets J Δ J_{\Delta} J Δ J s e p J_{\mathrm{sep}} J sep J ∂ J_{\partial} J ∂ I I I J J J
dist ( I s e p , J ) ≥ h and dist ( J s e p , I ) ≥ h . (216) \operatorname{dist}(I_{\mathrm{sep}},J)\geq h\qquad\text{and}\qquad\operatorname{dist}(J_{\mathrm{sep}},I)\geq h. \tag*{(216)} dist ( I sep , J ) ≥ h and dist ( J sep , I ) ≥ h . (216) For s > 0 s>0 s > 0
∂ s J : = { x ∈ X : dist ( x , J ) ≤ s , dist ( x , X ∖ J ) ≤ s } , \partial_sJ:=\{x\in X:\operatorname{dist}(x,J)\leq s,\ \operatorname{dist}(x,X\setminus J)\leq s\}, ∂ s J := { x ∈ X : dist ( x , J ) ≤ s , dist ( x , X ∖ J ) ≤ s } , and note that (216) implies the set inclusion
I ∂ ⊂ ( I ∩ ∂ h J ) ∪ ( I ∖ ⋃ B ∈ B ^ ξ B ) . I_{\partial}\subset(I\cap\partial_hJ)\cup\left(I\setminus\bigcup_{B\in\widehat{\mathcal{B}}_{\xi}}B\right). I ∂ ⊂ ( I ∩ ∂ h J ) ∪ I ∖ B ∈ B ξ ⋃ B . Indeed, a covered point x ∈ I ∩ J x \in I \cap J x ∈ I ∩ J B B B 5 B ⊄ J 5B \not\subset J 5 B ⊂ J 6 r ( B ) ≤ h 6r(B) \le h 6 r ( B ) ≤ h X ∖ J X \setminus J X ∖ J
By (213) , the scale of the family B ~ ξ \widetilde{\mathcal{B}}_{\xi} B ξ J J J I I I K ( I ) \mathcal{K}(I) K ( I ) k k k (213) for the 2 m + 1 2m+1 2 m + 1 J J J ℓ ( J ) ∼ m ℓ ( I ) \ell(J) \sim_m \ell(I) ℓ ( J ) ∼ m ℓ ( I ) K ( I ) \mathcal{K}(I) K ( I ) 2 m + 1 2m+1 2 m + 1 k ∈ K ( I ) k \in\mathcal{K}(I) k ∈ K ( I ) Proposition 6.13 with υ / ( 2 m + 1 ) \upsilon/(2m+1) υ / ( 2 m + 1 ) υ \upsilon υ Λ ~ \widetilde{\Lambda} Λ M M M m m m υ \upsilon υ B ~ ξ , k \widetilde{\mathcal{B}}_{\xi,k} B ξ , k h ≤ λ ℓ ( J ) h \le\lambda\ell(J) h ≤ λ ℓ ( J ) ℓ ( J ) ∼ m ℓ ( I ) \ell(J) \sim_m \ell(I) ℓ ( J ) ∼ m ℓ ( I ) B ~ ξ \widetilde{\mathcal{B}}_{\xi} B ξ K ( I ) \mathcal{K}(I) K ( I )
I ∂ ⊂ I ∩ ⋃ J ′ ∈ D ( w 2 ) ℓ ( J ′ ) ∼ m ℓ ( I ) ∂ λ ℓ ( J ′ ) J ′ ∪ ⋃ k ∈ K ( I ) ( I ∖ ⋃ B ∈ B ~ ξ , k B ) = : I ~ ∂ , 1 w 2 ∪ I ~ ∂ , 2 ξ = : I ~ ∂ w 2 , ξ . (217) \begin{aligned}
I_{\partial} \subset I \cap\bigcup_{\substack{J' \in\mathcal{D}(w_2)\\ \ell(J') \sim_m \ell(I)}} \partial_{\lambda\ell(J')}J' \cup\bigcup_{k \in\mathcal{K}(I)} \left(I \setminus\bigcup_{B \in\widetilde{\mathcal{B}}_{\xi,k}} B\right) \\
&=:\widetilde{I}^{w_2}_{\partial,1} \cup\widetilde{I}^{\xi}_{\partial,2} =: \widetilde{I}^{w_2,\xi}_{\partial}. \tag*{(217)}
\end{aligned} I ∂ ⊂ I ∩ J ′ ∈ D ( w 2 ) ℓ ( J ′ ) ∼ m ℓ ( I ) ⋃ ∂ λ ℓ ( J ′ ) J ′ ∪ k ∈ K ( I ) ⋃ I ∖ B ∈ B ξ , k ⋃ B =: I ∂ , 1 w 2 ∪ I ∂ , 2 ξ =: I ∂ w 2 , ξ . (217) Here, the union is over all the cubes J ′ ∈ D ( w 2 ) J' \in\mathcal{D}(w_2) J ′ ∈ D ( w 2 ) J J J
We also observe that, if s > 0 s>0 s > 0 J ′ ∈ D ( w 2 ) J' \in\mathcal{D}(w_2) J ′ ∈ D ( w 2 ) x ∈ ∂ s J ′ x \in\partial_sJ' x ∈ ∂ s J ′ J x ∈ D ( w 2 ) J_x \in\mathcal{D}(w_2) J x ∈ D ( w 2 ) J ′ J' J ′ x x x dist ( x , X ∖ J x ) ≤ s \operatorname{dist}(x, X \setminus J_x) \le s dist ( x , X ∖ J x ) ≤ s x ∈ J ′ x \in J' x ∈ J ′ J x = J ′ J_x=J' J x = J ′ J ′ ⊂ X ∖ J x J' \subset X \setminus J_x J ′ ⊂ X ∖ J x dist ( x , X ∖ J x ) ≤ dist ( x , J ′ ) ≤ s \operatorname{dist}(x, X \setminus J_x) \le\operatorname{dist}(x,J') \le s dist ( x , X ∖ J x ) ≤ dist ( x , J ′ ) ≤ s
Finally, we write
I Δ = ⋃ i ∈ I Δ B I , i and J Δ = ⋃ j ∈ J Δ B J , j . I_{\Delta}=\bigcup_{i \in\mathcal{I}_{\Delta}} B_{I,i}
\qquad\text{and}\qquad
J_{\Delta}=\bigcup_{j \in\mathcal{J}_{\Delta}} B_{J,j}. I Δ = i ∈ I Δ ⋃ B I , i and J Δ = j ∈ J Δ ⋃ B J , j . where { B I , i } i ∈ I Δ \{B_{I,i}\}_{i \in\mathcal{I}_{\Delta}} { B I , i } i ∈ I Δ B ∩ I B \cap I B ∩ I B ∈ C J B \in\mathcal{C}_J B ∈ C J { B J , j } j ∈ J Δ \{B_{J,j}\}_{j \in\mathcal{J}_{\Delta}} { B J , j } j ∈ J Δ B ′ ∩ J B' \cap J B ′ ∩ J B ′ ∈ B ~ ξ B' \in\widetilde{\mathcal{B}}_{\xi} B ′ ∈ B ξ 5 B ′ ⊂ I 5B' \subset I 5 B ′ ⊂ I B I , i = B ∩ I B_{I,i}=B \cap I B I , i = B ∩ I B J , j = B ′ ∩ J B_{J,j}=B' \cap J B J , j = B ′ ∩ J B = B ′ B=B' B = B ′ 5 B ⊂ I ∩ J 5B \subset I \cap J 5 B ⊂ I ∩ J B I , i = B J , j = B B_{I,i}=B_{J,j}=B B I , i = B J , j = B B ≠ B ′ B \ne B' B = B ′ (214) gives
dist ( B I , i , B J , j ) ≥ dist ( B , B ′ ) ≥ Λ ~ 8 max ( r ( B ) , r ( B ′ ) ) . \operatorname{dist}(B_{I,i},B_{J,j}) \ge\operatorname{dist}(B,B') \ge\frac{\widetilde{\Lambda}}{8}\max(r(B),r(B')). dist ( B I , i , B J , j ) ≥ dist ( B , B ′ ) ≥ 8 Λ max ( r ( B ) , r ( B ′ )) . To bound (206) we first consider
S_{\mathrm{sep}}(w) := \sum_{(I,J)\in\mathcal{D}(w)}
\left|\left\langle S_{|\nu_1|,\varepsilon}\bigl(\chi_{I_{\mathrm{sep}}\Delta_{1,I^*}f_w\bigr),\chi_{J\Delta_{2,J^*}g_w}\right\rangle\right|.}
Lemma 6.14. We have
S s e p ( w ) ≲ λ , m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . S_{\mathrm{sep}}(w) \lesssim_{\lambda,m} \|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. S sep ( w ) ≲ λ , m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . Proof. If ( I , J ) ∈ D ( w ) (I,J) \in\mathcal{D}(w) ( I , J ) ∈ D ( w ) x ∈ I s e p x \in I_{\mathrm{sep}} x ∈ I sep dist ( x , J ) ≳ λ , m ℓ ( I ) \operatorname{dist}(x,J) \gtrsim_{\lambda,m} \ell(I) dist ( x , J ) ≳ λ , m ℓ ( I ) (159) and the Cauchy–Schwarz inequality yield
\begin{aligned}
\left|\left\langle S_{|\nu_1|,\varepsilon}\bigl(\chi_{I_{\mathrm{sep}}\Delta_{1,I^*}f_w\bigr),\chi_{J\Delta_{2,J^*}g_w}\right\rangle\right|
\lesssim_{\lambda,m}
\frac{1}{\ell(I)^n}
\|\chi_{I_{\mathrm{sep}}\Delta_{1,I^*}f_w}\|_{L^1(|\nu_1|)}
\|\chi_{J\Delta_{2,J^*}g_w}\|_{L^1(|\nu_2|)}} \\
&\le
\frac{|\nu_1|(I_{\mathrm{sep}})^{1/2}|\nu_2|(J)^{1/2}}{\ell(I)^n}
\|\chi_{I_{\mathrm{sep}}\Delta_{1,I^*}f_w}\|_{L^2(|\nu_1|)}
\|\chi_{J\Delta_{2,J^*}g_w}\|_{L^2(|\nu_2|)}
\end{aligned}
≲ m ∥ χ I s e p Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≤ ∥ χ I Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) , \lesssim_m
\|\chi_{I_{\mathrm{sep}}\Delta_{1,I^*}f_w}\|_{L^2(|\nu_1|)}
\|\chi_{J\Delta_{2,J^*}g_w}\|_{L^2(|\nu_2|)}
\le
\|\chi_{I\Delta_{1,I^*}f_w}\|_{L^2(|\nu_1|)}
\|\chi_{J\Delta_{2,J^*}g_w}\|_{L^2(|\nu_2|)}, ≲ m ∥ χ I sep Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≤ ∥ χ I Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) , where we recall that ∣ ν 1 ∣ ( I ) ≲ ℓ ( I ) n |\nu_1|(I) \lesssim\ell(I)^n ∣ ν 1 ∣ ( I ) ≲ ℓ ( I ) n ∣ ν 2 ∣ ( J ) ≲ ℓ ( J ) n |\nu_2|(J) \lesssim\ell(J)^n ∣ ν 2 ∣ ( J ) ≲ ℓ ( J ) n I I I J J J
Hence, the Cauchy-Schwarz inequality, (207) , and (178) give
∑ ( I , J ) ∈ D ( ω ) ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I s e p Δ 1 , I ∗ f ω ) , χ J Δ 2 , J ∗ g ω ⟩ ∗ ∣ ≲ λ , m ∑ ( I , J ) ∈ D ( ω ) ∥ χ I Δ 1 , I ∗ f ω ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g ω ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f ω ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ω ∥ L 2 ( ∣ ν 2 ∣ ) . \sum_{(I,J)\in\mathcal{D}(\omega)} \left|\left\langle S_{|\nu_1|,\varepsilon}\left(\chi_{I_{\mathrm{sep}}}\Delta_{1,I}^{*}f_{\omega}\right),\chi_J\Delta_{2,J}^{*}g_{\omega}\right\rangle_{*}\right|
\lesssim_{\lambda,m}\sum_{(I,J)\in\mathcal{D}(\omega)}\left\|\chi_I\Delta_{1,I}^{*}f_{\omega}\right\|_{L^2(|\nu_1|)}\left\|\chi_J\Delta_{2,J}^{*}g_{\omega}\right\|_{L^2(|\nu_2|)}
\lesssim\left\|f_{\omega}\right\|_{L^2(|\nu_1|)}\left\|g_{\omega}\right\|_{L^2(|\nu_2|)}. ( I , J ) ∈ D ( ω ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ I sep Δ 1 , I ∗ f ω ) , χ J Δ 2 , J ∗ g ω ⟩ ∗ ≲ λ , m ( I , J ) ∈ D ( ω ) ∑ χ I Δ 1 , I ∗ f ω L 2 ( ∣ ν 1 ∣ ) χ J Δ 2 , J ∗ g ω L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f ω ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ω ∥ L 2 ( ∣ ν 2 ∣ ) . Next, we analyze the contribution in (206) corresponding to I ∂ I_{\partial} I ∂
S ∂ ( ω ) : = ∑ ( I , J ) ∈ D ( ω ) ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I ∂ Δ 1 , I ∗ f ω ) , χ J Δ 2 , J ∗ g ω ⟩ ∗ ∣ . S_{\partial}(\omega):=\sum_{(I,J)\in\mathcal{D}(\omega)}\left|\left\langle S_{|\nu_1|,\varepsilon}\left(\chi_{I_{\partial}}\Delta_{1,I}^{*}f_{\omega}\right),\chi_J\Delta_{2,J}^{*}g_{\omega}\right\rangle_{*}\right|. S ∂ ( ω ) := ( I , J ) ∈ D ( ω ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ I ∂ Δ 1 , I ∗ f ω ) , χ J Δ 2 , J ∗ g ω ⟩ ∗ . For m ∈ M + ( X ) m\in\mathcal{M}_{+}(X) m ∈ M + ( X ) ( ϕ j ) j ∈ Z (\phi_j)_{j\in\mathbb{Z}} ( ϕ j ) j ∈ Z m m m
∥ ( ϕ j ) j ∈ Z ∥ L 2 ( m ; ℓ 2 ) : = ∥ ( ∑ j ∈ Z ∣ ϕ j ∣ 2 ) 1 / 2 ∥ L 2 ( m ) . \left\|( \phi_j)_{j\in\mathbb{Z}}\right\|_{L^2(m;\ell^2)}:=\left\|\left(\sum_{j\in\mathbb{Z}}|\phi_j|^2\right)^{1/2}\right\|_{L^2(m)}. ∥ ( ϕ j ) j ∈ Z ∥ L 2 ( m ; ℓ 2 ) := j ∈ Z ∑ ∣ ϕ j ∣ 2 1/2 L 2 ( m ) . The following estimate follows from the Cauchy-Schwarz inequality.
Lemma 6.15. If ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) < ∞ \|(f_j)_{j\in\mathbb{Z}}\|_{L^2(|\nu_1|;\ell^2)}<\infty ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) < ∞ ∥ ( g j ) j ∈ Z ∥ L 2 ( ∣ ν 2 ∣ ; ℓ 2 ) < ∞ \|(g_j)_{j\in\mathbb{Z}}\|_{L^2(|\nu_2|;\ell^2)}<\infty ∥ ( g j ) j ∈ Z ∥ L 2 ( ∣ ν 2 ∣ ; ℓ 2 ) < ∞
∑ j ∈ Z ∣ ⟨ S ∣ ν 1 ∣ , ε f j , g j ⟩ ∗ ∣ ≤ ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) ∥ ( g j ) j ∈ Z ∥ L 2 ( ∣ ν 2 ∣ ; ℓ 2 ) . \sum_{j\in\mathbb{Z}}\left|\langle S_{|\nu_1|,\varepsilon}f_j,g_j\rangle_{*}\right|\leq\|S_{|\nu_1|,\varepsilon}\|_{L^2(|\nu_1|)\to L^2(|\nu_2|)}\|(f_j)_{j\in\mathbb{Z}}\|_{L^2(|\nu_1|;\ell^2)}\|(g_j)_{j\in\mathbb{Z}}\|_{L^2(|\nu_2|;\ell^2)}. j ∈ Z ∑ ⟨ S ∣ ν 1 ∣ , ε f j , g j ⟩ ∗ ≤ ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) ∥ ( g j ) j ∈ Z ∥ L 2 ( ∣ ν 2 ∣ ; ℓ 2 ) . Proof. By the Cauchy-Schwarz inequality, for every j ∈ Z j\in\mathbb{Z} j ∈ Z
∣ ⟨ S ∣ ν 1 ∣ , ε f j , g j ⟩ ∗ ∣ ≤ ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ∥ f j ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g j ∥ L 2 ( ∣ ν 2 ∣ ) . |\langle S_{|\nu_1|,\varepsilon}f_j,g_j\rangle_{*}|\leq\|S_{|\nu_1|,\varepsilon}\|_{L^2(|\nu_1|)\to L^2(|\nu_2|)}\|f_j\|_{L^2(|\nu_1|)}\|g_j\|_{L^2(|\nu_2|)}. ∣ ⟨ S ∣ ν 1 ∣ , ε f j , g j ⟩ ∗ ∣ ≤ ∥ S ∣ ν 1 ∣ , ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ∥ f j ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g j ∥ L 2 ( ∣ ν 2 ∣ ) . The lemma follows by summing over j j j ℓ 2 \ell^2 ℓ 2 ∑ j ∈ Z ∥ f j ∥ L 2 ( ∣ ν 1 ∣ ) 2 = ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) 2 \sum_{j\in\mathbb{Z}}\|f_j\|_{L^2(|\nu_1|)}^2=\|(f_j)_{j\in\mathbb{Z}}\|_{L^2(|\nu_1|;\ell^2)}^2 ∑ j ∈ Z ∥ f j ∥ L 2 ( ∣ ν 1 ∣ ) 2 = ∥ ( f j ) j ∈ Z ∥ L 2 ( ∣ ν 1 ∣ ; ℓ 2 ) 2 ( g j ) j ∈ Z (g_j)_{j\in\mathbb{Z}} ( g j ) j ∈ Z
The bound of S ∂ ( ω ) S_{\partial}(\omega) S ∂ ( ω ) L 2 ( ∣ ν 1 ∣ ) L^2(|\nu_1|) L 2 ( ∣ ν 1 ∣ )
Lemma 6.16. We have
E ~ ω E ^ ∑ I ∈ D ( ω 1 ) ∑ I ∗ ∈ D t r ∥ χ I ∂ ω 2 , ξ Δ 1 , I ∗ f ω ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≤ C [ ( 2 m + 1 ) λ η + υ ] P Ω 2 ( G ) ∥ f ω ∥ L 2 ( ∣ ν 1 ∣ ) 2 . \widetilde{\mathbb{E}}_{\omega}\widehat{\mathbb{E}}\sum_{I\in\mathcal{D}(\omega_1)}\sum_{I^{*}\in\mathcal{D}_{\mathrm{tr}}}\left\|\chi_{I_{\partial\omega_2,\xi}}\Delta_{1,I}^{*}f_{\omega}\right\|_{L^2(|\nu_1|)}^2
\leq\frac{C[(2m+1)\lambda^{\eta}+\upsilon]}{\mathbb{P}\Omega^2(\mathcal{G})}\|f_{\omega}\|_{L^2(|\nu_1|)}^2. E ω E I ∈ D ( ω 1 ) ∑ I ∗ ∈ D tr ∑ χ I ∂ ω 2 , ξ Δ 1 , I ∗ f ω L 2 ( ∣ ν 1 ∣ ) 2 ≤ P Ω 2 ( G ) C [( 2 m + 1 ) λ η + υ ] ∥ f ω ∥ L 2 ( ∣ ν 1 ∣ ) 2 . The constant C C C
Proof. Let ω 1 ∈ Ω \omega_1\in\Omega ω 1 ∈ Ω I ∈ D ( ω 1 ) I\in\mathcal{D}(\omega_1) I ∈ D ( ω 1 ) x ∈ I x\in I x ∈ I x ∈ ∂ λ ℓ ( J ′ ) J ′ x\in\partial_{\lambda\ell(J')}J' x ∈ ∂ λ ℓ ( J ′ ) J ′ J ′ ∈ D ( ω 2 ) J'\in\mathcal{D}(\omega_2) J ′ ∈ D ( ω 2 ) J x ∈ D ( ω 2 ) J_x\in\mathcal{D}(\omega_2) J x ∈ D ( ω 2 ) J ′ J' J ′ x x x (217) , we have dist ( x , X ∖ J x ) ≤ λ ℓ ( J ′ ) \operatorname{dist}(x,X\setminus J_x)\leq\lambda\ell(J') dist ( x , X ∖ J x ) ≤ λ ℓ ( J ′ ) x x x ∂ ( J x , λ ℓ ( J ′ ) ) \partial(J_x,\lambda\ell(J')) ∂ ( J x , λ ℓ ( J ′ )) ( 2 m + 1 ) (2m+1) ( 2 m + 1 ) J ′ J' J ′ ℓ ( J ′ ) ∼ m ℓ ( I ) \ell(J')\sim_m\ell(I) ℓ ( J ′ ) ∼ m ℓ ( I )
P ω 2 ( { x ∈ I ∂ ω 1 ω ~ 2 } ) ≤ C 2 ( 2 m + 1 ) λ η . (218) \mathbb{P}_{\omega_2}\left(\{x\in I_{\partial\omega_1}^{\widetilde{\omega}_2}\}\right)\leq C_2(2m+1)\lambda^\eta. \tag*{(218)} P ω 2 ( { x ∈ I ∂ ω 1 ω 2 } ) ≤ C 2 ( 2 m + 1 ) λ η . (218) Moreover, K ( I ) \mathcal{K}(I) K ( I ) ( 2 m + 1 ) (2m+1) ( 2 m + 1 ) υ / ( 2 m + 1 ) \upsilon/(2m+1) υ / ( 2 m + 1 ) (211) and the inclusion B ⊂ B ^ B\subset\widehat{B} B ⊂ B
P ^ ( { x ∈ I ∂ ω 2 ξ ~ } ) ≤ υ . (219) \widehat{\mathbb{P}}\left(\{x\in I_{\partial\omega_2}^{\widetilde{\xi}}\}\right)\leq\upsilon. \tag*{(219)} P ( { x ∈ I ∂ ω 2 ξ } ) ≤ υ . (219) We extend the martingale differences Δ 1 , Q \Delta_{1,Q} Δ 1 , Q w 1 ∈ Ω w_1 \in\Omega w 1 ∈ Ω (178) holds uniformly in w 1 w_1 w 1 (176) and Definition 6.9, we have
Δ 1 , I ∗ f w = χ { I ∗ ∈ D 1 G ( w ) , ℓ ( I ∗ ) ≥ ℓ K 1 } Δ 1 , I ∗ f . \Delta_{1,I^*}f_w=\chi_{\{I^*\in\mathcal{D}_1^G(w),\ \ell(I^*)\geq\ell K_1\}}\Delta_{1,I^*}f. Δ 1 , I ∗ f w = χ { I ∗ ∈ D 1 G ( w ) , ℓ ( I ∗ ) ≥ ℓ K 1 } Δ 1 , I ∗ f . In particular, ∣ Δ 1 , I ∗ f w ∣ ≤ ∣ Δ 1 , I ∗ f ∣ |\Delta_{1,I^*}f_w|\leq|\Delta_{1,I^*}f| ∣ Δ 1 , I ∗ f w ∣ ≤ ∣ Δ 1 , I ∗ f ∣ Δ 1 , I ∗ f \Delta_{1,I^*}f Δ 1 , I ∗ f I ∗ ∈ D 1 t r I^*\in\mathcal{D}_1^{\mathrm{tr}} I ∗ ∈ D 1 tr w 1 w_1 w 1 (218) , and (219) ,
E ~ w E ^ ∑ I : I ∗ ∈ D 1 t r ∥ χ I ∂ w ~ 2 , ξ Δ 1 , I ∗ f w ∥ 2 2 ≤ E w 1 P Ω 2 ( G ) ∑ I : I ∗ ∈ D 1 t r ∫ I E w 2 E ^ χ I ∂ w ~ 2 , ξ ( x ) ∣ Δ 1 , I ∗ f ( x ) ∣ 2 d ∣ ν 1 ∣ ( x ) ≤ C [ ( 2 m + 1 ) λ η + ν ] P Ω 2 ( G ) E w 1 ∑ I : I ∗ ∈ D 1 t r ∥ χ I Δ 1 , I ∗ f ∥ 2 2 ≲ ( 2 m + 1 ) λ η + ν P Ω 2 ( G ) ∥ f ∥ 2 2 , \begin{aligned}
\widetilde{\mathbb{E}}_w\widehat{\mathbb{E}}\sum_{I:I^*\in\mathcal{D}_1^{\mathrm{tr}}}\left\|\chi_{I_{\partial}^{\widetilde w_2,\xi}}\Delta_{1,I^*}f_w\right\|_2^2
\leq\frac{\mathbb{E}_{w_1}}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\sum_{I:I^*\in\mathcal{D}_1^{\mathrm{tr}}}\int_I\mathbb{E}_{w_2}\widehat{\mathbb{E}}\chi_{I_{\partial}^{\widetilde w_2,\xi}}(x)|\Delta_{1,I^*}f(x)|^2\,d|\nu_1|(x) \\
&\leq\frac{C[(2m+1)\lambda^\eta+\nu]}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\mathbb{E}_{w_1}\sum_{I:I^*\in\mathcal{D}_1^{\mathrm{tr}}}\|\chi_I\Delta_{1,I^*}f\|_2^2
\lesssim\frac{(2m+1)\lambda^\eta+\nu}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\|f\|_2^2,
\end{aligned} E w E I : I ∗ ∈ D 1 tr ∑ χ I ∂ w 2 , ξ Δ 1 , I ∗ f w 2 2 ≤ P Ω 2 ( G ) E w 1 I : I ∗ ∈ D 1 tr ∑ ∫ I E w 2 E χ I ∂ w 2 , ξ ( x ) ∣ Δ 1 , I ∗ f ( x ) ∣ 2 d ∣ ν 1 ∣ ( x ) ≤ P Ω 2 ( G ) C [( 2 m + 1 ) λ η + ν ] E w 1 I : I ∗ ∈ D 1 tr ∑ ∥ χ I Δ 1 , I ∗ f ∥ 2 2 ≲ P Ω 2 ( G ) ( 2 m + 1 ) λ η + ν ∥ f ∥ 2 2 , where the last inequality follows from (178) . □
We now prove the main estimate for S ∂ ( w ) S_{\partial}(w) S ∂ ( w )
Lemma 6.17. We have
S ∂ ( w ) ≲ m ( 1 + N ε ) ∥ ( ∑ I ∈ D ( w 1 ) , I ∗ ∈ D 1 t r ∣ χ I ∂ w ~ 2 , ξ Δ 1 , I ∗ f w ∣ 2 ) 1 / 2 ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (220) S_{\partial}(w)\lesssim_m(1+N_\varepsilon)\left\|\left(\sum_{\substack{I\in\mathcal{D}(w_1),\,I^*\in\mathcal{D}_1^{\mathrm{tr}}}}|\chi_{I_{\partial}^{\widetilde w_2,\xi}}\Delta_{1,I^*}f_w|^2\right)^{1/2}\right\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. \tag*{(220)} S ∂ ( w ) ≲ m ( 1 + N ε ) I ∈ D ( w 1 ) , I ∗ ∈ D 1 tr ∑ ∣ χ I ∂ w 2 , ξ Δ 1 , I ∗ f w ∣ 2 1/2 L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (220) Moreover, we have
E ~ w E ^ S ∂ ( w ) ≲ C 2 , m ( ( 2 m + 1 ) λ η + ν P Ω 2 ( G ) ) 1 / 2 ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (221) \widetilde{\mathbb{E}}_w\widehat{\mathbb{E}}S_{\partial}(w)\lesssim_{C_{2,m}}\left(\frac{(2m+1)\lambda^\eta+\nu}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\right)^{1/2}(1+N_\varepsilon)\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \tag*{(221)} E w E S ∂ ( w ) ≲ C 2 , m ( P Ω 2 ( G ) ( 2 m + 1 ) λ η + ν ) 1/2 ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (221) Proof. Let I ∈ D ( w 1 ) I\in\mathcal{D}(w_1) I ∈ D ( w 1 ) (217) , we have I ∂ ⊂ I ∂ w ~ 2 , ξ I_{\partial}\subset I_{\partial}^{\widetilde w_2,\xi} I ∂ ⊂ I ∂ w 2 , ξ J J J ( I , J ) ∈ D ( w ) (I,J)\in\mathcal{D}(w) ( I , J ) ∈ D ( w ) (184) , (207) , and the Cauchy-Schwarz inequality, we obtain
S ∂ ( w ) ≤ ∥ S ε ∥ o p ∑ ( I , J ) ∈ D ( w ) ∥ χ I ∂ w ~ 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) S_{\partial}(w)\leq\|S_\varepsilon\|_{\mathrm{op}}\sum_{(I,J)\in\mathcal{D}(w)}\|\chi_{I_{\partial}^{\widetilde w_2,\xi}}\Delta_{1,I^*}f_w\|_{L^2(|\nu_1|)}\|\chi_J\Delta_{2,J^*}g_w\|_{L^2(|\nu_2|)} S ∂ ( w ) ≤ ∥ S ε ∥ op ( I , J ) ∈ D ( w ) ∑ ∥ χ I ∂ w 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ m ( 1 + N ε ) ( ∑ I ∈ D ( w 1 ) I ∗ ∈ D 1 t r ∥ χ I ∂ w ~ 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ( ∑ J ∈ D ( w 2 ) J ∗ ∈ D 2 t r ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ) 1 / 2 . \lesssim_m(1+N_\varepsilon)\left(\sum_{\substack{I\in\mathcal{D}(w_1)\\ I^*\in\mathcal{D}_1^{\mathrm{tr}}}}\|\chi_{I_{\partial}^{\widetilde w_2,\xi}}\Delta_{1,I^*}f_w\|_{L^2(|\nu_1|)}^2\right)^{1/2}\left(\sum_{\substack{J\in\mathcal{D}(w_2)\\ J^*\in\mathcal{D}_2^{\mathrm{tr}}}}\|\chi_J\Delta_{2,J^*}g_w\|_{L^2(|\nu_2|)}^2\right)^{1/2}. ≲ m ( 1 + N ε ) I ∈ D ( w 1 ) I ∗ ∈ D 1 tr ∑ ∥ χ I ∂ w 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 J ∈ D ( w 2 ) J ∗ ∈ D 2 tr ∑ ∥ χ J Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 1/2 . Then, the bound (220) follows by Tonelli’s theorem and (178) .
Finally, (220) , (178) , the Cauchy-Schwarz inequality, and Lemma 6.16 yield
E ~ w E ^ S ∂ ( w ) ≲ m ( 1 + N ε ) ( E ~ w E ^ ∑ I ∈ D ( w 1 ) I ∗ ∈ D 1 t r ∥ χ I ∂ w ~ 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) \widetilde{\mathbb{E}}_w\widehat{\mathbb{E}}S_{\partial}(w)\lesssim_m(1+N_\varepsilon)\left(\widetilde{\mathbb{E}}_w\widehat{\mathbb{E}}\sum_{\substack{I\in\mathcal{D}(w_1)\\ I^*\in\mathcal{D}_1^{\mathrm{tr}}}}\|\chi_{I_{\partial}^{\widetilde w_2,\xi}}\Delta_{1,I^*}f_w\|_{L^2(|\nu_1|)}^2\right)^{1/2}\|g\|_{L^2(|\nu_2|)} E w E S ∂ ( w ) ≲ m ( 1 + N ε ) E w E I ∈ D ( w 1 ) I ∗ ∈ D 1 tr ∑ ∥ χ I ∂ w 2 , ξ Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) ≲ C 2 , m ( ( 2 m + 1 ) λ η + ν P Ω 2 ( G ) ) 1 / 2 ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . □ \lesssim_{C_{2,m}}\left(\frac{(2m+1)\lambda^\eta+\nu}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\right)^{1/2}(1+N_\varepsilon)\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \quad\square ≲ C 2 , m ( P Ω 2 ( G ) ( 2 m + 1 ) λ η + ν ) 1/2 ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . □ We are left with the study of the purely diagonal term
S Δ ( w ) : = ∑ ( I , J ) ∈ D ( w ) ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J Δ Δ 2 , J ∗ g w ⟩ ∗ ∣ . (222) S_{\Delta}(\mathrm{w}) := \sum_{(I,J)\in\mathcal{D}(\mathrm{w})} \left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_{I_{\Delta}}\Delta_{1,I^*}f_{\mathrm{w}}),\chi_{J_{\Delta}}\Delta_{2,J^*}g_{\mathrm{w}}\right\rangle_*\right|. \tag*{(222)} S Δ ( w ) := ( I , J ) ∈ D ( w ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J Δ Δ 2 , J ∗ g w ⟩ ∗ . (222) We further write
S Δ ( w ) ≤ ∑ ( I , J ) ∈ D ( w ) ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J Δ Δ 2 , J ∗ g w ⟩ ∗ ∣ S_{\Delta}(\mathrm{w}) \le\sum_{(I,J)\in\mathcal{D}(\mathrm{w})} \left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_{I_{\Delta}}\Delta_{1,I^*}f_{\mathrm{w}}),\chi_{J_{\Delta}}\Delta_{2,J^*}g_{\mathrm{w}}\right\rangle_*\right| S Δ ( w ) ≤ ( I , J ) ∈ D ( w ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J Δ Δ 2 , J ∗ g w ⟩ ∗ + ∑ ( I , J ) ∈ D ( w ) ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J ∖ J Δ Δ 2 , J ∗ g w ⟩ ∗ ∣ = : S Δ , 1 ( w ) + S Δ , 2 ( w ) . \qquad+ \sum_{(I,J)\in\mathcal{D}(\mathrm{w})} \left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_{I_{\Delta}}\Delta_{1,I^*}f_{\mathrm{w}}),\chi_{J\setminus J_{\Delta}}\Delta_{2,J^*}g_{\mathrm{w}}\right\rangle_*\right| =: S_{\Delta,1}(\mathrm{w}) + S_{\Delta,2}(\mathrm{w}). + ( I , J ) ∈ D ( w ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ I Δ Δ 1 , I ∗ f w ) , χ J ∖ J Δ Δ 2 , J ∗ g w ⟩ ∗ =: S Δ , 1 ( w ) + S Δ , 2 ( w ) . Since
S Δ , 2 = ∑ ( I , J ) ∈ D ( w ) ∣ ⟨ χ I Δ Δ 1 , I ∗ f w , S ∣ ν 2 ∣ , ε ∗ ( χ J ∖ J Δ Δ 2 , J ∗ g w ) ⟩ ∣ , S_{\Delta,2} = \sum_{(I,J)\in\mathcal{D}(\mathrm{w})} \left|\left\langle\chi_{I_{\Delta}}\Delta_{1,I^*}f_{\mathrm{w}},S^*_{|\nu_2|,\varepsilon}(\chi_{J\setminus J_{\Delta}}\Delta_{2,J^*}g_{\mathrm{w}})\right\rangle\right|, S Δ , 2 = ( I , J ) ∈ D ( w ) ∑ ⟨ χ I Δ Δ 1 , I ∗ f w , S ∣ ν 2 ∣ , ε ∗ ( χ J ∖ J Δ Δ 2 , J ∗ g w ) ⟩ , we can estimate S Δ , 2 ( w ) S_{\Delta,2}(\mathrm{w}) S Δ , 2 ( w ) S s e p ( w ) S_{\mathrm{sep}}(\mathrm{w}) S sep ( w ) S ∂ ( w ) S_{\partial}(\mathrm{w}) S ∂ ( w ) J ∖ J Δ = J s e p ∪ J ∂ J\setminus J_{\Delta}=J_{\mathrm{sep}}\cup J_{\partial} J ∖ J Δ = J sep ∪ J ∂
E ~ w E S Δ , 2 ( w ) ≤ [ C m , λ + C m ( ( 2 m + 1 ) λ + ν P Ω 2 ( G ) ) 1 / 2 ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . (223) \widetilde{\mathbb{E}}_{\mathrm{w}}\mathbb{E}S_{\Delta,2}(\mathrm{w}) \le\left[C_{m,\lambda}+C_m\left(\frac{(2m+1)^\lambda+\nu}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\right)^{1/2}(1+N_\varepsilon)\right]\lVert f\rVert_2\lVert g\rVert_2. \tag*{(223)} E w E S Δ , 2 ( w ) ≤ [ C m , λ + C m ( P Ω 2 ( G ) ( 2 m + 1 ) λ + ν ) 1/2 ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . (223) We will use the following definition.
Definition 6.18 (Nice balls). A ball B ∈ B ^ ξ B\in\widehat{\mathcal{B}}_\xi B ∈ B ξ 2 B ∩ F 1 ( w 1 ) ≠ ∅ 2B\cap F_1(w_1)\ne\varnothing 2 B ∩ F 1 ( w 1 ) = ∅ 2 B ∩ F 2 ( w 2 ) ≠ ∅ 2B\cap F_2(w_2)\ne\varnothing 2 B ∩ F 2 ( w 2 ) = ∅
If B B B Ξ w ≥ r ( B ) \Xi_{\mathrm{w}}\ge r(B) Ξ w ≥ r ( B ) B B B B B B 5 B ⊂ I ∩ J 5B\subset I\cap J 5 B ⊂ I ∩ J I I I J J J F i ( w i ) ⊂ X ∖ W i ( w i ) F_i(w_i)\subset X\setminus W_i(w_i) F i ( w i ) ⊂ X ∖ W i ( w i )
λ min ( ℓ ( I ) , ℓ ( J ) ) ≤ θ 5000 A 0 min ( r ( B 1 ) , r ( B 2 ) ) , (224) \lambda\min(\ell(I),\ell(J))\le\frac{\theta}{5000A_0}\min(r(B_1),r(B_2)), \tag*{(224)} λ min ( ℓ ( I ) , ℓ ( J )) ≤ 5000 A 0 θ min ( r ( B 1 ) , r ( B 2 )) , (224) then 2 B ∩ F i ( w i ) ≠ ∅ 2B\cap F_i(w_i)\ne\varnothing 2 B ∩ F i ( w i ) = ∅ F i ( w i ) ⊂ ( 1 − θ ) B i F_i(w_i)\subset(1-\theta)B_i F i ( w i ) ⊂ ( 1 − θ ) B i
1000 A 0 B ⊂ ( 1 − θ / 2 ) B 1 ∩ ( 1 − θ / 2 ) B 2 . (225) 1000A_0B\subset(1-\theta/2)B_1\cap(1-\theta/2)B_2. \tag*{(225)} 1000 A 0 B ⊂ ( 1 − θ /2 ) B 1 ∩ ( 1 − θ /2 ) B 2 . (225) Weak boundedness
We first prove the weak boundedness property for the original testing measures ν ^ i \widehat{\nu}_i ν i ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ ∥ T Ξ w , ε ∥ o p \lVert T_{\Xi_{\mathrm{w}},\varepsilon}\rVert_{\mathrm{op}} ∥ T Ξ w , ε ∥ op L 2 ( ∣ ν 1 ∣ ) L^2(|\nu_1|) L 2 ( ∣ ν 1 ∣ ) L 2 ( ∣ ν 2 ∣ ) L^2(|\nu_2|) L 2 ( ∣ ν 2 ∣ ) (186) , increasing c ♯ c_{\sharp} c ♯
T Ξ w , ♯ ν ^ 1 ≤ c ♯ and T Ξ w , ♯ ∗ ν ^ 2 ≤ c ♯ on X . T_{\Xi_{\mathrm{w},\sharp}}\widehat{\nu}_1\le c_{\sharp}\qquad\text{and}\qquad T^*_{\Xi_{\mathrm{w},\sharp}}\widehat{\nu}_2\le c_{\sharp}\quad\text{on }X. T Ξ w , ♯ ν 1 ≤ c ♯ and T Ξ w , ♯ ∗ ν 2 ≤ c ♯ on X . Before proving the weak boundedness lemmas, we state the following auxiliary result.
Lemma 6.19. There exists c > 0 c>0 c > 0 B B B a ∈ ( 0 , 1 / 4 ) a\in(0,1/4) a ∈ ( 0 , 1/4 )
sup y ∈ B ∣ T Ξ w , ∣ ν 1 ∣ , ε ( χ ( 1 + a ) B c b ^ 1 ) ( y ) ∣ ≤ c a − n + c ♯ (226) \sup_{y\in B}\left|T_{\Xi_{\mathrm{w}},|\nu_1|,\varepsilon}\bigl(\chi_{(1+a)B}c\widehat{b}_1\bigr)(y)\right|\le ca^{-n}+c_{\sharp} \tag*{(226)} y ∈ B sup T Ξ w , ∣ ν 1 ∣ , ε ( χ ( 1 + a ) B c b 1 ) ( y ) ≤ c a − n + c ♯ (226) and
sup y ∈ B ∣ T Ξ w , ∣ ν 2 ∣ , ε ∗ ( χ ( 1 + a ) B c b ^ 2 ) ( y ) ∣ ≤ c a − n + c ♯ . (227) \sup_{y\in B}\left|T^*_{\Xi_{\mathrm{w}},|\nu_2|,\varepsilon}\bigl(\chi_{(1+a)B}c\widehat{b}_2\bigr)(y)\right|\le ca^{-n}+c_{\sharp}. \tag*{(227)} y ∈ B sup T Ξ w , ∣ ν 2 ∣ , ε ∗ ( χ ( 1 + a ) B c b 2 ) ( y ) ≤ c a − n + c ♯ . (227) Proof. Write r = r ( B ) r=r(B) r = r ( B ) y ∈ B y\in B y ∈ B (153) ,
∣ ν ^ i ∣ ( B ( y , R ) ) ≲ R n , for R ≥ Ξ w ( y ) , i = 1 , 2. (228) |\widehat{\nu}_i|(B(y,R))\lesssim R^n,\qquad\text{for }R\geq\Xi_w(y),\ i=1,2. \tag*{(228)} ∣ ν i ∣ ( B ( y , R )) ≲ R n , for R ≥ Ξ w ( y ) , i = 1 , 2. (228) Since ( 1 + a ) B ⊂ B ( y , 3 r ) (1+a)B\subset B(y,3r) ( 1 + a ) B ⊂ B ( y , 3 r )
T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ ( ( 1 + a ) B ) c b ^ 1 ) ( y ) = T Ξ w , max { ε , 3 r } ν ^ 1 ( y ) + ∫ ( ( 1 + a ) B ) c ∩ { z : ε < d ( y , z ) ≤ 3 r } K Ξ w ( y , z ) d ν ^ 1 ( z ) . (229) T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}\bigl(\chi_{((1+a)B)^c}\widehat{b}_1\bigr)(y)
=
T_{\Xi_w,\max\{\varepsilon,3r\}}\widehat{\nu}_1(y)
+
\int_{((1+a)B)^c\cap\{z:\varepsilon<d(y,z)\leq3r\}}K_{\Xi_w}(y,z)\,d\widehat{\nu}_1(z). \tag*{(229)} T Ξ w , ∣ ν 1 ∣ , ε ( χ (( 1 + a ) B ) c b 1 ) ( y ) = T Ξ w , m a x { ε , 3 r } ν 1 ( y ) + ∫ (( 1 + a ) B ) c ∩ { z : ε < d ( y , z ) ≤ 3 r } K Ξ w ( y , z ) d ν 1 ( z ) . (229) The first term is bounded by c ♯ c_{\sharp} c ♯ (186) . To bound the second term, note that d ( y , z ) ≥ a r d(y,z)\geq ar d ( y , z ) ≥ a r ( ( 1 + a ) B ) c ((1+a)B)^c (( 1 + a ) B ) c (70) gives
∫ ( ( 1 + a ) B ) c ∩ B ( y , 4 r ) ∣ K Ξ w ( y , z ) ∣ d ∣ ν ^ 1 ∣ ( z ) ≲ ∣ ν ^ 1 ∣ ( B ( y , 4 max { r , Ξ w ( y ) } ) ) max { a r , Ξ w ( y ) } n (230) \int_{((1+a)B)^c\cap B(y,4r)}|K_{\Xi_w}(y,z)|\,d|\widehat{\nu}_1|(z)
\lesssim
\frac{|\widehat{\nu}_1|(B(y,4\max\{r,\Xi_w(y)\}))}{\max\{ar,\Xi_w(y)\}^n} \tag*{(230)} ∫ (( 1 + a ) B ) c ∩ B ( y , 4 r ) ∣ K Ξ w ( y , z ) ∣ d ∣ ν 1 ∣ ( z ) ≲ max { a r , Ξ w ( y ) } n ∣ ν 1 ∣ ( B ( y , 4 max { r , Ξ w ( y )})) (230) ≲ max { r , Ξ w ( y ) } n max { a r , Ξ w ( y ) } n ≲ a − n . \lesssim
\frac{\max\{r,\Xi_w(y)\}^n}{\max\{ar,\Xi_w(y)\}^n}
\lesssim a^{-n}. ≲ max { a r , Ξ w ( y ) } n max { r , Ξ w ( y ) } n ≲ a − n . This proves (226) . The estimate (227) follows by the same argument applied to T ∗ T^* T ∗ ν ^ 1 \widehat{\nu}_1 ν 1 ν ^ 2 \widehat{\nu}_2 ν 2
Lemma 6.20 (Key suppressed WBP). Let a ∈ ( 0 , 1 / 4 ) a\in(0,1/4) a ∈ ( 0 , 1/4 ) B ⊂ X B\subset X B ⊂ X M M M ∣ ν ^ 1 ∣ |\widehat{\nu}_1| ∣ ν 1 ∣ ∣ ν ^ 2 ∣ |\widehat{\nu}_2| ∣ ν 2 ∣ C w b > 0 C_{wb}>0 C w b > 0
∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ ≤ a − n C w b ∣ ν ^ 1 ∣ ( B ) 1 / 2 ∣ ν ^ 2 ∣ ( B ) 1 / 2 + C w b ( M a ) 1 / 2 ∥ T Ξ w , ε ∥ o p ∣ ν ^ 1 ∣ ( 5 B ) 1 / 2 ∣ ν ^ 2 ∣ ( 5 B ) 1 / 2 , (231) \left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
\leq
a^{-n}C_{wb}|\widehat{\nu}_1|(B)^{1/2}|\widehat{\nu}_2|(B)^{1/2}
+
C_{wb}(Ma)^{1/2}\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op}}|\widehat{\nu}_1|(5B)^{1/2}|\widehat{\nu}_2|(5B)^{1/2}, \tag*{(231)} ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ ≤ a − n C w b ∣ ν 1 ∣ ( B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 + C w b ( M a ) 1/2 ∥ T Ξ w , ε ∥ op ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 , (231) where C w b C_{wb} C w b Theorem 1.3 , but not on a a a M M M w w w ε \varepsilon ε
Proof. For B B B a a a
∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ ≤ ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε b ^ 1 , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ + ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ B c b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ = : I 1 + I 2 . (232) \left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
\leq
\left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}\widehat{b}_1,\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
+
\left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_{B^c}\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
=:I_1+I_2. \tag*{(232)} ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ ≤ ⟨ T Ξ w , ∣ ν 1 ∣ , ε b 1 , χ B b 2 ⟩ ∣ ν 2 ∣ + ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ B c b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ =: I 1 + I 2 . (232) The term I 1 I_1 I 1 (186) and, since T Ξ w , ∣ ν ^ 1 ∣ , ε b ^ 1 = T Ξ w , ε ν 1 T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}\widehat{b}_1=T_{\Xi_w,\varepsilon}\nu_1 T Ξ w , ∣ ν 1 ∣ , ε b 1 = T Ξ w , ε ν 1
I 1 ≤ ∫ B ∣ T Ξ w , ε ν ^ 1 ∣ d ∣ ν ^ 2 ∣ ≤ c ♯ ∣ ν ^ 2 ∣ ( B ) . (233) I_1\leq\int_B|T_{\Xi_w,\varepsilon}\widehat{\nu}_1|\,d|\widehat{\nu}_2|
\leq c_{\sharp}|\widehat{\nu}_2|(B). \tag*{(233)} I 1 ≤ ∫ B ∣ T Ξ w , ε ν 1 ∣ d ∣ ν 2 ∣ ≤ c ♯ ∣ ν 2 ∣ ( B ) . (233) To deal with the second term, we write
I 2 ≤ ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ ( 1 + a ) B ∖ B b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ + ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ ( ( 1 + a ) B ) c b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ = : I 2 ′ + I 2 ′ ′ . I_2\leq
\left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_{(1+a)B\setminus B}\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
+
\left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_{((1+a)B)^c}\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
=:I'_2+I''_2. I 2 ≤ ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ ( 1 + a ) B ∖ B b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ + ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ (( 1 + a ) B ) c b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ =: I 2 ′ + I 2 ′′ . Since B B B M M M ∣ ν ^ 1 ∣ |\widehat{\nu}_1| ∣ ν 1 ∣
I 2 ′ ≤ ∥ T Ξ w , ε ∥ o p ∥ χ ( 1 + a ) B ∖ B b ^ 1 ∥ L 2 ( ∣ ν ^ 1 ∣ ) ∥ χ B b ^ 2 ∥ L 2 ( ∣ ν ^ 2 ∣ ) ≤ a 1 / 2 M 1 / 2 ∥ T Ξ w , ε ∥ o p ∣ ν ^ 1 ∣ ( 5 B ) 1 / 2 ∣ ν ^ 2 ∣ ( 5 B ) 1 / 2 . (234) \begin{aligned}
I'_2\leq
\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op}}
\|\chi_{(1+a)B\setminus B}\widehat{b}_1\|_{L^2(|\widehat{\nu}_1|)}
\|\chi_B\widehat{b}_2\|_{L^2(|\widehat{\nu}_2|)} \\
&\leq
a^{1/2}M^{1/2}\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op}}
|\widehat{\nu}_1|(5B)^{1/2}|\widehat{\nu}_2|(5B)^{1/2}. \tag*{(234)}
\end{aligned} I 2 ′ ≤ ∥ T Ξ w , ε ∥ op ∥ χ ( 1 + a ) B ∖ B b 1 ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ B b 2 ∥ L 2 ( ∣ ν 2 ∣ ) ≤ a 1/2 M 1/2 ∥ T Ξ w , ε ∥ op ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 . (234) On the other hand, we bound the second summand by
I 2 ′ ′ ≤ sup y ∈ B ∣ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ ( ( 1 + a ) B ) c b ^ 1 ) ( y ) ∣ ∣ ν ^ 2 ∣ ( B ) . (235) I''_2\leq
\sup_{y\in B}\left|T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}\bigl(\chi_{((1+a)B)^c}\widehat{b}_1\bigr)(y)\right|
|\widehat{\nu}_2|(B). \tag*{(235)} I 2 ′′ ≤ y ∈ B sup T Ξ w , ∣ ν 1 ∣ , ε ( χ (( 1 + a ) B ) c b 1 ) ( y ) ∣ ν 2 ∣ ( B ) . (235) Gathering (232) , (233) , (234) , (235) , and Lemma 6.19 , we obtain, for some c ′ > 0 c' > 0 c ′ > 0
∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ ≤ c ′ a − n ∣ ν ^ 2 ∣ ( B ) + a 1 / 2 M 1 / 2 ∥ T Ξ w , ε ∥ o p , ∣ ν ^ 1 ∣ ( 5 B ) 1 / 2 ∣ ν ^ 2 ∣ ( 5 B ) 1 / 2 . (236) \left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
\leq c'a^{-n}|\widehat{\nu}_2|(B)
+a^{1/2}M^{1/2}\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op},|\widehat{\nu}_1|}(5B)^{1/2}|\widehat{\nu}_2|(5B)^{1/2}. \tag*{(236)} ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ ≤ c ′ a − n ∣ ν 2 ∣ ( B ) + a 1/2 M 1/2 ∥ T Ξ w , ε ∥ op , ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 . (236) Since the operator norms of T Ξ w , ∣ ν ^ 1 ∣ , ε T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon} T Ξ w , ∣ ν 1 ∣ , ε T Ξ w , ∣ ν ^ 2 ∣ , ε ∗ T_{\Xi_w,|\widehat{\nu}_2|,\varepsilon}^{*} T Ξ w , ∣ ν 2 ∣ , ε ∗ T Ξ w , ∣ ν ^ 2 ∣ , ε ∗ T_{\Xi_w,|\widehat{\nu}_2|,\varepsilon}^{*} T Ξ w , ∣ ν 2 ∣ , ε ∗
∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∣ ν ^ 2 ∣ ∣ ≤ c ′ a − n ∣ ν ^ 1 ∣ ( B ) + a 1 / 2 M 1 / 2 ∥ T Ξ w , ε ∥ o p , ∣ ν ^ 1 ∣ ( 5 B ) 1 / 2 ∣ ν ^ 2 ∣ ( 5 B ) 1 / 2 . (237) \left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_{|\widehat{\nu}_2|}\right|
\leq c'a^{-n}|\widehat{\nu}_1|(B)
+a^{1/2}M^{1/2}\|T_{\Xi_w,\varepsilon}\|_{\mathrm{op},|\widehat{\nu}_1|}(5B)^{1/2}|\widehat{\nu}_2|(5B)^{1/2}. \tag*{(237)} ⟨ T Ξ w , ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∣ ν 2 ∣ ≤ c ′ a − n ∣ ν 1 ∣ ( B ) + a 1/2 M 1/2 ∥ T Ξ w , ε ∥ op , ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 . (237) Taking the smaller of (236) and (237) , and using
min { ∣ ν ^ 1 ∣ ( B ) , ∣ ν ^ 2 ∣ ( B ) } ≤ ∣ ν ^ 1 ∣ ( B ) 1 / 2 ∣ ν ^ 2 ∣ ( B ) 1 / 2 , \min\{|\widehat{\nu}_1|(B),|\widehat{\nu}_2|(B)\}
\leq|\widehat{\nu}_1|(B)^{1/2}|\widehat{\nu}_2|(B)^{1/2}, min { ∣ ν 1 ∣ ( B ) , ∣ ν 2 ∣ ( B )} ≤ ∣ ν 1 ∣ ( B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 , proves (231) . □
We now prove the analogue of Lemma 6.20 for the localized operator S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε
Lemma 6.21 (Weak boundedness for the localized operator). Let a ∈ ( 0 , 1 / 4 ) a\in(0,1/4) a ∈ ( 0 , 1/4 ) B ⊂ X B\subset X B ⊂ X M Λ M_{\Lambda} M Λ ∣ ν 1 ∣ |\nu_1| ∣ ν 1 ∣ ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣
∣ ⟨ S ∣ ν 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ ∣ ≤ C a − n ∣ ν 1 ∣ ( B ) 1 / 2 ∣ ν 2 ∣ ( B ) 1 / 2 + C ( M Λ a ) 1 / 2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1 / 2 ∣ ν 2 ∣ ( 5 B ) 1 / 2 , (238) \left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_*\right|
\leq Ca^{-n}|\nu_1|(B)^{1/2}|\nu_2|(B)^{1/2}
+C(M_{\Lambda}a)^{1/2}(1+N_{\varepsilon})|\nu_1|(5B)^{1/2}|\nu_2|(5B)^{1/2}, \tag*{(238)} ⟨ S ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∗ ≤ C a − n ∣ ν 1 ∣ ( B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 + C ( M Λ a ) 1/2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 , (238) where C > 0 C>0 C > 0 a a a M Λ M_{\Lambda} M Λ B B B w w w ε \varepsilon ε
Proof. To simplify the notation, we write S : = S ∣ ν 1 ∣ , ε S:=S_{|\nu_1|,\varepsilon} S := S ∣ ν 1 ∣ , ε r : = r ( B ) r:=r(B) r := r ( B ) x B x_B x B B B B
⟨ S ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ = ⟨ S b ^ 1 , χ B b ^ 2 ⟩ ∗ − ⟨ S ( χ ( 1 + a ) B ∖ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ − ⟨ S ( χ ( ( 1 + a ) B ) c b ^ 1 ) , χ B b ^ 2 ⟩ ∗ = : J 1 − J 2 − J 3 . (239) \left\langle S(\chi_B\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_*
=
\left\langle S\widehat{b}_1,\chi_B\widehat{b}_2\right\rangle_*
-\left\langle S(\chi_{(1+a)B\setminus B}\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_*
-\left\langle S(\chi_{((1+a)B)^c}\widehat{b}_1),\chi_B\widehat{b}_2\right\rangle_*
=: \mathcal{J}_1-\mathcal{J}_2-\mathcal{J}_3. \tag*{(239)} ⟨ S ( χ B b 1 ) , χ B b 2 ⟩ ∗ = ⟨ S b 1 , χ B b 2 ⟩ ∗ − ⟨ S ( χ ( 1 + a ) B ∖ B b 1 ) , χ B b 2 ⟩ ∗ − ⟨ S ( χ (( 1 + a ) B ) c b 1 ) , χ B b 2 ⟩ ∗ =: J 1 − J 2 − J 3 . (239) Since L ε ( x , y ) = 0 L_\varepsilon(x,y)=0 L ε ( x , y ) = 0 d ( x , y ) ≤ ε d(x,y)\leq\varepsilon d ( x , y ) ≤ ε S b ^ 1 = S ∣ ν 1 ∣ , ε , ρ b ^ 1 S\widehat{b}_1=S_{|\nu_1|,\varepsilon,\rho}\widehat{b}_1 S b 1 = S ∣ ν 1 ∣ , ε , ρ b 1 ρ < ε \rho<\varepsilon ρ < ε ∣ b ^ 2 ∣ = 1 |\widehat{b}_2|=1 ∣ b 2 ∣ = 1 ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ (187) gives
∣ J 1 ∣ ≤ ∫ B ∣ S b ^ 1 ∣ d ∣ ν 2 ∣ ≤ c ♯ ∣ ν 2 ∣ ( B ) . |\mathcal{J}_1|\leq\int_B|S\widehat{b}_1|\,d|\nu_2|\leq c_{\sharp}|\nu_2|(B). ∣ J 1 ∣ ≤ ∫ B ∣ S b 1 ∣ d ∣ ν 2 ∣ ≤ c ♯ ∣ ν 2 ∣ ( B ) . By (18) with λ = a \lambda=a λ = a ∣ ν 1 ∣ ( ( 1 + a ) B ∖ B ) ≤ M Λ a ∣ ν 1 ∣ ( 5 B ) |\nu_1|((1+a)B\setminus B)\leq M_{\Lambda}a|\nu_1|(5B) ∣ ν 1 ∣ (( 1 + a ) B ∖ B ) ≤ M Λ a ∣ ν 1 ∣ ( 5 B ) (184) yields
∣ J 2 ∣ ≤ ∥ S ε ∥ o p ∣ ν 1 ∣ ( ( 1 + a ) B ∖ B ) 1 / 2 ∣ ν 2 ∣ ( B ) 1 / 2 ≲ ( M Λ a ) 1 / 2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1 / 2 ∣ ν 2 ∣ ( 5 B ) 1 / 2 . |\mathcal{J}_2|
\leq\|S_\varepsilon\|_{\mathrm{op}}|\nu_1|((1+a)B\setminus B)^{1/2}|\nu_2|(B)^{1/2}
\lesssim(M_{\Lambda}a)^{1/2}(1+N_\varepsilon)|\nu_1|(5B)^{1/2}|\nu_2|(5B)^{1/2}. ∣ J 2 ∣ ≤ ∥ S ε ∥ op ∣ ν 1 ∣ (( 1 + a ) B ∖ B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 ≲ ( M Λ a ) 1/2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 . To bound J 3 \mathcal{J}_3 J 3 y ∈ B y\in B y ∈ B ( 1 + a ) B ⊂ B ( y , 3 r ) (1+a)B\subset B(y,3r) ( 1 + a ) B ⊂ B ( y , 3 r ) d ( y , z ) > a r d(y,z)>ar d ( y , z ) > a r z ∉ ( 1 + a ) B z\notin(1+a)B z ∈ / ( 1 + a ) B
S ( χ ( ( 1 + a ) B ) c b ^ 1 ) ( y ) = S ∣ ν 1 ∣ , ε , 3 r b ^ 1 ( y ) + ∫ ( ( 1 + a ) B ) c ∩ { z : d ( y , z ) ≤ 3 r } L ε ( y , z ) b ^ 1 ( z ) d ∣ ν 1 ∣ ( z ) , S(\chi_{((1+a)B)^c}\widehat{b}_1)(y)
=S_{|\nu_1|,\varepsilon,3r}\widehat{b}_1(y)
+\int_{((1+a)B)^c\cap\{z:d(y,z)\leq3r\}}L_\varepsilon(y,z)\widehat{b}_1(z)\,d|\nu_1|(z), S ( χ (( 1 + a ) B ) c b 1 ) ( y ) = S ∣ ν 1 ∣ , ε , 3 r b 1 ( y ) + ∫ (( 1 + a ) B ) c ∩ { z : d ( y , z ) ≤ 3 r } L ε ( y , z ) b 1 ( z ) d ∣ ν 1 ∣ ( z ) , and, by (159) and (154) , the absolute value of the last integral is bounded by a constant times
∣ ν 1 ∣ ( B ( y , 4 r ) ) max { a r , Ξ w ( y ) } n ≤ ( m + E 0 ) ( 4 r + Ξ w ( y ) ) n max { a r , Ξ w ( y ) } n ≲ a − n . \frac{|\nu_1|(B(y,4r))}{\max\{ar,\Xi_w(y)\}^{n}}
\leq(m+\mathcal{E}_0)\frac{(4r+\Xi_w(y))^{n}}{\max\{ar,\Xi_w(y)\}^{n}}
\lesssim a^{-n}. max { a r , Ξ w ( y ) } n ∣ ν 1 ∣ ( B ( y , 4 r )) ≤ ( m + E 0 ) max { a r , Ξ w ( y ) } n ( 4 r + Ξ w ( y ) ) n ≲ a − n . Therefore, by (187) , we obtain
∣ J 3 ∣ ≤ sup y ∈ B ∣ S ( χ ( ( 1 + a ) B ) c b ^ 1 ) ( y ) ∣ ∣ ν 2 ∣ ( B ) ≲ ( c ♯ + a − n ) ∣ ν 2 ∣ ( B ) . |\mathcal{J}_3|
\leq\sup_{y\in B}|S(\chi_{((1+a)B)^c}\widehat{b}_1)(y)|\,|\nu_2|(B)
\lesssim(c_{\sharp}+a^{-n})|\nu_2|(B). ∣ J 3 ∣ ≤ y ∈ B sup ∣ S ( χ (( 1 + a ) B ) c b 1 ) ( y ) ∣ ∣ ν 2 ∣ ( B ) ≲ ( c ♯ + a − n ) ∣ ν 2 ∣ ( B ) . Since a < 1 a<1 a < 1 (239) and obtain
∣ ⟨ S ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ ∣ ≤ C a − n ∣ ν 2 ∣ ( B ) + C ( M Λ a ) 1 / 2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1 / 2 ∣ ν 2 ∣ ( 5 B ) 1 / 2 . (240) \left|\left\langle S(\chi_B\widehat b_1),\chi_B\widehat b_2\right\rangle_*\right|\le Ca^{-n}|\nu_2|(B)+C(M_\Lambda a)^{1/2}(1+N_\varepsilon)|\nu_1|(5B)^{1/2}|\nu_2|(5B)^{1/2}. \tag*{(240)} ⟨ S ( χ B b 1 ) , χ B b 2 ⟩ ∗ ≤ C a − n ∣ ν 2 ∣ ( B ) + C ( M Λ a ) 1/2 ( 1 + N ε ) ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 . (240) Moreover, ⟨ S ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ = ⟨ χ B b ^ 1 , S ∣ ν 2 ∣ , ε ∗ ( χ B b ^ 2 ) ⟩ \left\langle S(\chi_B\widehat b_1),\chi_B\widehat b_2\right\rangle_*=\left\langle\chi_B\widehat b_1,S^*_{|\nu_2|,\varepsilon}(\chi_B\widehat b_2)\right\rangle ⟨ S ( χ B b 1 ) , χ B b 2 ⟩ ∗ = ⟨ χ B b 1 , S ∣ ν 2 ∣ , ε ∗ ( χ B b 2 ) ⟩ S S S S ∣ ν 2 ∣ , ε ∗ S^*_{|\nu_2|,\varepsilon} S ∣ ν 2 ∣ , ε ∗ S ∣ ν 2 ∣ , ε ∗ S^*_{|\nu_2|,\varepsilon} S ∣ ν 2 ∣ , ε ∗ (187) and the M Λ M_\Lambda M Λ B B B ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ (240) with ∣ ν 1 ∣ ( B ) |\nu_1|(B) ∣ ν 1 ∣ ( B ) ∣ ν 2 ∣ ( B ) |\nu_2|(B) ∣ ν 2 ∣ ( B ) min { ∣ ν 1 ∣ ( B ) , ∣ ν 2 ∣ ( B ) } ≤ ∣ ν 1 ∣ ( B ) 1 / 2 ∣ ν 2 ∣ ( B ) 1 / 2 \min\{|\nu_1|(B),|\nu_2|(B)\}\le|\nu_1|(B)^{1/2}|\nu_2|(B)^{1/2} min { ∣ ν 1 ∣ ( B ) , ∣ ν 2 ∣ ( B )} ≤ ∣ ν 1 ∣ ( B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 (238) . □
We recall that, in the following lemma, the finite depth and separation constants of the almost-covering families depend on m m m ϑ \vartheta ϑ
Lemma 6.22. There exist C m , λ , ϑ , C m , ϑ > 0 C_{m,\lambda,\vartheta}, C_{m,\vartheta}>0 C m , λ , ϑ , C m , ϑ > 0 a a a ε \varepsilon ε w w w ξ \xi ξ
∣ S Δ , 1 ( w ) ∣ ≤ [ C m , λ , ϑ ( 1 + a − n ) + C m , ϑ ( M Λ a ) 1 / 2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . |S_{\Delta,1}(w)|\le\left[C_{m,\lambda,\vartheta}(1+a^{-n})+C_{m,\vartheta}(M_\Lambda a)^{1/2}(1+N_\varepsilon)\right]\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. ∣ S Δ , 1 ( w ) ∣ ≤ [ C m , λ , ϑ ( 1 + a − n ) + C m , ϑ ( M Λ a ) 1/2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . Here, M Λ M_\Lambda M Λ B ^ ξ \widehat{\mathcal B}_\xi B ξ
Proof. Using I Δ = ⋃ i B I , i I_\Delta=\bigcup_i B_{I,i} I Δ = ⋃ i B I , i J Δ = ⋃ j B J , j J_\Delta=\bigcup_j B_{J,j} J Δ = ⋃ j B J , j
S Δ , 1 ( w ) ≤ ∑ ( I , J ) ∈ D ( w ) ∑ i , j ∣ ⟨ S ∣ ν 1 ∣ , ε ( χ B I , i Δ 1 , I ∗ f w ) , χ B J , j Δ 2 , J ∗ g w ⟩ ∗ ∣ = : S Δ ′ + S Δ ′ ′ + S Δ ′ ′ ′ . S_{\Delta,1}(w)\le\sum_{(I,J)\in\mathcal D(w)}\sum_{i,j}\left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_{B_{I,i}}\Delta_{1,I}^*f_w),\chi_{B_{J,j}}\Delta_{2,J}^*g_w\right\rangle_*\right|=:S'_\Delta+S''_\Delta+S'''_\Delta. S Δ , 1 ( w ) ≤ ( I , J ) ∈ D ( w ) ∑ i , j ∑ ⟨ S ∣ ν 1 ∣ , ε ( χ B I , i Δ 1 , I ∗ f w ) , χ B J , j Δ 2 , J ∗ g w ⟩ ∗ =: S Δ ′ + S Δ ′′ + S Δ ′′′ . where S Δ ′ S'_\Delta S Δ ′ S Δ ′ ′ S''_\Delta S Δ ′′ S Δ ′ ′ ′ S'''_\Delta S Δ ′′′ B , B ′ ∈ B ^ ξ B,B'\in\widehat{\mathcal B}_\xi B , B ′ ∈ B ξ B B B B B B (210) and (213) , every B ∈ B ^ ξ B\in\widehat{\mathcal B}_\xi B ∈ B ξ
r ( B ) ≥ c m , ϑ ϑ k ≥ c m , ϑ ϑ λ min ( ℓ ( I ) , ℓ ( J ) ) . (241) r(B)\ge c_{m,\vartheta}\vartheta^k\ge c_{m,\vartheta}\vartheta^\lambda\min(\ell(I),\ell(J)). \tag*{(241)} r ( B ) ≥ c m , ϑ ϑ k ≥ c m , ϑ ϑ λ min ( ℓ ( I ) , ℓ ( J )) . (241) If B ≠ B ′ B\ne B' B = B ′ (214) and (241) give
dist ( B I , i , B J , j ) ≥ c m , λ , ϑ min ( ℓ ( I ) , ℓ ( J ) ) . (242) \operatorname{dist}(B_{I,i},B_{J,j})\ge c_{m,\lambda,\vartheta}\min(\ell(I),\ell(J)). \tag*{(242)} dist ( B I , i , B J , j ) ≥ c m , λ , ϑ min ( ℓ ( I ) , ℓ ( J )) . (242) Since the balls in B ^ ξ \widehat{\mathcal B}_\xi B ξ ℓ ( I ) ∼ m ℓ ( J ) \ell(I)\sim_m\ell(J) ℓ ( I ) ∼ m ℓ ( J ) (242) and (159) imply that, for fixed ( I , J ) (I,J) ( I , J ) C m , λ , ϑ ℓ ( I ) − n ∥ χ B I , i Δ 1 , I ∗ f w ∥ L 1 ( ∣ ν 1 ∣ ) ∥ χ B J , j Δ 2 , J ∗ g w ∥ L 1 ( ∣ ν 2 ∣ ) C_{m,\lambda,\vartheta}\ell(I)^{-n}\|\chi_{B_{I,i}}\Delta_{1,I}^*f_w\|_{L^1(|\nu_1|)}\|\chi_{B_{J,j}}\Delta_{2,J}^*g_w\|_{L^1(|\nu_2|)} C m , λ , ϑ ℓ ( I ) − n ∥ χ B I , i Δ 1 , I ∗ f w ∥ L 1 ( ∣ ν 1 ∣ ) ∥ χ B J , j Δ 2 , J ∗ g w ∥ L 1 ( ∣ ν 2 ∣ ) (178) give
S Δ ′ ≤ C m , λ , ϑ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (243) S'_\Delta\le C_{m,\lambda,\vartheta}\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \tag*{(243)} S Δ ′ ≤ C m , λ , ϑ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (243) If B = B ′ B=B' B = B ′ 5 B ⊂ I ∩ J 5B\subset I\cap J 5 B ⊂ I ∩ J B B B B B B 2 B ∩ F i ( w i ) = ∅ 2B\cap F_i(w_i)=\varnothing 2 B ∩ F i ( w i ) = ∅ i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } Ξ w ≥ dist ( ⋅ , F i ( w i ) ) ≥ r ( B ) \Xi_w\ge\operatorname{dist}(\cdot,F_i(w_i))\ge r(B) Ξ w ≥ dist ( ⋅ , F i ( w i )) ≥ r ( B ) B B B (159) gives ∣ L ε ∣ ≲ r ( B ) − n |L_\varepsilon|\lesssim r(B)^{-n} ∣ L ε ∣ ≲ r ( B ) − n B × B B\times B B × B ∣ ν 1 ∣ ( I ) ≲ ℓ ( I ) n |\nu_1|(I)\lesssim\ell(I)^n ∣ ν 1 ∣ ( I ) ≲ ℓ ( I ) n ∣ ν 2 ∣ ( J ) ≲ ℓ ( J ) n |\nu_2|(J)\lesssim\ell(J)^n ∣ ν 2 ∣ ( J ) ≲ ℓ ( J ) n (241) yield
∣ ⟨ S ∣ ν 1 ∣ , ε ( χ B Δ 1 , I ∗ f w ) , χ B Δ 2 , J ∗ g w ⟩ ∗ ∣ ≤ C m , λ , ϑ ∥ χ B Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ B Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (244) \left|\left\langle S_{|\nu_1|,\varepsilon}(\chi_B\Delta_{1,I}^*f_w),\chi_B\Delta_{2,J}^*g_w\right\rangle_*\right|\le C_{m,\lambda,\vartheta}\|\chi_B\Delta_{1,I}^*f_w\|_{L^2(|\nu_1|)}\|\chi_B\Delta_{2,J}^*g_w\|_{L^2(|\nu_2|)}. \tag*{(244)} ⟨ S ∣ ν 1 ∣ , ε ( χ B Δ 1 , I ∗ f w ) , χ B Δ 2 , J ∗ g w ⟩ ∗ ≤ C m , λ , ϑ ∥ χ B Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ χ B Δ 2 , J ∗ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (244) Since the balls are disjoint and each cube occurs in at most C m C_m C m D ( w ) \mathcal D(w) D ( w ) (178) give
S Δ ′ ′ ≤ C m , λ , ϑ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (245) S''_\Delta\le C_{m,\lambda,\vartheta}\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. \tag*{(245)} S Δ ′′ ≤ C m , λ , ϑ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (245) If B B B 2 B 2B 2 B F 1 ( w 1 ) F_1(w_1) F 1 ( w 1 ) F 2 ( w 2 ) F_2(w_2) F 2 ( w 2 ) 2 B ⊂ I ∩ J 2B\subset I\cap J 2 B ⊂ I ∩ J F i ( w i ) ∩ W i ( w i ) = ∅ F_i(w_i)\cap\mathcal W_i(w_i)=\varnothing F i ( w i ) ∩ W i ( w i ) = ∅ I I I J J J (175) , Δ 1 , I ∗ f w = D I 1 f w b ^ 1 \Delta_{1,I}^*f_w=D^1_{I}f_w\widehat b_1 Δ 1 , I ∗ f w = D I 1 f w b 1 I I I Δ 2 , J ∗ g w = D J 2 g w b ^ 2 \Delta_{2,J}^*g_w=D^2_{J}g_w\widehat b_2 Δ 2 , J ∗ g w = D J 2 g w b 2 J J J
⟨ S ∣ ν 1 ∣ , ε ( χ B Δ 1 , I ∗ f w ) , χ B Δ 2 , J ∗ g w ⟩ ∗ = D I 1 f w D J 2 g w ⟨ S ∣ ν 1 ∣ , ε ( χ B b ^ 1 ) , χ B b ^ 2 ⟩ ∗ . (246) \left\langle S_{|\nu_1|,\varepsilon}(\chi_B\Delta_{1,I}^*f_w),\chi_B\Delta_{2,J}^*g_w\right\rangle_*
=D^1_If_wD^2_Jg_w\left\langle S_{|\nu_1|,\varepsilon}(\chi_B\widehat b_1),\chi_B\widehat b_2\right\rangle_*. \tag*{(246)} ⟨ S ∣ ν 1 ∣ , ε ( χ B Δ 1 , I ∗ f w ) , χ B Δ 2 , J ∗ g w ⟩ ∗ = D I 1 f w D J 2 g w ⟨ S ∣ ν 1 ∣ , ε ( χ B b 1 ) , χ B b 2 ⟩ ∗ . (246) and Lemma 6.21 applies to the last pairing. The balls in B ^ ξ \widehat{\mathcal{B}}_{\xi} B ξ (214) , so, by geometric doubling, the dilates 5 B 5B 5 B C m , υ C_{m,\upsilon} C m , υ 5 B ⊂ I ∩ J 5B \subset I \cap J 5 B ⊂ I ∩ J
∑ B ( ∣ ν 1 ∣ ( B ) 1 / 2 ∣ ν 2 ∣ ( B ) 1 / 2 + ∣ ν 1 ∣ ( 5 B ) 1 / 2 ∣ ν 2 ∣ ( 5 B ) 1 / 2 ) ≤ C m , υ ∣ ν 1 ∣ ( I ∩ J ) 1 / 2 ∣ ν 2 ∣ ( I ∩ J ) 1 / 2 . \sum_{B}\left(|\nu_{1}|(B)^{1/2}|\nu_{2}|(B)^{1/2}+|\nu_{1}|(5B)^{1/2}|\nu_{2}|(5B)^{1/2}\right)\leq C_{m,\upsilon}|\nu_{1}|(I\cap J)^{1/2}|\nu_{2}|(I\cap J)^{1/2}. B ∑ ( ∣ ν 1 ∣ ( B ) 1/2 ∣ ν 2 ∣ ( B ) 1/2 + ∣ ν 1 ∣ ( 5 B ) 1/2 ∣ ν 2 ∣ ( 5 B ) 1/2 ) ≤ C m , υ ∣ ν 1 ∣ ( I ∩ J ) 1/2 ∣ ν 2 ∣ ( I ∩ J ) 1/2 . For fixed I I I I ∩ J I\cap J I ∩ J 2 m + 1 2m+1 2 m + 1 J J J I I I J J J ∣ D I 1 f w ∣ 2 ∣ ν 1 ∣ ( I ) = ∥ χ I Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 |D^{1}_{I}f_{w}|^{2}|\nu_{1}|(I)=\|\chi_{I}\Delta_{1,I^{*}}f_{w}\|_{L^{2}(|\nu_{1}|)}^{2} ∣ D I 1 f w ∣ 2 ∣ ν 1 ∣ ( I ) = ∥ χ I Δ 1 , I ∗ f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 J J J (178) give
∑ ( I , J ) ∈ D t r t r ( w ) ∣ D I 1 f w D J 2 g w ∣ ∣ ν 1 ∣ ( I ∩ J ) 1 / 2 ∣ ν 2 ∣ ( I ∩ J ) 1 / 2 ≤ ( 2 m + 1 ) ( ∑ I ∣ D I 1 f w ∣ 2 ∣ ν 1 ∣ ( I ) ) 1 / 2 ( ∑ J ∣ D J 2 g w ∣ 2 ∣ ν 2 ∣ ( J ) ) 1 / 2 ≲ m ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . \begin{aligned}
\sum_{(I,J)\in\mathcal{D}_{\mathrm{tr}}^{\mathrm{tr}}(w)}|D^{1}_{I}f_{w}D^{2}_{J}g_{w}|\,|\nu_{1}|(I\cap J)^{1/2}|\nu_{2}|(I\cap J)^{1/2} \\
&\leq(2m+1)\left(\sum_{I}|D^{1}_{I}f_{w}|^{2}|\nu_{1}|(I)\right)^{1/2}\left(\sum_{J}|D^{2}_{J}g_{w}|^{2}|\nu_{2}|(J)\right)^{1/2}\lesssim_{m}\|f\|_{L^{2}(|\nu_{1}|)}\|g\|_{L^{2}(|\nu_{2}|)}.
\end{aligned} ( I , J ) ∈ D tr tr ( w ) ∑ ∣ D I 1 f w D J 2 g w ∣ ∣ ν 1 ∣ ( I ∩ J ) 1/2 ∣ ν 2 ∣ ( I ∩ J ) 1/2 ≤ ( 2 m + 1 ) ( I ∑ ∣ D I 1 f w ∣ 2 ∣ ν 1 ∣ ( I ) ) 1/2 ( J ∑ ∣ D J 2 g w ∣ 2 ∣ ν 2 ∣ ( J ) ) 1/2 ≲ m ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . which, combined with (246) , Lemma 6.21, and the previous bound, yields
S Δ ′ ′ ′ ≤ C m , υ [ a − n + ( M Λ a ) 1 / 2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (247) \mathcal{S}^{\prime\prime\prime}_{\Delta}\leq C_{m,\upsilon}\left[a^{-n}+(M_{\Lambda}a)^{1/2}(1+N_{\varepsilon})\right]\|f\|_{L^{2}(|\nu_{1}|)}\|g\|_{L^{2}(|\nu_{2}|)}. \tag*{(247)} S Δ ′′′ ≤ C m , υ [ a − n + ( M Λ a ) 1/2 ( 1 + N ε ) ] ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . (247) Gathering (243) , (245) , and (247) , we conclude the proof. □
Now, we write
ρ m ( λ , υ ) : = C m ( ( 2 m + 1 ) λ η + υ P Ω 2 ( G ) ) 1 / 2 . \rho_{m}(\lambda,\upsilon):=C_{m}\left(\frac{(2m+1)\lambda^{\eta}+\upsilon}{\mathbb{P}\Omega^{2}(\mathcal{G})}\right)^{1/2}. ρ m ( λ , υ ) := C m ( P Ω 2 ( G ) ( 2 m + 1 ) λ η + υ ) 1/2 . and, combining the preceding lemma with (223) , we obtain
E ~ w E S Δ ( w ) ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) 1 / 2 + ρ m ( λ , υ ) ) ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . (248) \widetilde{\mathbb{E}}_{w}\mathbb{E}\mathcal{S}_{\Delta}(w)\leq\left[C_{m,\lambda,\upsilon}(1+a^{-n})+(C_{m,\upsilon}(M_{\Lambda}a)^{1/2}+\rho_{m}(\lambda,\upsilon))(1+N_{\varepsilon})\right]\|f\|_{2}\|g\|_{2}. \tag*{(248)} E w E S Δ ( w ) ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) 1/2 + ρ m ( λ , υ )) ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . (248) Proposition 6.23. There are constants C m , λ , υ C_{m,\lambda,\upsilon} C m , λ , υ C m , υ C_{m,\upsilon} C m , υ a a a ε \varepsilon ε
E ~ w ∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) 1 / 2 + ρ m ( λ , υ ) ) ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . \widetilde{\mathbb{E}}_{w}|S_{2}(w)+S_{3}(w)|\leq\left[C_{m,\lambda,\upsilon}(1+a^{-n})+(C_{m,\upsilon}(M_{\Lambda}a)^{1/2}+\rho_{m}(\lambda,\upsilon))(1+N_{\varepsilon})\right]\|f\|_{2}\|g\|_{2}. E w ∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) 1/2 + ρ m ( λ , υ )) ( 1 + N ε ) ] ∥ f ∥ 2 ∥ g ∥ 2 . Proof. By (205) , (206) , and the decomposition I = I s e p ∪ I ∂ ∪ I Δ I=I_{\mathrm{sep}}\cup I_{\partial}\cup I_{\Delta} I = I sep ∪ I ∂ ∪ I Δ ξ \xi ξ ∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ S s e p ( w ) + S ∂ ( w ) + S Δ ( w ) |S_{2}(w)+S_{3}(w)|\leq S_{\mathrm{sep}}(w)+S_{\partial}(w)+S_{\Delta}(w) ∣ S 2 ( w ) + S 3 ( w ) ∣ ≤ S sep ( w ) + S ∂ ( w ) + S Δ ( w ) ξ \xi ξ (178) , (221) , and (248) . □
Estimate of S 4 S_{4} S 4
We split
S 4 ( w ) = ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q ∩ R ≠ ∅ ℓ ( Q ) < ℓ − m ℓ ( R ) ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ + ∑ Q ∈ D 1 t r , R ∈ D 2 t r Q ∩ R ≠ ∅ ℓ ( Q ) > ℓ − m ℓ ( R ) ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ = : S 4 ′ + S 4 ′ ′ . (249) S_{4}(w)=\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_{1},R\in\mathcal{D}^{\mathrm{tr}}_{2}\\Q\cap R\neq\varnothing\\\ell(Q)<\ell^{-m}\ell(R)}}\langle S_{|\nu_{1}|,\varepsilon}(\Delta_{1,Q}f_{w}),\Delta_{2,R}g_{w}\rangle_{*}
+\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_{1},R\in\mathcal{D}^{\mathrm{tr}}_{2}\\Q\cap R\neq\varnothing\\\ell(Q)>\ell^{-m}\ell(R)}}\langle S_{|\nu_{1}|,\varepsilon}(\Delta_{1,Q}f_{w}),\Delta_{2,R}g_{w}\rangle_{*}
=:S'_{4}+S''_{4}. \tag*{(249)} S 4 ( w ) = Q ∈ D 1 tr , R ∈ D 2 tr Q ∩ R = ∅ ℓ ( Q ) < ℓ − m ℓ ( R ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ + Q ∈ D 1 tr , R ∈ D 2 tr Q ∩ R = ∅ ℓ ( Q ) > ℓ − m ℓ ( R ) ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , Δ 2 , R g w ⟩ ∗ =: S 4 ′ + S 4 ′′ . (249) Note that it suffices to bound the first sum since, by (203) , the second one can be estimated by the same arguments for S ∣ ν 2 ∣ , ε ∗ S^{*}_{|\nu_{2}|,\varepsilon} S ∣ ν 2 ∣ , ε ∗ S ∣ ν 1 ∣ , ε S_{|\nu_{1}|,\varepsilon} S ∣ ν 1 ∣ , ε
Let Q ∈ D 1 t r Q \in\mathbb{D}_1^{\mathrm{tr}} Q ∈ D 1 tr R ∈ D 2 t r R \in\mathbb{D}_2^{\mathrm{tr}} R ∈ D 2 tr Q ∩ R ≠ ∅ Q \cap R \ne\varnothing Q ∩ R = ∅ ℓ ( Q ) < ℓ m ℓ ( R ) \ell(Q) < \ell^{m}\ell(R) ℓ ( Q ) < ℓ m ℓ ( R ) f w f_w f w g w g_w g w Q Q Q R R R (249) are good. Since the generations are discrete, ℓ ( Q ) ≤ ℓ m + 1 ℓ ( R ) \ell(Q) \le\ell^{m+1}\ell(R) ℓ ( Q ) ≤ ℓ m + 1 ℓ ( R ) U U U R R R dist ( Q , U ) > ℓ ( Q ) α ℓ ( U ) 1 − α \operatorname{dist}(Q,U) > \ell(Q)^\alpha\ell(U)^{1-\alpha} dist ( Q , U ) > ℓ ( Q ) α ℓ ( U ) 1 − α dist ( Q , X ∖ U ) > ℓ ( Q ) α ℓ ( U ) 1 − α \operatorname{dist}(Q,X\setminus U) > \ell(Q)^\alpha\ell(U)^{1-\alpha} dist ( Q , X ∖ U ) > ℓ ( Q ) α ℓ ( U ) 1 − α Q Q Q R R R R Q R_Q R Q Q Q Q Q Q Q
dist ( Q , X ∖ R Q ) > ℓ ( Q ) α ℓ ( R Q ) 1 − α . \operatorname{dist}(Q,X\setminus R_Q) > \ell(Q)^\alpha\ell(R_Q)^{1-\alpha}. dist ( Q , X ∖ R Q ) > ℓ ( Q ) α ℓ ( R Q ) 1 − α . We use this notation and define the families
S 4 , t r ′ : = { ( Q , R ) ∈ D 1 t r × D 2 t r : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q transit } \mathcal{S}'_{4,\mathrm{tr}}:=\left\{(Q,R)\in\mathbb{D}_1^{\mathrm{tr}}\times\mathbb{D}_2^{\mathrm{tr}}:Q\subset R,\ \ell(Q)<\ell^{m}\ell(R),\ R_Q\ \text{transit}\right\} S 4 , tr ′ := { ( Q , R ) ∈ D 1 tr × D 2 tr : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q transit } and
S 4 , t e r m ′ : = { ( Q , R ) ∈ D 1 t r × D 2 t r : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q terminal } . \mathcal{S}'_{4,\mathrm{term}}:=\left\{(Q,R)\in\mathbb{D}_1^{\mathrm{tr}}\times\mathbb{D}_2^{\mathrm{tr}}:Q\subset R,\ \ell(Q)<\ell^{m}\ell(R),\ R_Q\ \text{terminal}\right\}. S 4 , term ′ := { ( Q , R ) ∈ D 1 tr × D 2 tr : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q terminal } . Hence, we further split the sum S 4 ′ S'_4 S 4 ′
S 4 ′ = ∑ ( Q , R ) ∈ S 4 , t r ′ + ∑ ( Q , R ) ∈ S 4 , t e r m ′ = : S 4 , t r ′ + S 4 , t e r m ′ . S'_4=\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}+\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{term}}=:S'_{4,\mathrm{tr}}+S'_{4,\mathrm{term}}.} S 4 ′ = ( Q , R ) ∈ S 4 , tr ′ ∑ + ( Q , R ) ∈ S 4 , term ′ =: S 4 , tr ′ + S 4 , term ′ . ∑ The sum over S 4 , t r ′ \mathcal{S}'_{4,\mathrm{tr}} S 4 , tr ′ To bound the contribution of transit cubes, we apply (203) and write
S 4 , t r ′ = ∑ ( Q , R ) ∈ S 4 , t r ′ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ R Q Δ 2 , R g w ) ⟩ + ∑ ( Q , R ) ∈ S 4 , t r ′ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q Δ 2 , R g w ) ⟩ . (250) \begin{aligned}
S'_{4,\mathrm{tr}}={}&\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}\left\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}\left(\chi_{R_Q}\Delta_{2,R}g_w\right)\right\rangle\\
&+\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}\left\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}\left(\chi_{X\setminus R_Q}\Delta_{2,R}g_w\right)\right\rangle. \tag*{(250)}
\end{aligned} S 4 , tr ′ = ( Q , R ) ∈ S 4 , tr ′ ∑ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ R Q Δ 2 , R g w ) ⟩ + ( Q , R ) ∈ S 4 , tr ′ ∑ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q Δ 2 , R g w ) ⟩ . (250) To bound the far away term, we note that
dist ( Q , X ∖ R Q ) > ℓ ( Q ) α ℓ ( R Q ) 1 − α = ℓ 1 − α ℓ ( Q ) α ℓ ( R ) 1 − α \operatorname{dist}(Q,X\setminus R_Q)>\ell(Q)^\alpha\ell(R_Q)^{1-\alpha}=\ell^{1-\alpha}\ell(Q)^\alpha\ell(R)^{1-\alpha} dist ( Q , X ∖ R Q ) > ℓ ( Q ) α ℓ ( R Q ) 1 − α = ℓ 1 − α ℓ ( Q ) α ℓ ( R ) 1 − α and that D ( Q , R ) ≈ ℓ ( R ) D(Q,R)\approx\ell(R) D ( Q , R ) ≈ ℓ ( R ) Q ⊂ R Q\subset R Q ⊂ R S 1 ′ S'_1 S 1 ′ [77 Lemma 5.17] , Lemma 6.12 with c ′ = ℓ 1 − α c'=\ell^{1-\alpha} c ′ = ℓ 1 − α (201) combined with Schur’s test, and (178) give
∑ ( Q , R ) ∈ S 4 , t r ′ ∣ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q Δ 2 , R g w ) ⟩ ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (251) \sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}\left|\left\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}\left(\chi_{X\setminus R_Q}\Delta_{2,R}g_w\right)\right\rangle\right|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. \tag*{(251)} ( Q , R ) ∈ S 4 , tr ′ ∑ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q Δ 2 , R g w ) ⟩ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (251) To estimate the first sum in (250) , we define
c R , Q ( g w ) : = ⟨ g w ⟩ 2 , R Q ⟨ b ^ 2 ⟩ 2 , R Q − ⟨ g w ⟩ 2 , R ⟨ b ^ 2 ⟩ 2 , R , for g w ∈ L 2 ( ∣ ν 2 ∣ ) . c_{R,Q}(g_w):=\frac{\langle g_w\rangle_{2,R_Q}}{\langle\widehat b_2\rangle_{2,R_Q}}-\frac{\langle g_w\rangle_{2,R}}{\langle\widehat b_2\rangle_{2,R}},\qquad\text{for }g_w\in L^2(|\nu_2|). c R , Q ( g w ) := ⟨ b 2 ⟩ 2 , R Q ⟨ g w ⟩ 2 , R Q − ⟨ b 2 ⟩ 2 , R ⟨ g w ⟩ 2 , R , for g w ∈ L 2 ( ∣ ν 2 ∣ ) . So, for g w ∈ L 2 ( ∣ ν 2 ∣ ) g_w\in L^2(|\nu_2|) g w ∈ L 2 ( ∣ ν 2 ∣ ) ( Q , R ) ∈ S 4 , t r ′ (Q,R)\in\mathcal{S}'_{4,\mathrm{tr}} ( Q , R ) ∈ S 4 , tr ′
∥ Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≥ ∫ R Q ∣ Δ 2 , R g w ∣ 2 d ∣ ν 2 ∣ = ∣ c R , Q ( g w ) ∣ 2 ∫ R Q ∣ b ^ 2 ∣ 2 d ∣ ν 2 ∣ = ∣ c R , Q ( g w ) ∣ 2 ∣ ν 2 ∣ ( R Q ) (252) \begin{aligned}
\|\Delta_{2,R}g_w\|_{L^2(|\nu_2|)}^2\ge\int_{R_Q}|\Delta_{2,R}g_w|^2\,d|\nu_2| \\
&=|c_{R,Q}(g_w)|^2\int_{R_Q}|\widehat b_2|^2\,d|\nu_2|=|c_{R,Q}(g_w)|^2|\nu_2|(R_Q) \tag*{(252)}
\end{aligned} ∥ Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≥ ∫ R Q ∣ Δ 2 , R g w ∣ 2 d ∣ ν 2 ∣ = ∣ c R , Q ( g w ) ∣ 2 ∫ R Q ∣ b 2 ∣ 2 d ∣ ν 2 ∣ = ∣ c R , Q ( g w ) ∣ 2 ∣ ν 2 ∣ ( R Q ) (252) and
χ R Q Δ 2 , R g w = c R , Q ( g w ) χ R Q b ^ 2 . (253) \chi_{R_Q}\Delta_{2,R}g_w=c_{R,Q}(g_w)\chi_{R_Q}\widehat b_2. \tag*{(253)} χ R Q Δ 2 , R g w = c R , Q ( g w ) χ R Q b 2 . (253) We recall that the parameter m ^ \widehat{m} m (50) and (51) . In the present argument, the measures are ∣ ν i ∣ = ∣ ν ^ i ∣ + μ ∣ X ∖ B i |\nu_i|=|\widehat{\nu}_i|+\mu|_{X\setminus B_i} ∣ ν i ∣ = ∣ ν i ∣ + μ ∣ X ∖ B i S ∣ ν 1 ∣ , ε S_{|\nu_1|,\varepsilon} S ∣ ν 1 ∣ , ε (157) . Thus, the high- and low-density stopping conditions give
m ^ − 1 μ ( Q ) ≤ ∣ ν i ∣ ( Q ) ≤ m ^ μ ( Q ) , Q ∈ D i t r , Q ⊂ Q 2 ( w i ) , i = 1 , 2. \widehat{m}^{-1}\mu(Q)\leq|\nu_i|(Q)\leq\widehat{m}\mu(Q),\qquad Q\in\mathcal{D}^{\mathrm{tr}}_i,\ Q\subset Q_2(w_i),\ i=1,2. m − 1 μ ( Q ) ≤ ∣ ν i ∣ ( Q ) ≤ m μ ( Q ) , Q ∈ D i tr , Q ⊂ Q 2 ( w i ) , i = 1 , 2. We first record the consequence of these inequalities that is needed below.
Lemma 6.24. Let F ⊂ D 1 t r \mathcal{F}\subset\mathcal{D}^{\mathrm{tr}}_1 F ⊂ D 1 tr Q 2 ( w 1 ) Q_2(w_1) Q 2 ( w 1 ) P ∈ D 2 t r P\in\mathcal{D}^{\mathrm{tr}}_2 P ∈ D 2 tr P ⊂ Q 2 ( w 2 ) P\subset Q_2(w_2) P ⊂ Q 2 ( w 2 )
∑ Q ∈ F Q ⊂ P ∣ ν 1 ∣ ( Q ) ≤ m ^ 2 ∣ ν 2 ∣ ( P ) . \sum_{\substack{Q\in\mathcal{F}\\Q\subset P}}|\nu_1|(Q)\leq\widehat{m}^{\,2}|\nu_2|(P). Q ∈ F Q ⊂ P ∑ ∣ ν 1 ∣ ( Q ) ≤ m 2 ∣ ν 2 ∣ ( P ) . The same assertion holds with the indices 1 and 2 interchanged.
Proof. Since all the cubes in the sum are transit and F \mathcal{F} F
∑ Q ∈ F Q ⊂ P ∣ ν 1 ∣ ( Q ) ≤ m ^ ∑ Q ∈ F Q ⊂ P μ ( Q ) ≤ m ^ μ ( P ) ≤ m ^ 2 ∣ ν 2 ∣ ( P ) . \sum_{\substack{Q\in\mathcal{F}\\Q\subset P}}|\nu_1|(Q)\leq\widehat{m}\sum_{\substack{Q\in\mathcal{F}\\Q\subset P}}\mu(Q)\leq\widehat{m}\mu(P)\leq\widehat{m}^{\,2}|\nu_2|(P). Q ∈ F Q ⊂ P ∑ ∣ ν 1 ∣ ( Q ) ≤ m Q ∈ F Q ⊂ P ∑ μ ( Q ) ≤ m μ ( P ) ≤ m 2 ∣ ν 2 ∣ ( P ) . In the following two lemmas, we use the coefficients c R , Q ( g w ) c_{R,Q}(g_w) c R , Q ( g w ) (252) , with the averages taken with respect to ∣ ν 2 ∣ |\nu_2| ∣ ν 2 ∣ (253) , the local term in (250) , for the auxiliary operator, is
∑ ( Q , R ) ∈ S 4 , t r ′ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ R Q Δ 2 , R g w ) ⟩ = ∑ ( Q , R ) ∈ S 4 , t r ′ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b ^ 2 ⟩ − ∑ ( Q , R ) ∈ S 4 , t r ′ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b ^ 2 ) ⟩ = : S ^ 1 − S ^ 2 . \begin{aligned}
\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}\left\langle\Delta_{1,Q}f_w,S^{*}_{|\nu_2|,\varepsilon}(\chi_{R_Q}\Delta_{2,R}g_w)\right\rangle\\
&=\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}c_{R,Q}(g_w)\left\langle\Delta_{1,Q}f_w,S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\right\rangle-\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}c_{R,Q}(g_w)\left\langle\Delta_{1,Q}f_w,S^{*}_{|\nu_2|,\varepsilon}(\chi_{X\setminus R_Q}\widehat{b}_2)\right\rangle\\
&=:\widehat{S}_1-\widehat{S}_2.
\end{aligned} ( Q , R ) ∈ S 4 , tr ′ ∑ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ R Q Δ 2 , R g w ) ⟩ = ( Q , R ) ∈ S 4 , tr ′ ∑ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b 2 ⟩ − ( Q , R ) ∈ S 4 , tr ′ ∑ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b 2 ) ⟩ =: S 1 − S 2 . Lemma 6.25. We have that
∣ S ^ 2 ∣ ≲ ℓ m γ / 2 m ^ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , |\widehat{S}_2|\lesssim\ell^{m\gamma/2}\widehat{m}\,\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}, ∣ S 2 ∣ ≲ ℓ mγ /2 m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , where the implicit constant depends only on the structural parameters.
Proof. We denote by z Q z_Q z Q Q Q Q [77 display (5.49)] , the kernel estimate (158) , the identity ∫ Δ 1 , Q f w d ∣ ν 1 ∣ = 0 \int\Delta_{1,Q}f_w\,d|\nu_1|=0 ∫ Δ 1 , Q f w d ∣ ν 1 ∣ = 0 ∣ b ^ 2 ∣ ≡ 1 |\widehat{b}_2|\equiv1 ∣ b 2 ∣ ≡ 1
∣ ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b ^ 2 ) ⟩ ∣ ≲ ∫ X ∖ R Q ℓ ( Q ) γ d ( x , z Q ) n + γ d ∣ ν 2 ∣ ( x ) ∥ Δ 1 , Q f w ∥ L 1 ( ∣ ν 1 ∣ ) . (254) \left|\left\langle\Delta_{1,Q}f_w,S^{*}_{|\nu_2|,\varepsilon}(\chi_{X\setminus R_Q}\widehat{b}_2)\right\rangle\right|\lesssim\int_{X\setminus R_Q}\frac{\ell(Q)^\gamma}{d(x,z_Q)^{n+\gamma}}\,d|\nu_2|(x)\,\|\Delta_{1,Q}f_w\|_{L^1(|\nu_1|)}. \tag*{(254)} ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b 2 ) ⟩ ≲ ∫ X ∖ R Q d ( x , z Q ) n + γ ℓ ( Q ) γ d ∣ ν 2 ∣ ( x ) ∥ Δ 1 , Q f w ∥ L 1 ( ∣ ν 1 ∣ ) . (254) Here the goodness parameter m m m (33) and the separation from X ∖ R Q X\setminus R_Q X ∖ R Q R Q R_Q R Q R R R ∣ ν 2 ∣ ( B ( z Q , t ) ) ≲ t n |\nu_2|(B(z_Q,t))\lesssim t^n ∣ ν 2 ∣ ( B ( z Q , t )) ≲ t n t ≥ ℓ ( R Q ) t\geq\ell(R_Q) t ≥ ℓ ( R Q ) Q Q Q
dist ( Q , X ∖ R Q ) ≥ ℓ ( Q ) α ℓ ( R Q ) 1 − α . \operatorname{dist}(Q,X\setminus R_Q)\geq\ell(Q)^\alpha\ell(R_Q)^{1-\alpha}. dist ( Q , X ∖ R Q ) ≥ ℓ ( Q ) α ℓ ( R Q ) 1 − α . Thus, since α ( n + γ ) = γ / 2 \alpha(n+\gamma)=\gamma/2 α ( n + γ ) = γ /2
∫ X ∖ R Q ℓ ( Q ) γ d ( x , z Q ) n + γ d ∣ ν 2 ∣ ( x ) ≲ ℓ ( Q ) γ ℓ ( R Q ) n ( ℓ ( Q ) α ℓ ( R Q ) 1 − α ) n + γ + ∑ k = 0 ∞ ℓ ( Q ) γ ( 2 k + 1 ℓ ( R Q ) ) n ( 2 k ℓ ( R Q ) ) n + γ \int_{X\setminus R_Q}\frac{\ell(Q)^\gamma}{d(x,z_Q)^{n+\gamma}}\,d|\nu_2|(x)\lesssim
\frac{\ell(Q)^\gamma\ell(R_Q)^n}{\bigl(\ell(Q)^\alpha\ell(R_Q)^{1-\alpha}\bigr)^{n+\gamma}}
+\sum_{k=0}^{\infty}\frac{\ell(Q)^\gamma(2^{k+1}\ell(R_Q))^n}{(2^k\ell(R_Q))^{n+\gamma}} ∫ X ∖ R Q d ( x , z Q ) n + γ ℓ ( Q ) γ d ∣ ν 2 ∣ ( x ) ≲ ( ℓ ( Q ) α ℓ ( R Q ) 1 − α ) n + γ ℓ ( Q ) γ ℓ ( R Q ) n + k = 0 ∑ ∞ ( 2 k ℓ ( R Q ) ) n + γ ℓ ( Q ) γ ( 2 k + 1 ℓ ( R Q ) ) n ≲ ℓ ( Q ) γ / 2 ℓ ( R Q ) γ / 2 . \lesssim\frac{\ell(Q)^{\gamma/2}}{\ell(R_Q)^{\gamma/2}}. ≲ ℓ ( R Q ) γ /2 ℓ ( Q ) γ /2 . We combine this bound with (254) and apply the Cauchy-Schwarz inequality to obtain
∣ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b ^ 2 ) ⟩ ∣ ≲ ∣ c R , Q ( g w ) ∣ ℓ ( Q ) γ / 2 ℓ ( R Q ) γ / 2 ∣ ν 1 ∣ ( Q ) 1 / 2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) . (255) \left|c_{R,Q}(g_w)\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}(\chi_{X\setminus R_Q}\widehat b_2)\rangle\right|
\lesssim|c_{R,Q}(g_w)|\frac{\ell(Q)^{\gamma/2}}{\ell(R_Q)^{\gamma/2}}|\nu_1|(Q)^{1/2}\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}. \tag*{(255)} c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ ( χ X ∖ R Q b 2 )⟩ ≲ ∣ c R , Q ( g w ) ∣ ℓ ( R Q ) γ /2 ℓ ( Q ) γ /2 ∣ ν 1 ∣ ( Q ) 1/2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) . (255) Fix j ≥ m + 1 j\ge m+1 j ≥ m + 1 R R R S S S R R R Q Q Q R Q = S R_Q=S R Q = S c R , Q ( g w ) c_{R,Q}(g_w) c R , Q ( g w ) (253) gives
∣ c R , Q ( g w ) ∣ ∣ ν 2 ∣ ( S ) 1 / 2 = ∥ χ S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) . |c_{R,Q}(g_w)||\nu_2|(S)^{1/2}=\|\chi_S\Delta_{2,R}g_w\|_{L^2(|\nu_2|)}. ∣ c R , Q ( g w ) ∣∣ ν 2 ∣ ( S ) 1/2 = ∥ χ S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) . The cubes Q Q Q R Q = S R_Q=S R Q = S ℓ ( Q ) = ℓ j ℓ ( R ) \ell(Q)=\ell^j\ell(R) ℓ ( Q ) = ℓ j ℓ ( R )
∑ Q ∈ D 1 t r : R Q = S ℓ ( Q ) = ℓ j ℓ ( R ) ∣ c R , Q ( g w ) ∣ ℓ ( Q ) γ / 2 ℓ ( S ) γ / 2 ∣ ν 1 ∣ ( Q ) 1 / 2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ≤ ℓ ( j − 1 ) γ / 2 m ^ ∥ χ S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) ( ∑ Q ∈ D 1 t r : R Q = S ℓ ( Q ) = ℓ j ℓ ( R ) ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 . (256) \sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1:R_Q=S\\\ell(Q)=\ell^j\ell(R)}}|c_{R,Q}(g_w)|\frac{\ell(Q)^{\gamma/2}}{\ell(S)^{\gamma/2}}|\nu_1|(Q)^{1/2}\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}
\le\ell^{(j-1)\gamma/2}\widehat{\mathfrak m}\|\chi_S\Delta_{2,R}g_w\|_{L^2(|\nu_2|)}
\left(\sum_{\substack{Q\in\mathcal{D}^{\mathrm{tr}}_1:R_Q=S\\\ell(Q)=\ell^j\ell(R)}}\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}^2\right)^{1/2}. \tag*{(256)} Q ∈ D 1 tr : R Q = S ℓ ( Q ) = ℓ j ℓ ( R ) ∑ ∣ c R , Q ( g w ) ∣ ℓ ( S ) γ /2 ℓ ( Q ) γ /2 ∣ ν 1 ∣ ( Q ) 1/2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ≤ ℓ ( j − 1 ) γ /2 m ∥ χ S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) Q ∈ D 1 tr : R Q = S ℓ ( Q ) = ℓ j ℓ ( R ) ∑ ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 . (256) For fixed j j j Q Q Q ( R , S ) (R,S) ( R , S ) R R R (256) over R R R (178) , we obtain
∣ S ^ 2 ∣ ≲ m ^ ∑ j ≥ m + 1 ℓ ( j − 1 ) γ / 2 ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ℓ m γ / 2 m ^ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . |\widehat S_2|\lesssim\widehat{\mathfrak m}\sum_{j\ge m+1}\ell^{(j-1)\gamma/2}\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}
\lesssim\ell^{m\gamma/2}\widehat{\mathfrak m}\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. ∣ S 2 ∣ ≲ m j ≥ m + 1 ∑ ℓ ( j − 1 ) γ /2 ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ℓ mγ /2 m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . It remains to estimate S ^ 1 \widehat S_1 S 1
Lemma 6.26. We have that
∣ S ^ 1 ∣ = ∣ ∑ ( Q , R ) ∈ S 4 , t r ′ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b ^ 2 ⟩ ∣ ≲ c ♯ m ^ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . |\widehat S_1|=\left|\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}c_{R,Q}(g_w)\langle\Delta_{1,Q}f_w,S^*_{|\nu_2|,\varepsilon}\widehat b_2\rangle\right|\lesssim c^\sharp\widehat{\mathfrak m}\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. ∣ S 1 ∣ = ( Q , R ) ∈ S 4 , tr ′ ∑ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b 2 ⟩ ≲ c ♯ m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . Proof. Let g ~ = g w − E 2 g w \widetilde g=g_w-E_2g_w g = g w − E 2 g w ( Q , R ) ∈ S 4 , t r ′ (Q,R)\in\mathcal{S}'_{4,\mathrm{tr}} ( Q , R ) ∈ S 4 , tr ′
c R , Q ( g w ) = ⟨ g w ⟩ 2 , R Q ⟨ b ^ 2 ⟩ 2 , R Q − ⟨ g w ⟩ 2 , R ⟨ b ^ 2 ⟩ 2 , R = ⟨ g ~ ⟩ 2 , R Q ⟨ b ^ 2 ⟩ 2 , R Q − ⟨ g ~ ⟩ 2 , R ⟨ b ^ 2 ⟩ 2 , R . (257) c_{R,Q}(g_w)=\frac{\langle g_w\rangle_{2,R_Q}}{\langle\widehat b_2\rangle_{2,R_Q}}-\frac{\langle g_w\rangle_{2,R}}{\langle\widehat b_2\rangle_{2,R}}=\frac{\langle\widetilde g\rangle_{2,R_Q}}{\langle\widehat b_2\rangle_{2,R_Q}}-\frac{\langle\widetilde g\rangle_{2,R}}{\langle\widehat b_2\rangle_{2,R}}. \tag*{(257)} c R , Q ( g w ) = ⟨ b 2 ⟩ 2 , R Q ⟨ g w ⟩ 2 , R Q − ⟨ b 2 ⟩ 2 , R ⟨ g w ⟩ 2 , R = ⟨ b 2 ⟩ 2 , R Q ⟨ g ⟩ 2 , R Q − ⟨ b 2 ⟩ 2 , R ⟨ g ⟩ 2 , R . (257) We denote by F Π \mathcal{F}_{\Pi} F Π Q ∈ D 1 t r Q\in\mathcal{D}^{\mathrm{tr}}_1 Q ∈ D 1 tr R R R ( Q , R ) ∈ S 4 , t r ′ (Q,R)\in\mathcal{S}'_{4,\mathrm{tr}} ( Q , R ) ∈ S 4 , tr ′ Q ∈ F Π Q\in\mathcal{F}_{\Pi} Q ∈ F Π R ( Q ) R(Q) R ( Q ) R Q R_Q R Q 3 Q 2 ( w 2 ) Q_2(w_2) Q 2 ( w 2 ) Q 2 ( w 2 ) Q_2(w_2) Q 2 ( w 2 ) g ~ \widetilde g g (257) gives
∑ R : ( Q , R ) ∈ S 4 , t r ′ c R , Q ( g w ) = ⟨ g ~ ⟩ 2 , R ( Q ) ⟨ b ^ 2 ⟩ 2 , R ( Q ) . \sum_{R:(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}c_{R,Q}(g_w)=\frac{\langle\widetilde g\rangle_{2,R(Q)}}{\langle\widehat b_2\rangle_{2,R(Q)}}. R : ( Q , R ) ∈ S 4 , tr ′ ∑ c R , Q ( g w ) = ⟨ b 2 ⟩ 2 , R ( Q ) ⟨ g ⟩ 2 , R ( Q ) . Thus, we define the paraproduct
Π Ξ w ( ϕ ) : = ∑ Q ∈ F Π ⟨ ϕ ⟩ 2 , R ( Q ) ⟨ b ^ 2 ⟩ 2 , R ( Q ) Δ 1 , Q ∗ S ∣ ν 2 ∣ , ε ∗ b ^ 2 , \Pi_{\Xi_w}(\phi) := \sum_{Q\in\mathcal{F}_{\Pi}} \frac{\langle\phi\rangle_{2,R(Q)}}{\langle\widehat{b}_2\rangle_{2,R(Q)}} \Delta_{1,Q}^{*}S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2, Π Ξ w ( ϕ ) := Q ∈ F Π ∑ ⟨ b 2 ⟩ 2 , R ( Q ) ⟨ ϕ ⟩ 2 , R ( Q ) Δ 1 , Q ∗ S ∣ ν 2 ∣ , ε ∗ b 2 , for ϕ ∈ L 2 ( ∣ ν 2 ∣ ) \phi\in L^2(|\nu_2|) ϕ ∈ L 2 ( ∣ ν 2 ∣ )
and, arguing as in [77 p. 166] , we obtain the identity
S ^ 1 = ∑ ( Q , R ) ∈ S 4 , t r ′ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b ^ 2 ⟩ = ⟨ f w , Π Ξ w ( g ~ ) ⟩ . (258) \widehat{S}_1=\sum_{(Q,R)\in\mathcal{S}'_{4,\mathrm{tr}}}c_{R,Q}(g_w)\langle\Delta_{1,Q}f_w,S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\rangle=\langle f_w,\Pi_{\Xi_w}(\widetilde{g})\rangle. \tag*{(258)} S 1 = ( Q , R ) ∈ S 4 , tr ′ ∑ c R , Q ( g w ) ⟨ Δ 1 , Q f w , S ∣ ν 2 ∣ , ε ∗ b 2 ⟩ = ⟨ f w , Π Ξ w ( g )⟩ . (258) We first consider finite sums and obtain the corresponding bounded operator by passage to the limit. To prove that Π Ξ w \Pi_{\Xi_w} Π Ξ w L 2 ( ∣ ν 2 ∣ ) L^2(|\nu_2|) L 2 ( ∣ ν 2 ∣ ) L 2 ( ∣ ν 1 ∣ ) L^2(|\nu_1|) L 2 ( ∣ ν 1 ∣ ) φ ∈ L 2 ( ∣ ν 1 ∣ ) \varphi\in L^2(|\nu_1|) φ ∈ L 2 ( ∣ ν 1 ∣ )
a Q : = { ∣ ⟨ Δ 1 , Q φ , S ∣ ν 2 ∣ , ε ∗ b ^ 2 ⟩ ∣ 2 ∥ Δ 1 , Q φ ∥ L 2 ( ∣ ν 1 ∣ ) , if Δ 1 , Q φ ≠ 0 , 0 , if Δ 1 , Q φ = 0 , A R : = ∑ Q ∈ F Π R ( Q ) = R a Q . a_Q:=\begin{cases}
\frac{|\langle\Delta_{1,Q}\varphi,S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\rangle|^2}{\|\Delta_{1,Q}\varphi\|_{L^2(|\nu_1|)}},&\text{if }\Delta_{1,Q}\varphi\ne0,\\
0,&\text{if }\Delta_{1,Q}\varphi=0,
\end{cases}
\qquad
A_R:=\sum_{\substack{Q\in\mathcal{F}_{\Pi}\\R(Q)=R}}a_Q. a Q := ⎩ ⎨ ⎧ ∥ Δ 1 , Q φ ∥ L 2 ( ∣ ν 1 ∣ ) ∣ ⟨ Δ 1 , Q φ , S ∣ ν 2 ∣ , ε ∗ b 2 ⟩ ∣ 2 , 0 , if Δ 1 , Q φ = 0 , if Δ 1 , Q φ = 0 , A R := Q ∈ F Π R ( Q ) = R ∑ a Q . By (176) and the Cauchy-Schwarz inequality,
a Q ≤ ∥ Δ 1 , Q ∗ S ∣ ν 2 ∣ , ε ∗ b ^ 2 ∥ L 2 ( ∣ ν 1 ∣ ) . a_Q\leq\|\Delta^{*}_{1,Q}S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\|_{L^2(|\nu_1|)}. a Q ≤ ∥ Δ 1 , Q ∗ S ∣ ν 2 ∣ , ε ∗ b 2 ∥ L 2 ( ∣ ν 1 ∣ ) . For S ∈ D 2 t r S\in\mathbb{D}^{\mathrm{tr}}_2 S ∈ D 2 tr F Π + ( S ) \mathcal{F}^{+}_{\Pi}(S) F Π + ( S ) F Π \mathcal{F}_{\Pi} F Π S S S (187) , give
∑ R ∈ D 2 t r R ⊂ S A R ≤ ∑ P ∈ F Π + ( S ) ∑ Q ∈ F Π Q ⊂ P ∥ Δ 1 , Q ∗ ( χ P S ∣ ν 2 ∣ , ε ∗ b ^ 2 ) ∥ L 2 ( ∣ ν 1 ∣ ) \sum_{\substack{R\in\mathbb{D}^{\mathrm{tr}}_2\\R\subset S}}A_R
\leq\sum_{P\in\mathcal{F}^{+}_{\Pi}(S)}\sum_{\substack{Q\in\mathcal{F}_{\Pi}\\Q\subset P}}\|\Delta^{*}_{1,Q}(\chi_P S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2)\|_{L^2(|\nu_1|)} R ∈ D 2 tr R ⊂ S ∑ A R ≤ P ∈ F Π + ( S ) ∑ Q ∈ F Π Q ⊂ P ∑ ∥ Δ 1 , Q ∗ ( χ P S ∣ ν 2 ∣ , ε ∗ b 2 ) ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ∑ P ∈ F Π + ( S ) ∥ χ P S ∣ ν 2 ∣ , ε ∗ b ^ 2 ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≲ c ♯ 2 ∑ P ∈ F Π + ( S ) ∣ ν 1 ∣ ( P ) ≤ c ♯ 2 m ^ 2 ∣ ν 2 ∣ ( S ) , \lesssim\sum_{P\in\mathcal{F}^{+}_{\Pi}(S)}\|\chi_P S^{*}_{|\nu_2|,\varepsilon}\widehat{b}_2\|_{L^2(|\nu_1|)}^2
\lesssim c_{\sharp}^{2}\sum_{P\in\mathcal{F}^{+}_{\Pi}(S)}|\nu_1|(P)
\leq c_{\sharp}^{2}\widehat{m}^{2}|\nu_2|(S), ≲ P ∈ F Π + ( S ) ∑ ∥ χ P S ∣ ν 2 ∣ , ε ∗ b 2 ∥ L 2 ( ∣ ν 1 ∣ ) 2 ≲ c ♯ 2 P ∈ F Π + ( S ) ∑ ∣ ν 1 ∣ ( P ) ≤ c ♯ 2 m 2 ∣ ν 2 ∣ ( S ) , where in the last inequality we used Lemma 6.24 . If S S S R ⊂ S R\subset S R ⊂ S R R R ∣ ⟨ b ^ 2 ⟩ 2 , R ∣ ≳ 1 |\langle\widehat{b}_2\rangle_{2,R}|\gtrsim1 ∣ ⟨ b 2 ⟩ 2 , R ∣ ≳ 1 (178) , and the Carleson embedding theorem yield, for ψ ∈ L 2 ( ∣ ν 2 ∣ ) \psi\in L^2(|\nu_2|) ψ ∈ L 2 ( ∣ ν 2 ∣ )
∣ ⟨ φ , Π Ξ w ψ ⟩ ∣ ≤ ( ∑ Q ∈ F Π ∥ Δ 1 , Q φ ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ( ∑ R ∈ D 2 t r ⟨ ψ ⟩ 2 , R 2 ⟨ b ^ 2 ⟩ 2 , R 2 A R ) 1 / 2 |\langle\varphi,\Pi_{\Xi_w}\psi\rangle|
\leq\left(\sum_{Q\in\mathcal{F}_{\Pi}}\|\Delta_{1,Q}\varphi\|_{L^2(|\nu_1|)}^2\right)^{1/2}
\left(\sum_{R\in\mathbb{D}^{\mathrm{tr}}_2}\frac{\langle\psi\rangle_{2,R}^2}{\langle\widehat{b}_2\rangle_{2,R}^2}A_R\right)^{1/2} ∣ ⟨ φ , Π Ξ w ψ ⟩ ∣ ≤ Q ∈ F Π ∑ ∥ Δ 1 , Q φ ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 R ∈ D 2 tr ∑ ⟨ b 2 ⟩ 2 , R 2 ⟨ ψ ⟩ 2 , R 2 A R 1/2 ≲ c ♯ m ^ ∥ φ ∥ L 2 ( ∣ ν 1 ∣ ) ∥ ψ ∥ L 2 ( ∣ ν 2 ∣ ) . (259) \lesssim c_{\sharp}\widehat{m}\|\varphi\|_{L^2(|\nu_1|)}\|\psi\|_{L^2(|\nu_2|)}. \tag*{(259)} ≲ c ♯ m ∥ φ ∥ L 2 ( ∣ ν 1 ∣ ) ∥ ψ ∥ L 2 ( ∣ ν 2 ∣ ) . (259) Thus, by (258) , (259) , and the bound for E 2 E_2 E 2
∣ S ^ 1 ∣ ≲ c ♯ m ^ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w − E 2 g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ c ♯ m ^ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . |\widehat{S}_1|\lesssim c_{\sharp}\widehat{m}\|f_w\|_{L^2(|\nu_1|)}\|g_w-E_2g_w\|_{L^2(|\nu_2|)}
\lesssim c_{\sharp}\widehat{m}\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. ∣ S 1 ∣ ≲ c ♯ m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w − E 2 g w ∥ L 2 ( ∣ ν 2 ∣ ) ≲ c ♯ m ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . We gather (250) , (251) , and Lemmas 6.25 and 6.26, and obtain
∣ S 4 , t r ′ ∣ ≲ ( 1 + ( 1 + c ♯ ) m ^ ) ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (260) |S'_{4,\mathrm{tr}}|\lesssim\left(1+(1+c_{\sharp})\widehat{m}\right)\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. \tag*{(260)} ∣ S 4 , tr ′ ∣ ≲ ( 1 + ( 1 + c ♯ ) m ) ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (260) The sum over S 4 , t e r m ′ \mathcal{S}'_{4,\mathrm{term}} S 4 , term ′ Recall that
S 4 , t e r m ′ : = { ( Q , R ) ∈ D 1 t r × D 2 t r : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q terminal } . \mathcal{S}'_{4,\mathrm{term}}:=\{(Q,R)\in\mathbb{D}^{\mathrm{tr}}_1\times\mathbb{D}^{\mathrm{tr}}_2:Q\subset R,\ \ell(Q)<\ell^m\ell(R),\ R_Q\text{ terminal}\}. S 4 , term ′ := {( Q , R ) ∈ D 1 tr × D 2 tr : Q ⊂ R , ℓ ( Q ) < ℓ m ℓ ( R ) , R Q terminal } . We reindex this sum by the terminal child, which we now denote by R R R R ^ \widehat{R} R R ∈ D 2 t e r m R\in\mathbb{D}^{\mathrm{term}}_2 R ∈ D 2 term R ^ ∈ D 2 t r \widehat{R}\in\mathbb{D}^{\mathrm{tr}}_2 R ∈ D 2 tr Q ⊂ R Q\subset R Q ⊂ R ℓ ( Q ) < ℓ m ℓ ( R ^ ) \ell(Q)<\ell^m\ell(\widehat{R}) ℓ ( Q ) < ℓ m ℓ ( R ) ℓ ( Q ) ≤ ℓ m ℓ ( R ) \ell(Q)\leq\ell^m\ell(R) ℓ ( Q ) ≤ ℓ m ℓ ( R ) R R R
We use the following Whitney decomposition. We prove it in Appendix § 9 .
Lemma 6.27. Let ( X , d ) (X,d) ( X , d ) D D D D \mathbb{D} D Theorem 2.9 . For every Q ∈ D Q\in\mathbb{D} Q ∈ D X ∖ Q ≠ ∅ X\setminus Q\ne\varnothing X ∖ Q = ∅ W ( Q ) \mathcal{W}(Q) W ( Q ) S ∈ D S\in\mathbb{D} S ∈ D Q Q Q
Cubes in W ( Q ) \mathcal{W}(Q) W ( Q ) Q Q Q
There exist C W > c W > 1 C_W>c_W>1 C W > c W > 1
c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) ≤ C W ℓ ( S ) , S ∈ W ( Q ) . c_W\ell(S)\leq\operatorname{dist}(S,X\setminus Q)\leq C_W\ell(S),\qquad S\in\mathcal{W}(Q). c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) ≤ C W ℓ ( S ) , S ∈ W ( Q ) . There exists c > 0 c>0 c > 0
∑ S ∈ W ( Q ) χ 2 S ( x ) ≤ c , x ∈ X , (261) \sum_{S\in\mathcal{W}(Q)}\chi_{2S}(x)\leq c,\qquad x\in X, \tag*{(261)} S ∈ W ( Q ) ∑ χ 2 S ( x ) ≤ c , x ∈ X , (261) where 2 S = { x ∈ X : dist ( x , S ) < ℓ ( S ) } 2S=\{x\in X:\operatorname{dist}(x,S)<\ell(S)\} 2 S = { x ∈ X : dist ( x , S ) < ℓ ( S )}
The constants depend only on D D D Theorem 2.9 .
For each good cube Q Q Q S ∈ W ( R ) S\in\mathcal{W}(R) S ∈ W ( R ) z Q z_Q z Q dist ( Q , X ∖ R ) > 0 \operatorname{dist}(Q,X\setminus R)>0 dist ( Q , X ∖ R ) > 0
ℓ ( Q ) α ℓ ( R ) 1 − α ≤ dist ( Q , X ∖ R ) ≤ dist ( z Q , X ∖ R ) ≤ ( C W + 2 C 1 ) ℓ ( S ) , \ell(Q)^\alpha\ell(R)^{1-\alpha}\leq\operatorname{dist}(Q,X\setminus R)\leq\operatorname{dist}(z_Q,X\setminus R)\leq(C_W+2C_1)\ell(S), ℓ ( Q ) α ℓ ( R ) 1 − α ≤ dist ( Q , X ∖ R ) ≤ dist ( z Q , X ∖ R ) ≤ ( C W + 2 C 1 ) ℓ ( S ) , where diam ( S ) ≤ 2 C 1 ℓ ( S ) \operatorname{diam}(S)\leq2C_1\ell(S) diam ( S ) ≤ 2 C 1 ℓ ( S ) ℓ ( Q ) ≤ ℓ m ℓ ( R ) \ell(Q)\leq\ell^m\ell(R) ℓ ( Q ) ≤ ℓ m ℓ ( R ) m m m
diam ( Q ) ≤ 2 C 1 ℓ ( Q ) ≤ 1 4 ℓ ( S ) and ℓ ( Q ) α ℓ ( R ) 1 − α ≲ ℓ ( S ) . (262) \operatorname{diam}(Q)\leq2C_1\ell(Q)\leq\frac{1}{4}\ell(S)\qquad\text{and}\qquad\ell(Q)^\alpha\ell(R)^{1-\alpha}\lesssim\ell(S). \tag*{(262)} diam ( Q ) ≤ 2 C 1 ℓ ( Q ) ≤ 4 1 ℓ ( S ) and ℓ ( Q ) α ℓ ( R ) 1 − α ≲ ℓ ( S ) . (262) In particular, Q ⊂ 2 S Q\subset2S Q ⊂ 2 S Q Q Q S S S 2 S ⊂ R 2S\subset R 2 S ⊂ R
We can therefore write
S 4 , t e r m ′ = ∑ R , S , Q ⟨ S ∣ ν 1 ∣ ε ( Δ 1 , Q f w ) , χ 2 S Δ 2 , R g ^ w ⟩ ∗ + ∑ R , S , Q ⟨ S ∣ ν 1 ∣ ε ( Δ 1 , Q f w ) , χ R ^ ∖ 2 S Δ 2 , R g ^ w ⟩ ∗ = : S ^ 1 t e r m + S ^ 2 t e r m . \mathcal{S}'_{4,\mathrm{term}}
=
\sum_{R,S,Q}\left\langle S|\nu_1|_{\varepsilon}(\Delta_{1,Q}f_w),\chi_{2S}\Delta_{2,R}\widehat{g}_w\right\rangle_*
+
\sum_{R,S,Q}\left\langle S|\nu_1|_{\varepsilon}(\Delta_{1,Q}f_w),\chi_{\widehat{R}\setminus2S}\Delta_{2,R}\widehat{g}_w\right\rangle_*
=:\widehat{\mathcal{S}}^{\mathrm{term}}_1+\widehat{\mathcal{S}}^{\mathrm{term}}_2. S 4 , term ′ = R , S , Q ∑ ⟨ S ∣ ν 1 ∣ ε ( Δ 1 , Q f w ) , χ 2 S Δ 2 , R g w ⟩ ∗ + R , S , Q ∑ ⟨ S ∣ ν 1 ∣ ε ( Δ 1 , Q f w ) , χ R ∖ 2 S Δ 2 , R g w ⟩ ∗ =: S 1 term + S 2 term . Here R R R S ∈ W ( R ) S\in\mathcal{W}(R) S ∈ W ( R ) Q Q Q Q ⊂ R Q\subset R Q ⊂ R ℓ ( Q ) ≤ ℓ m ℓ ( R ) \ell(Q)\leq\ell^m\ell(R) ℓ ( Q ) ≤ ℓ m ℓ ( R ) z Q ∈ S z_Q\in S z Q ∈ S
Lemma 6.28. If m m m α \alpha α
∣ S ^ 1 t e r m ∣ + ∣ S ^ 2 t e r m ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , \left|\widehat{\mathcal{S}}^{\mathrm{term}}_1\right|+\left|\widehat{\mathcal{S}}^{\mathrm{term}}_2\right|
\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}, S 1 term + S 2 term ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , uniformly in w w w ε \varepsilon ε
Proof. To bound S ^ 1 t e r m \widehat{S}_1^{\mathrm{term}} S 1 term R R R S ∈ W ( R ) S \in\mathcal{W}(R) S ∈ W ( R ) g ′ : = χ 2 S Δ 2 , R ^ g w g' := \chi_{2S}\Delta_{2,\widehat{R}}g_w g ′ := χ 2 S Δ 2 , R g w R ⊂ W 2 ( w 2 ) R \subset\mathcal{W}_2(w_2) R ⊂ W 2 ( w 2 ) F 2 ( w 2 ) ∩ W 2 ( w 2 ) = ∅ F_2(w_2) \cap\mathcal{W}_2(w_2) = \varnothing F 2 ( w 2 ) ∩ W 2 ( w 2 ) = ∅ (149) and the second property in Lemma 6.27 , we have
Ξ w ( x ) ≥ dist ( x , F 2 ( w 2 ) ) ≥ dist ( x , X ∖ R ) ≥ ( c W − 1 ) ℓ ( S ) , for x ∈ 2 S . \Xi_w(x) \ge\operatorname{dist}(x,F_2(w_2)) \ge\operatorname{dist}(x,X\setminus R) \ge(c_W-1)\ell(S), \qquad\text{for }x\in2S. Ξ w ( x ) ≥ dist ( x , F 2 ( w 2 )) ≥ dist ( x , X ∖ R ) ≥ ( c W − 1 ) ℓ ( S ) , for x ∈ 2 S . We set t S : = ℓ ( S ) + inf 2 S Ξ w t_S := \ell(S) + \inf_{2S}\Xi_w t S := ℓ ( S ) + inf 2 S Ξ w
t S ≤ c W ( c W − 1 ) − 1 inf 2 S Ξ w . t_S \le c_W(c_W-1)^{-1}\inf_{2S}\Xi_w. t S ≤ c W ( c W − 1 ) − 1 2 S inf Ξ w . Hence, (159) yields
∣ L ε ( x , y ) ∣ ≲ t S − n , for x , y ∈ 2 S . (263) |L_\varepsilon(x,y)| \lesssim t_S^{-n}, \qquad\text{for }x,y\in2S. \tag*{(263)} ∣ L ε ( x , y ) ∣ ≲ t S − n , for x , y ∈ 2 S . (263) Moreover, diam ( 2 S ) ≤ ( 2 C 1 + 2 ) ℓ ( S ) \operatorname{diam}(2S) \le(2C_1+2)\ell(S) diam ( 2 S ) ≤ ( 2 C 1 + 2 ) ℓ ( S ) Ξ w \Xi_w Ξ w 1 1 1 z ∈ 2 S z\in2S z ∈ 2 S 2 S ⊂ B ( z , ( 2 C 1 + 3 ) ℓ ( S ) ) 2S\subset B(z,(2C_1+3)\ell(S)) 2 S ⊂ B ( z , ( 2 C 1 + 3 ) ℓ ( S )) Ξ w ( z ) ≤ t S + ( 2 C 1 + 2 ) ℓ ( S ) \Xi_w(z)\le t_S+(2C_1+2)\ell(S) Ξ w ( z ) ≤ t S + ( 2 C 1 + 2 ) ℓ ( S ) (154) gives ∣ ν i ∣ ( 2 S ) ≲ t S n |\nu_i|(2S)\lesssim t_S^n ∣ ν i ∣ ( 2 S ) ≲ t S n i = 1 , 2 i=1,2 i = 1 , 2 (263) and the Cauchy-Schwarz inequality, we obtain
∥ χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ∥ L 2 ( ∣ ν 1 ∣ ) ≲ t S − n ∣ ν 1 ∣ ( 2 S ) 1 / 2 ∣ ν 2 ∣ ( 2 S ) 1 / 2 ∥ g ′ ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ g ′ ∥ L 2 ( ∣ ν 2 ∣ ) . \|\chi_{2S}S^*_{|\nu_2|,\varepsilon}g'\|_{L^2(|\nu_1|)}
\lesssim t_S^{-n}|\nu_1|(2S)^{1/2}|\nu_2|(2S)^{1/2}\|g'\|_{L^2(|\nu_2|)}
\lesssim\|g'\|_{L^2(|\nu_2|)}. ∥ χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ∥ L 2 ( ∣ ν 1 ∣ ) ≲ t S − n ∣ ν 1 ∣ ( 2 S ) 1/2 ∣ ν 2 ∣ ( 2 S ) 1/2 ∥ g ′ ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ g ′ ∥ L 2 ( ∣ ν 2 ∣ ) . The cubes Q Q Q ( R , S ) (R,S) ( R , S ) 2 S 2S 2 S (176) , the Cauchy-Schwarz inequality, and the square-function bound for the adjoint martingale differences stated after (179) , we have
∑ Q ∣ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′ ⟩ ∗ ∣ = ∑ Q ∣ ⟨ Δ 1 , Q f w , Δ 1 , Q ∗ ( χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ) ⟩ ∣ ≤ ( ∑ Q ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ( ∑ Q ∥ Δ 1 , Q ∗ ( χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ) ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ≲ ( ∑ Q ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 ) 1 / 2 ∥ χ 2 S Δ 2 , R ^ g w ∥ L 2 ( ∣ ν 2 ∣ ) , \begin{aligned}
\sum_Q \left|\left\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_w),g'\right\rangle_*\right|
&=\sum_Q\left|\left\langle\Delta_{1,Q}f_w,\Delta^*_{1,Q}(\chi_{2S}S^*_{|\nu_2|,\varepsilon}g')\right\rangle\right|\\
&\le\left(\sum_Q\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}^2\right)^{1/2}
\left(\sum_Q\|\Delta^*_{1,Q}(\chi_{2S}S^*_{|\nu_2|,\varepsilon}g')\|_{L^2(|\nu_1|)}^2\right)^{1/2}\\
&\lesssim\left(\sum_Q\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}^2\right)^{1/2}
\|\chi_{2S}\Delta_{2,\widehat{R}}g_w\|_{L^2(|\nu_2|)},
\end{aligned} Q ∑ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′ ⟩ ∗ = Q ∑ ⟨ Δ 1 , Q f w , Δ 1 , Q ∗ ( χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ) ⟩ ≤ Q ∑ ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 Q ∑ ∥ Δ 1 , Q ∗ ( χ 2 S S ∣ ν 2 ∣ , ε ∗ g ′ ) ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 ≲ Q ∑ ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) 2 1/2 ∥ χ 2 S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) , where the sums run over the cubes Q Q Q ( R , S ) (R,S) ( R , S ) Q Q Q ( R , S ) (R,S) ( R , S ) W ( R ) \mathcal{W}(R) W ( R ) (261) and the inclusion 2 S ⊂ R 2S\subset R 2 S ⊂ R R R R
∑ R , S ∥ χ 2 S Δ 2 , R ^ g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≲ ∑ R ∥ χ R Δ 2 , R ^ g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≤ ∑ P ∈ D 2 t r ∥ Δ 2 , P g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 . \sum_{R,S}\|\chi_{2S}\Delta_{2,\widehat{R}}g_w\|_{L^2(|\nu_2|)}^2
\lesssim\sum_R\|\chi_R\Delta_{2,\widehat{R}}g_w\|_{L^2(|\nu_2|)}^2
\le\sum_{P\in\mathcal{D}^{\mathrm{tr}}_2}\|\Delta_{2,P}g_w\|_{L^2(|\nu_2|)}^2. R , S ∑ ∥ χ 2 S Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≲ R ∑ ∥ χ R Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 ≤ P ∈ D 2 tr ∑ ∥ Δ 2 , P g w ∥ L 2 ( ∣ ν 2 ∣ ) 2 . Finally, summing the previous bound over ( R , S ) (R,S) ( R , S ) (178) give
∣ S ^ 1 t e r m ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . |\widehat{S}_1^{\mathrm{term}}|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. ∣ S 1 term ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . To bound S ^ 2 t e r m \widehat{S}_2^{\mathrm{term}} S 2 term ( R , S , Q ) (R,S,Q) ( R , S , Q ) g ′ ′ : = χ R ^ ∖ 2 S Δ 2 , R ^ g w g'' := \chi_{\widehat{R}\setminus2S}\Delta_{2,\widehat{R}}g_w g ′′ := χ R ∖ 2 S Δ 2 , R g w z Q ∈ S z_Q\in S z Q ∈ S (262) gives
dist ( Q , supp g ′ ′ ) ≥ ℓ ( S ) − diam ( Q ) ≥ 3 4 ℓ ( S ) ≳ ℓ ( Q ) α ℓ ( R ) 1 − α . \operatorname{dist}(Q,\operatorname{supp}g'')\ge\ell(S)-\operatorname{diam}(Q)\ge\frac{3}{4}\ell(S)\gtrsim\ell(Q)^\alpha\ell(R)^{1-\alpha}. dist ( Q , supp g ′′ ) ≥ ℓ ( S ) − diam ( Q ) ≥ 4 3 ℓ ( S ) ≳ ℓ ( Q ) α ℓ ( R ) 1 − α . In particular, diam ( Q ) ≤ dist ( Q , supp g ′ ′ ) / 3 \operatorname{diam}(Q)\le\operatorname{dist}(Q,\operatorname{supp}g'')/3 diam ( Q ) ≤ dist ( Q , supp g ′′ ) /3 (158) and the identity ∫ Δ 1 , Q f w d ∣ ν 1 ∣ = 0 \int\Delta_{1,Q}f_w\,d|\nu_1|=0 ∫ Δ 1 , Q f w d ∣ ν 1 ∣ = 0
∣ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′ ′ ⟩ ∗ ∣ ≲ ℓ ( Q ) γ dist ( Q , supp g ′ ′ ) n + γ ∥ Δ 1 , Q f w ∥ L 1 ( ∣ ν 1 ∣ ) ∥ g ′ ′ ∥ L 1 ( ∣ ν 2 ∣ ) . \left|\left\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_w),g''\right\rangle_*\right|
\lesssim\frac{\ell(Q)^\gamma}{\operatorname{dist}(Q,\operatorname{supp}g'')^{n+\gamma}}\|\Delta_{1,Q}f_w\|_{L^1(|\nu_1|)}\|g''\|_{L^1(|\nu_2|)}. ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′′ ⟩ ∗ ≲ dist ( Q , supp g ′′ ) n + γ ℓ ( Q ) γ ∥ Δ 1 , Q f w ∥ L 1 ( ∣ ν 1 ∣ ) ∥ g ′′ ∥ L 1 ( ∣ ν 2 ∣ ) . Since α ( n + γ ) = γ / 2 \alpha(n+\gamma)=\gamma/2 α ( n + γ ) = γ /2 D ( Q , R ^ ) ≈ ℓ ( R ^ ) ≈ ℓ ( R ) D(Q,\widehat{R})\approx\ell(\widehat{R})\approx\ell(R) D ( Q , R ) ≈ ℓ ( R ) ≈ ℓ ( R )
ℓ ( Q ) γ dist ( Q , supp g ′ ′ ) n + γ ≲ ℓ ( Q ) γ / 2 ℓ ( R ) n + γ / 2 ≈ ℓ ( Q ) γ / 2 ℓ ( R ^ ) γ / 2 D ( Q , R ^ ) n + γ , \frac{\ell(Q)^\gamma}{\operatorname{dist}(Q,\operatorname{supp}g'')^{n+\gamma}}
\lesssim\frac{\ell(Q)^{\gamma/2}}{\ell(R)^{n+\gamma/2}}
\approx\frac{\ell(Q)^{\gamma/2}\ell(\widehat{R})^{\gamma/2}}{D(Q,\widehat{R})^{n+\gamma}}, dist ( Q , supp g ′′ ) n + γ ℓ ( Q ) γ ≲ ℓ ( R ) n + γ /2 ℓ ( Q ) γ /2 ≈ D ( Q , R ) n + γ ℓ ( Q ) γ /2 ℓ ( R ) γ /2 , so the Cauchy-Schwarz inequality yields
∣ ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′ ′ ⟩ ∗ ∣ ≲ ℓ ( Q ) γ / 2 ℓ ( R ^ ) γ / 2 D ( Q , R ^ ) n + γ ∣ ν 1 ∣ ( Q ) 1 / 2 ∣ ν 2 ∣ ( R ^ ) 1 / 2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ Δ 2 , R ^ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (264) \left|\left\langle S_{|\nu_1|,\varepsilon}(\Delta_{1,Q}f_w),g''\right\rangle_{*}\right|
\lesssim\frac{\ell(Q)^{\gamma/2}\ell(\widehat{R})^{\gamma/2}}{D(Q,\widehat{R})^{n+\gamma}}|\nu_1|(Q)^{1/2}|\nu_2|(\widehat{R})^{1/2}\|\Delta_{1,Q}f_w\|_{L^2(|\nu_1|)}\|\Delta_{2,\widehat{R}}g_w\|_{L^2(|\nu_2|)}. \tag*{(264)} ⟨ S ∣ ν 1 ∣ , ε ( Δ 1 , Q f w ) , g ′′ ⟩ ∗ ≲ D ( Q , R ) n + γ ℓ ( Q ) γ /2 ℓ ( R ) γ /2 ∣ ν 1 ∣ ( Q ) 1/2 ∣ ν 2 ∣ ( R ) 1/2 ∥ Δ 1 , Q f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ Δ 2 , R g w ∥ L 2 ( ∣ ν 2 ∣ ) . (264) Each pair ( Q , R ^ ) (Q,\widehat{R}) ( Q , R ) S 2 t e r m S_2^{\mathrm{term}} S 2 term ℓ ( Q ) ≤ ℓ ( R ^ ) \ell(Q)\leq\ell(\widehat{R}) ℓ ( Q ) ≤ ℓ ( R ) (264) , (201) combined with Schur’s test, and (178) give ∣ S 2 t e r m ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) |S_2^{\mathrm{term}}|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)} ∣ S 2 term ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ )
As a consequence of Lemma 6.28, we obtain
∣ S 4 , t e r m ′ ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) (265) |S'_{4,\mathrm{term}}|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)} \tag*{(265)} ∣ S 4 , term ′ ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) (265) and applying the same estimates to S ∣ ν 2 ∣ , ε ∗ S^*_{|\nu_2|,\varepsilon} S ∣ ν 2 ∣ , ε ∗ (203) , we obtain
∣ S 4 ′ ′ ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (266) |S''_4|\lesssim\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}. \tag*{(266)} ∣ S 4 ′′ ∣ ≲ ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) . (266) Then, we gather (260) , (265) , and (266) , and obtain a constant c S 4 > 0 c_{S_4}>0 c S 4 > 0
∣ S 4 ( w ) ∣ = ∣ S 4 ′ + S 4 ′ ′ ∣ ≤ ∣ S 4 , t r ′ ∣ + ∣ S 4 , t e r m ′ ∣ + ∣ S 4 ′ ′ ∣ ≤ c S 4 ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , (267) |S_4(w)|=|S'_4+S''_4|\leq|S'_{4,\mathrm{tr}}|+|S'_{4,\mathrm{term}}|+|S''_4|\leq c_{S_4}\|f_w\|_{L^2(|\nu_1|)}\|g_w\|_{L^2(|\nu_2|)}, \tag*{(267)} ∣ S 4 ( w ) ∣ = ∣ S 4 ′ + S 4 ′′ ∣ ≤ ∣ S 4 , tr ′ ∣ + ∣ S 4 , term ′ ∣ + ∣ S 4 ′′ ∣ ≤ c S 4 ∥ f w ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g w ∥ L 2 ( ∣ ν 2 ∣ ) , (267) where c S 4 c_{S_4} c S 4 m m m w w w λ \lambda λ υ \upsilon υ a a a ε \varepsilon ε
Finally, (204) , Proposition 6.23, and (267) give
E ~ w ∣ I 4 ( w ) ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) ) 1 / 2 + ρ m ( λ , υ ) ] ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . \widetilde{\mathbb E}_w|I_4(w)|\leq\left[C_{m,\lambda,\upsilon}(1+a^{-n})+(C_{m,\upsilon}(M_\Lambda a))^{1/2}+\rho_m(\lambda,\upsilon)\right](1+N_\varepsilon)\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}. E w ∣ I 4 ( w ) ∣ ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) ) 1/2 + ρ m ( λ , υ ) ] ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . By (185) , (181) , (190) , (192) , (194) , and (195) , we obtain
E ~ w ∣ ⟨ T Ξ w , ∣ ν ^ 1 ∣ , ε f w , g w ⟩ ∣ ν ^ 2 ∣ ∣ ≤ E ~ w ∣ ⟨ S ∣ ν 1 ∣ , ε f w , g w ⟩ ∗ ∣ + C ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ E ~ w ∣ I 4 ( w ) ∣ + C ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) ) 1 / 2 + ρ m ( λ , υ ) ] ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . \begin{aligned}
\widetilde{\mathbb E}_w\left|\left\langle T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}f_w,g_w\right\rangle_{|\widehat{\nu}_2|}\right|
\leq\widetilde{\mathbb E}_w\left|\left\langle S_{|\nu_1|,\varepsilon}f_w,g_w\right\rangle_*\right|+C\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)} \\
&\leq\widetilde{\mathbb E}_w|I_4(w)|+C\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)} \\
&\leq\left[C_{m,\lambda,\upsilon}(1+a^{-n})+(C_{m,\upsilon}(M_\Lambda a))^{1/2}+\rho_m(\lambda,\upsilon)\right](1+N_\varepsilon)\|f\|_{L^2(|\nu_1|)}\|g\|_{L^2(|\nu_2|)}.
\end{aligned} E w ⟨ T Ξ w , ∣ ν 1 ∣ , ε f w , g w ⟩ ∣ ν 2 ∣ ≤ E w ⟨ S ∣ ν 1 ∣ , ε f w , g w ⟩ ∗ + C ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ E w ∣ I 4 ( w ) ∣ + C ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) ≤ [ C m , λ , υ ( 1 + a − n ) + ( C m , υ ( M Λ a ) ) 1/2 + ρ m ( λ , υ ) ] ( 1 + N ε ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) . Letting K 1 , K 2 → ∞ K_1,K_2\to\infty K 1 , K 2 → ∞ (183) , since the zero extensions of f f f g g g ∣ ν i ∣ |\nu_i| ∣ ν i ∣ ∣ ν ^ i ∣ |\widehat{\nu}_i| ∣ ν i ∣
For fixed m m m ρ m ( λ , υ ) → 0 \rho_m(\lambda,\upsilon)\to0 ρ m ( λ , υ ) → 0 λ , υ → 0 \lambda,\upsilon\to0 λ , υ → 0 λ , υ \lambda,\upsilon λ , υ M Λ M_\Lambda M Λ C m , υ C_{m,\upsilon} C m , υ C m , υ ( M Λ a ) 1 / 2 → 0 C_{m,\upsilon}(M_\Lambda a)^{1/2}\to0 C m , υ ( M Λ a ) 1/2 → 0 a → 0 a\to0 a → 0 C m , λ , υ ( 1 + a − n ) C_{m,\lambda,\upsilon}(1+a^{-n}) C m , λ , υ ( 1 + a − n ) N ε N_\varepsilon N ε
Proof of the main theorem
Averaged suppressed operators and the probabilistic argument
Under the normalized measure-testing assumptions of Proposition 3.1 and within the framework of the mixed corona decomposition of § 4 , let R i ∈ Top i ′ ′ R_i\in\operatorname{Top}''_i R i ∈ Top i ′′ i = 1 , 2 i=1,2 i = 1 , 2 Tree ~ ( R 1 , R 2 ) ≠ ∅ \widetilde{\operatorname{Tree}}(R_1,R_2)\ne\varnothing Tree ( R 1 , R 2 ) = ∅ R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 R 2 ⊂ R 1 R_2\subset R_1 R 2 ⊂ R 1 R 1 ⊂ R 2 R_1\subset R_2 R 1 ⊂ R 2 (137) and (138) , we defined the suppression
Φ ~ R 1 , R 2 ( x ) : = max ( Φ 1 , R 1 ( x ) , Φ 2 , R 2 ( x ) ) , \widetilde{\Phi}_{R_1,R_2}(x):=\max\left(\Phi_{1,R_1}(x),\Phi_{2,R_2}(x)\right), Φ R 1 , R 2 ( x ) := max ( Φ 1 , R 1 ( x ) , Φ 2 , R 2 ( x ) ) , and, for R ∈ Top R \in\operatorname{Top} R ∈ Top Φ ~ R ( x ) : = Φ ~ R 1 , R 2 ( x ) \widetilde{\Phi}_{R}(x) := \widetilde{\Phi}_{R_{1},R_{2}}(x) Φ R ( x ) := Φ R 1 , R 2 ( x ) Φ w i \Phi_{w_i} Φ w i Φ i \Phi_i Φ i Φ \Phi Φ Φ i , R i , w i \Phi_{i,R_i,w_i} Φ i , R i , w i Φ i , R i \Phi_{i,R_i} Φ i , R i Φ ~ R \widetilde{\Phi}_{R} Φ R i = 1 , 2 i=1,2 i = 1 , 2 (148) and (149) we introduced
Ξ i : = Ξ i , R i , w i : = max { Φ i , R i , Φ ~ i , R i , w i } and Ξ w : = max { Ξ 1 , R 1 , w 1 , Ξ 2 , R 2 , w 2 } . \Xi_i := \Xi_{i,R_i,w_i} := \max\left\{\Phi_{i,R_i},\widetilde{\Phi}_{i,R_i,w_i}\right\}
\qquad\text{and}\qquad
\Xi_w := \max\left\{\Xi_{1,R_1,w_1},\Xi_{2,R_2,w_2}\right\}. Ξ i := Ξ i , R i , w i := max { Φ i , R i , Φ i , R i , w i } and Ξ w := max { Ξ 1 , R 1 , w 1 , Ξ 2 , R 2 , w 2 } . Now we define the averaged suppressed SIO . Namely, given ε > 0 \varepsilon>0 ε > 0 m \mathfrak m m X X X
T ~ m , ε f ( x ) : = ∫ T Ξ w , m , ε f ( x ) d P G Ω 2 ( w ) , x ∈ X . (268) \widetilde{T}_{\mathfrak m,\varepsilon}f(x):=\int T_{\Xi_w,\mathfrak m,\varepsilon}f(x)\,d\mathbb{P}_{\mathcal G}^{\Omega^2}(w),\qquad x\in X. \tag*{(268)} T m , ε f ( x ) := ∫ T Ξ w , m , ε f ( x ) d P G Ω 2 ( w ) , x ∈ X . (268) Initially, we take f f f ∣ ν ^ i ∣ |\widehat{\nu}_i| ∣ ν i ∣ f ∈ L 2 ( ∣ ν ^ i ∣ ) f\in L^2(|\widehat{\nu}_i|) f ∈ L 2 ( ∣ ν i ∣ ) T ~ \widetilde{T} T
K ~ ( x , y ) : = ∫ K Ξ w ( x , y ) d P G Ω 2 ( w ) . (269) \widetilde{K}(x,y):=\int K_{\Xi_w}(x,y)\,d\mathbb{P}_{\mathcal G}^{\Omega^2}(w). \tag*{(269)} K ( x , y ) := ∫ K Ξ w ( x , y ) d P G Ω 2 ( w ) . (269) Given a non-negative 1-Lipschitz function Ψ : X → R \Psi:X\to\mathbb{R} Ψ : X → R
T Ψ , m , ε f ( x ) : = ∫ d ( x , y ) ≥ max ( Ψ ( x ) , ε ) K ( x , y ) f ( y ) d m ( y ) = ∫ K Ψ , ε ( x , y ) f ( y ) d m ( y ) , T_{\Psi,\mathfrak m,\varepsilon}f(x):=\int_{d(x,y)\geq\max(\Psi(x),\varepsilon)}K(x,y)f(y)\,d\mathfrak m(y)=\int K_{\Psi,\varepsilon}(x,y)f(y)\,d\mathfrak m(y), T Ψ , m , ε f ( x ) := ∫ d ( x , y ) ≥ m a x ( Ψ ( x ) , ε ) K ( x , y ) f ( y ) d m ( y ) = ∫ K Ψ , ε ( x , y ) f ( y ) d m ( y ) , where
K Ψ , ε ( x , y ) : = K ( x , y ) χ { d ( x , y ) ≥ max ( Ψ ( x ) , ε ) } . K_{\Psi,\varepsilon}(x,y):=K(x,y)\chi_{\{d(x,y)\geq\max(\Psi(x),\varepsilon)\}}. K Ψ , ε ( x , y ) := K ( x , y ) χ { d ( x , y ) ≥ m a x ( Ψ ( x ) , ε )} . Then, in the case Ψ = Ξ w \Psi=\Xi_w Ψ = Ξ w
T ~ m , ε f ( x ) : = ∫ T Ξ w , m , ε f ( x ) d P G Ω 2 ( w ) , x ∈ X . \widetilde{T}_{\mathfrak m,\varepsilon}f(x):=\int T_{\Xi_w,\mathfrak m,\varepsilon}f(x)\,d\mathbb{P}_{\mathcal G}^{\Omega^2}(w),\qquad x\in X. T m , ε f ( x ) := ∫ T Ξ w , m , ε f ( x ) d P G Ω 2 ( w ) , x ∈ X . We also set
T ~ m , ♯ , ε f ( x ) : = sup ρ ≥ ε ∣ T ~ m , ρ f ( x ) ∣ and T ~ m , ♯ , ε f ( x ) : = sup ρ ≥ ε ∣ T ~ m , ρ f ( x ) ∣ . \widetilde{T}_{\mathfrak m,\sharp,\varepsilon}f(x):=\sup_{\rho\geq\varepsilon}\left|\widetilde{T}_{\mathfrak m,\rho}f(x)\right|
\qquad\text{and}\qquad
\widetilde{T}_{\mathfrak m,\sharp,\varepsilon}f(x):=\sup_{\rho\geq\varepsilon}\left|\widetilde{T}_{\mathfrak m,\rho}f(x)\right|. T m , ♯ , ε f ( x ) := ρ ≥ ε sup T m , ρ f ( x ) and T m , ♯ , ε f ( x ) := ρ ≥ ε sup T m , ρ f ( x ) . We define
ν x , ε ( t ) : = P G Ω 2 ( { w ∈ Ω 2 : max ( Ξ w ( x ) , ε ) ≤ t } ) , (270) \nu_{x,\varepsilon}(t):=\mathbb{P}_{\mathcal G}^{\Omega^2}\left(\left\{w\in\Omega^2:\max(\Xi_w(x),\varepsilon)\leq t\right\}\right), \tag*{(270)} ν x , ε ( t ) := P G Ω 2 ( { w ∈ Ω 2 : max ( Ξ w ( x ) , ε ) ≤ t } ) , (270) for t ∈ R t\in\mathbb{R} t ∈ R max ( Ξ w ( x ) , ε ) \max(\Xi_w(x),\varepsilon) max ( Ξ w ( x ) , ε ) w 1 w_1 w 1 w 2 w_2 w 2 G \mathcal G G
A i ( t ) : = { w i ∈ Ω : t ≥ max ( Φ w i ( x ) , Φ i ( x ) , ε ) } , for i = 1 , 2 , t ∈ R , A_i(t):=\left\{w_i\in\Omega:t\geq\max(\Phi_{w_i}(x),\Phi_i(x),\varepsilon)\right\},\qquad\text{for }i=1,2,\ t\in\mathbb{R}, A i ( t ) := { w i ∈ Ω : t ≥ max ( Φ w i ( x ) , Φ i ( x ) , ε ) } , for i = 1 , 2 , t ∈ R , so (270) yields
ν x , ε ( t ) = P G Ω 2 ( A 1 ( t ) × A 2 ( t ) ) . \nu_{x,\varepsilon}(t)=\mathbb{P}_{\mathcal G}^{\Omega^2}\left(A_1(t)\times A_2(t)\right). ν x , ε ( t ) = P G Ω 2 ( A 1 ( t ) × A 2 ( t ) ) . By Fubini’s theorem and (270) , T ~ \widetilde{T} T
K ~ ( x , y ) = ∫ K ( x , y ) χ { d ( x , y ) ≥ max ( Ξ w ( x ) , ε ) } d P G Ω 2 ( w ) = K ( x , y ) ν x , ε ( d ( x , y ) ) . (271) \widetilde{K}(x,y)=\int K(x,y)\chi_{\{d(x,y)\geq\max(\Xi_w(x),\varepsilon)\}}\,d\mathbb{P}_{\mathcal G}^{\Omega^2}(w)=K(x,y)\nu_{x,\varepsilon}(d(x,y)). \tag*{(271)} K ( x , y ) = ∫ K ( x , y ) χ { d ( x , y ) ≥ m a x ( Ξ w ( x ) , ε )} d P G Ω 2 ( w ) = K ( x , y ) ν x , ε ( d ( x , y )) . (271) By (269) and the kernel estimates for K Ξ w K_{\Xi_w} K Ξ w K ~ \widetilde{K} K Lemma 3.13 , which depend only on n n n γ \gamma γ K K K K ~ ( x , y ) \widetilde{K}(x,y) K ( x , y ) (271) .
We recall that δ p r \delta_{\mathrm{pr}} δ pr Lemma 4.2 . Then, we choose the error κ \kappa κ (172) so that 0 < κ < δ p r 2 / 2 0 < \kappa< \delta_{\mathrm{pr}}^{2}/2 0 < κ < δ pr 2 /2
c G : = δ p r 2 − P Ω 2 ( Ω 2 ∖ G ) P Ω 2 ( G ) ≥ δ p r 2 − κ P Ω 2 ( G ) ≥ δ p r 2 2 . c_{\mathcal{G}} := \frac{\delta_{\mathrm{pr}}^{2}-\mathbb{P}^{\Omega^{2}}(\Omega^{2}\setminus\mathcal{G})}{\mathbb{P}^{\Omega^{2}}(\mathcal{G})} \ge\frac{\delta_{\mathrm{pr}}^{2}-\kappa}{\mathbb{P}^{\Omega^{2}}(\mathcal{G})} \ge\frac{\delta_{\mathrm{pr}}^{2}}{2}. c G := P Ω 2 ( G ) δ pr 2 − P Ω 2 ( Ω 2 ∖ G ) ≥ P Ω 2 ( G ) δ pr 2 − κ ≥ 2 δ pr 2 . Here we used that P Ω 2 ( G ) ≤ 1 \mathbb{P}^{\Omega^{2}}(\mathcal{G}) \le1 P Ω 2 ( G ) ≤ 1 P Ω 2 ( Ω 2 ∖ G ) < κ \mathbb{P}^{\Omega^{2}}(\Omega^{2}\setminus\mathcal{G}) < \kappa P Ω 2 ( Ω 2 ∖ G ) < κ (172) for every goodness parameter m ≥ m 0 m \ge m_{0} m ≥ m 0 m 0 = m 0 ( κ ) m_{0}=m_{0}(\kappa) m 0 = m 0 ( κ ) Lemma A.4 . In the rest of this section, we assume that m ≥ m 0 m \ge m_{0} m ≥ m 0
Lemma 7.1. For ε > 0 \varepsilon>0 ε > 0
∥ T Φ , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ≤ 3 c G − 1 ∥ T ~ ∣ ν ^ 1 ∣ , ♯ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) (272) \left\|T_{\Phi,|\widehat{\nu}_{1}|,\varepsilon}\right\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}
\le3c_{\mathcal{G}}^{-1}
\left\|\widetilde{T}_{|\widehat{\nu}_{1}|,\sharp,\varepsilon}\right\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)} \tag*{(272)} T Φ , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≤ 3 c G − 1 T ∣ ν 1 ∣ , ♯ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) (272) and, for every w ∈ Ω 2 w\in\Omega^{2} w ∈ Ω 2
∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ≤ 3 c G − 1 ∥ T ~ ∣ ν ^ 1 ∣ , ♯ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) . (273) \left\|T_{\Xi_{w},|\widehat{\nu}_{1}|,\varepsilon}\right\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}
\le3c_{\mathcal{G}}^{-1}
\left\|\widetilde{T}_{|\widehat{\nu}_{1}|,\sharp,\varepsilon}\right\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}. \tag*{(273)} T Ξ w , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≤ 3 c G − 1 T ∣ ν 1 ∣ , ♯ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) . (273) Proof. The function v x , ε v_{x,\varepsilon} v x , ε (270) satisfies the following properties:
If 0 ≤ t < max ( Φ ( x ) , ε ) 0\le t<\max(\Phi(x),\varepsilon) 0 ≤ t < max ( Φ ( x ) , ε ) v x , ε ( t ) = 0 v_{x,\varepsilon}(t)=0 v x , ε ( t ) = 0
If t ≥ max ( Φ ( x ) , ε ) t\ge\max(\Phi(x),\varepsilon) t ≥ max ( Φ ( x ) , ε ) v x , ε ( t ) ≥ c G v_{x,\varepsilon}(t)\ge c_{\mathcal{G}} v x , ε ( t ) ≥ c G
The function v x , ε ( ⋅ ) v_{x,\varepsilon}(\cdot) v x , ε ( ⋅ )
Properties (1) and (3) follow from the definition of v x , ε v_{x,\varepsilon} v x , ε Ξ w ≥ Φ \Xi_{w}\ge\Phi Ξ w ≥ Φ t ≥ max ( Φ ( x ) , ε ) t\ge\max(\Phi(x),\varepsilon) t ≥ max ( Φ ( x ) , ε ) Φ i ( x ) \Phi_{i}(x) Φ i ( x ) (99) , for every integer j ≥ 1 j\ge1 j ≥ 1 Γ i , j ⊂ Ω \Gamma_{i,j}\subset\Omega Γ i , j ⊂ Ω i = 1 , 2 i=1,2 i = 1 , 2
P Ω ( Γ i , j ) > δ p r , sup w i ∈ Γ i , j Φ w i ( x ) < t + 1 / j . \mathbb{P}^{\Omega}(\Gamma_{i,j})>\delta_{\mathrm{pr}}, \qquad\sup_{w_{i}\in\Gamma_{i,j}}\Phi_{w_{i}}(x)<t+1/j. P Ω ( Γ i , j ) > δ pr , w i ∈ Γ i , j sup Φ w i ( x ) < t + 1/ j . Thus Γ i , j ⊂ A i ( t + 1 / j ) \Gamma_{i,j}\subset A_{i}(t+1/j) Γ i , j ⊂ A i ( t + 1/ j ) A i ( t + 1 / j ) ↓ A i ( t ) A_{i}(t+1/j)\downarrow A_{i}(t) A i ( t + 1/ j ) ↓ A i ( t ) P Ω ( A i ( t ) ) ≥ δ p r \mathbb{P}^{\Omega}(A_{i}(t))\ge\delta_{\mathrm{pr}} P Ω ( A i ( t )) ≥ δ pr
v x , ε ( t ) = P Ω 2 ( ( A 1 ( t ) × A 2 ( t ) ) ∩ G ) P Ω 2 ( G ) ≥ P Ω ( A 1 ( t ) ) P Ω ( A 2 ( t ) ) − P Ω 2 ( Ω 2 ∖ G ) P Ω 2 ( G ) ≥ c G . \begin{aligned}
v_{x,\varepsilon}(t)
=\frac{\mathbb{P}^{\Omega^{2}}\left((A_{1}(t)\times A_{2}(t))\cap\mathcal{G}\right)}
{\mathbb{P}^{\Omega^{2}}(\mathcal{G})} \\
&\ge\frac{\mathbb{P}^{\Omega}(A_{1}(t))\mathbb{P}^{\Omega}(A_{2}(t))-\mathbb{P}^{\Omega^{2}}(\Omega^{2}\setminus\mathcal{G})}
{\mathbb{P}^{\Omega^{2}}(\mathcal{G})}
\ge c_{\mathcal{G}}.
\end{aligned} v x , ε ( t ) = P Ω 2 ( G ) P Ω 2 ( ( A 1 ( t ) × A 2 ( t )) ∩ G ) ≥ P Ω 2 ( G ) P Ω ( A 1 ( t )) P Ω ( A 2 ( t )) − P Ω 2 ( Ω 2 ∖ G ) ≥ c G . Let us turn to the proof of (272) . We denote Φ ε ( x ) : = max ( Φ ( x ) , ε ) \Phi^{\varepsilon}(x):=\max(\Phi(x),\varepsilon) Φ ε ( x ) := max ( Φ ( x ) , ε ) 1 1 1 v x , ε ( t ) v_{x,\varepsilon}(t) v x , ε ( t ) t < Φ ε ( x ) t<\Phi^{\varepsilon}(x) t < Φ ε ( x )
K ~ ( x , y ) = K ( x , y ) v x , ε ( d ( x , y ) ) χ { d ( x , y ) ≥ Φ ε ( x ) } = v x , ε ( d ( x , y ) ) K Φ , ε ( x , y ) . \widetilde{K}(x,y)=K(x,y)v_{x,\varepsilon}(d(x,y))\chi_{\{d(x,y)\ge\Phi^{\varepsilon}(x)\}}
=v_{x,\varepsilon}(d(x,y))K_{\Phi,\varepsilon}(x,y). K ( x , y ) = K ( x , y ) v x , ε ( d ( x , y )) χ { d ( x , y ) ≥ Φ ε ( x )} = v x , ε ( d ( x , y )) K Φ , ε ( x , y ) . Thus,
K Φ , ε ( x , y ) = 1 v x , ε ( d ( x , y ) ) K ~ ( x , y ) χ { d ( x , y ) ≥ Φ ε ( x ) } , (274) K_{\Phi,\varepsilon}(x,y)
=\frac{1}{v_{x,\varepsilon}(d(x,y))}\widetilde{K}(x,y)\chi_{\{d(x,y)\ge\Phi^{\varepsilon}(x)\}}, \tag*{(274)} K Φ , ε ( x , y ) = v x , ε ( d ( x , y )) 1 K ( x , y ) χ { d ( x , y ) ≥ Φ ε ( x )} , (274) where the reciprocal is only used on { d ( x , y ) ≥ Φ ε ( x ) } \{d(x,y)\ge\Phi^{\varepsilon}(x)\} { d ( x , y ) ≥ Φ ε ( x )} 1 / v x , ε 1/v_{x,\varepsilon} 1/ v x , ε [ Φ ε ( x ) , ∞ ) [\Phi^{\varepsilon}(x),\infty) [ Φ ε ( x ) , ∞ ) σ \sigma σ R \mathbb{R} R
σ ( ( a , b ] ) = 1 v x , ε ( a ) − 1 v x , ε ( b ) , Φ ε ( x ) ≤ a < b , σ ( ( − ∞ , Φ ε ( x ) ] ) = 0. \sigma((a,b])=\frac{1}{v_{x,\varepsilon}(a)}-\frac{1}{v_{x,\varepsilon}(b)}, \qquad\Phi^{\varepsilon}(x)\le a<b, \qquad\sigma((-\infty,\Phi^{\varepsilon}(x)])=0. σ (( a , b ]) = v x , ε ( a ) 1 − v x , ε ( b ) 1 , Φ ε ( x ) ≤ a < b , σ (( − ∞ , Φ ε ( x )]) = 0. Observe that σ \sigma σ v x , ε ( t ) → 1 v_{x,\varepsilon}(t) \to1 v x , ε ( t ) → 1 t → ∞ t \to\infty t → ∞
σ ( R ) = 1 v x , ε ( Φ ε ( x ) ) − 1 ≤ c G − 1 . (275) \sigma(\mathbb{R})=\frac{1}{v_{x,\varepsilon}(\Phi^\varepsilon(x))}-1\le c_{\mathscr{G}}^{-1}. \tag*{(275)} σ ( R ) = v x , ε ( Φ ε ( x )) 1 − 1 ≤ c G − 1 . (275) Now, for d ( x , y ) ≥ Φ ε ( x ) d(x,y)\ge\Phi^\varepsilon(x) d ( x , y ) ≥ Φ ε ( x )
∫ t > Φ ε ( x ) K ~ ( x , y ) χ { d ( x , y ) ≥ t } d σ ( t ) = K ~ ( x , y ) σ ( ( Φ ε ( x ) , d ( x , y ) ] ) = K ~ ( x , y ) ( 1 v x , ε ( Φ ε ( x ) ) − 1 v x , ε ( d ( x , y ) ) ) . \int_{t>\Phi^\varepsilon(x)}\widetilde{K}(x,y)\chi_{\{d(x,y)\ge t\}}\,d\sigma(t)
=\widetilde{K}(x,y)\sigma\bigl((\Phi^\varepsilon(x),d(x,y)]\bigr)
=\widetilde{K}(x,y)\left(\frac{1}{v_{x,\varepsilon}(\Phi^\varepsilon(x))}-\frac{1}{v_{x,\varepsilon}(d(x,y))}\right). ∫ t > Φ ε ( x ) K ( x , y ) χ { d ( x , y ) ≥ t } d σ ( t ) = K ( x , y ) σ ( ( Φ ε ( x ) , d ( x , y )] ) = K ( x , y ) ( v x , ε ( Φ ε ( x )) 1 − v x , ε ( d ( x , y )) 1 ) . Therefore, by (274) ,
K Φ , ε ( x , y ) = K ~ ( x , y ) v x , ε ( Φ ε ( x ) ) − ∫ t > Φ ε ( x ) K ~ ( x , y ) χ { d ( x , y ) ≥ t } d σ ( t ) . K_{\Phi,\varepsilon}(x,y)=\frac{\widetilde{K}(x,y)}{v_{x,\varepsilon}(\Phi^\varepsilon(x))}-\int_{t>\Phi^\varepsilon(x)}\widetilde{K}(x,y)\chi_{\{d(x,y)\ge t\}}\,d\sigma(t). K Φ , ε ( x , y ) = v x , ε ( Φ ε ( x )) K ( x , y ) − ∫ t > Φ ε ( x ) K ( x , y ) χ { d ( x , y ) ≥ t } d σ ( t ) . For t ≥ ε t\ge\varepsilon t ≥ ε v x , ε ( u ) χ { u ≥ t } = v x , t ( u ) v_{x,\varepsilon}(u)\chi_{\{u\ge t\}}=v_{x,t}(u) v x , ε ( u ) χ { u ≥ t } = v x , t ( u ) u ∈ R u\in\mathbb{R} u ∈ R K ~ ( x , y ) χ { d ( x , y ) ≥ t } \widetilde{K}(x,y)\chi_{\{d(x,y)\ge t\}} K ( x , y ) χ { d ( x , y ) ≥ t } T ~ ∣ ν ^ 1 ∣ , t \widetilde{T}_{|\widehat{\nu}_1|,t} T ∣ ν 1 ∣ , t f ( y ) f(y) f ( y ) ∣ ν ^ 1 ∣ |\widehat{\nu}_1| ∣ ν 1 ∣ y y y
T Φ , ∣ ν ^ 1 ∣ , ε f ( x ) = T ~ ∣ ν ^ 1 ∣ , ε f ( x ) v x , ε ( Φ ε ( x ) ) − ∫ t > Φ ε ( x ) T ~ ∣ ν ^ 1 ∣ , t f ( x ) d σ ( t ) . T_{\Phi,|\widehat{\nu}_1|,\varepsilon}f(x)
=\frac{\widetilde{T}_{|\widehat{\nu}_1|,\varepsilon}f(x)}{v_{x,\varepsilon}(\Phi^\varepsilon(x))}
-\int_{t>\Phi^\varepsilon(x)}\widetilde{T}_{|\widehat{\nu}_1|,t}f(x)\,d\sigma(t). T Φ , ∣ ν 1 ∣ , ε f ( x ) = v x , ε ( Φ ε ( x )) T ∣ ν 1 ∣ , ε f ( x ) − ∫ t > Φ ε ( x ) T ∣ ν 1 ∣ , t f ( x ) d σ ( t ) . From the preceding identity and (275) , we infer that
∣ T Φ , ∣ ν ^ 1 ∣ , ε f ( x ) ∣ ≤ c G − 1 ∣ T ~ ∣ ν ^ 1 ∣ , ε f ( x ) ∣ + c G − 1 sup ρ ≥ ε ∣ T ~ ∣ ν ^ 1 ∣ , ρ f ( x ) ∣ ≤ 3 c G − 1 T ~ ∣ ν ^ 1 ∣ , ♯ , ε f ( x ) . \left|T_{\Phi,|\widehat{\nu}_1|,\varepsilon}f(x)\right|
\le c_{\mathscr{G}}^{-1}\left|\widetilde{T}_{|\widehat{\nu}_1|,\varepsilon}f(x)\right|
+c_{\mathscr{G}}^{-1}\sup_{\rho\ge\varepsilon}\left|\widetilde{T}_{|\widehat{\nu}_1|,\rho}f(x)\right|
\le3c_{\mathscr{G}}^{-1}\widetilde{T}_{|\widehat{\nu}_1|,\sharp,\varepsilon}f(x). T Φ , ∣ ν 1 ∣ , ε f ( x ) ≤ c G − 1 T ∣ ν 1 ∣ , ε f ( x ) + c G − 1 ρ ≥ ε sup T ∣ ν 1 ∣ , ρ f ( x ) ≤ 3 c G − 1 T ∣ ν 1 ∣ , ♯ , ε f ( x ) . Taking the L 2 ( ∣ ν ^ 2 ∣ ) L^2(|\widehat{\nu}_2|) L 2 ( ∣ ν 2 ∣ ) (272) .
Let us now turn our attention to (273) . By the definition of K Ξ w , ε K_{\Xi_w,\varepsilon} K Ξ w , ε Ξ w ≥ Φ \Xi_w\ge\Phi Ξ w ≥ Φ
K Ξ w , ε ( x , y ) = K Φ , ε ( x , y ) χ { d ( x , y ) ≥ max ( Ξ w ( x ) , ε ) } = K ~ ( x , y ) v x , ε ( d ( x , y ) ) χ { d ( x , y ) ≥ max ( Ξ w ( x ) , ε ) } . K_{\Xi_w,\varepsilon}(x,y)=K_{\Phi,\varepsilon}(x,y)\chi_{\{d(x,y)\ge\max(\Xi_w(x),\varepsilon)\}}
=\frac{\widetilde{K}(x,y)}{v_{x,\varepsilon}(d(x,y))}\chi_{\{d(x,y)\ge\max(\Xi_w(x),\varepsilon)\}}. K Ξ w , ε ( x , y ) = K Φ , ε ( x , y ) χ { d ( x , y ) ≥ m a x ( Ξ w ( x ) , ε )} = v x , ε ( d ( x , y )) K ( x , y ) χ { d ( x , y ) ≥ m a x ( Ξ w ( x ) , ε )} . This is the same identity as (274) , with Φ \Phi Φ Ξ w \Xi_w Ξ w σ \sigma σ ( max ( Ξ w ( x ) , ε ) , ∞ ) (\max(\Xi_w(x),\varepsilon),\infty) ( max ( Ξ w ( x ) , ε ) , ∞ )
∣ T Ξ w , ∣ ν ^ 1 ∣ , ε f ( x ) ∣ ≤ 3 c G − 1 T ~ ∣ ν ^ 1 ∣ , ♯ , ε f ( x ) . \left|T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}f(x)\right|
\le3c_{\mathscr{G}}^{-1}\widetilde{T}_{|\widehat{\nu}_1|,\sharp,\varepsilon}f(x). T Ξ w , ∣ ν 1 ∣ , ε f ( x ) ≤ 3 c G − 1 T ∣ ν 1 ∣ , ♯ , ε f ( x ) . Taking the L 2 ( ∣ ν ^ 2 ∣ ) L^2(|\widehat{\nu}_2|) L 2 ( ∣ ν 2 ∣ ) (273) and completes the proof of the lemma. □
Using maximal function estimates, the bounds in Lemma 7.1 imply analogous estimates in terms of T T T T ~ \widetilde{T} T
Lemma 7.2. If ε > 0 \varepsilon>0 ε > 0
∥ T Φ , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ≲ 1 + sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) , (276) \left\|T_{\Phi,|\widehat{\nu}_1|,\varepsilon}\right\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}
\lesssim1+\sup_{\rho\ge\varepsilon}\left\|\widetilde{T}_{|\widehat{\nu}_1|,\rho}\right\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}, \tag*{(276)} T Φ , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≲ 1 + ρ ≥ ε sup T ∣ ν 1 ∣ , ρ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) , (276) and, for every w ∈ Ω 2 w\in\Omega^2 w ∈ Ω 2
∥ T Ξ w , ∣ ν ^ 1 ∣ , ε ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ≲ 1 + sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) , (277) \left\|T_{\Xi_w,|\widehat{\nu}_1|,\varepsilon}\right\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}
\lesssim1+\sup_{\rho\ge\varepsilon}\left\|\widetilde{T}_{|\widehat{\nu}_1|,\rho}\right\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)}, \tag*{(277)} T Ξ w , ∣ ν 1 ∣ , ε L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≲ 1 + ρ ≥ ε sup T ∣ ν 1 ∣ , ρ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) , (277) where the implicit constants depend on the structural parameters and c G − 1 c_{\mathscr{G}}^{-1} c G − 1 w w w ε \varepsilon ε λ \lambda λ υ \upsilon υ a a a κ \kappa κ m m m c G − 1 ≤ 2 δ p r − 2 c_{\mathscr{G}}^{-1}\le2\delta_{\mathrm{pr}}^{-2} c G − 1 ≤ 2 δ pr − 2
Proof. In this proof, we write ∥ ⋅ ∥ : = ∥ ⋅ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) \|\cdot\|:=\|\cdot\|_{L^2(|\widehat{\nu}_1|)\to L^2(|\widehat{\nu}_2|)} ∥ ⋅ ∥ := ∥ ⋅ ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) Ψ ∈ { Φ , Ξ w } \Psi\in\{\Phi,\Xi_w\} Ψ ∈ { Φ , Ξ w } Lemma 5.3 , we have max { R 1 , R 2 } ≤ Φ ≤ Ψ \max\{R_1,R_2\}\le\Phi\le\Psi max { R 1 , R 2 } ≤ Φ ≤ Ψ R i R_i R i (52) associated with ν ^ i \widehat{\nu}_i ν i Lemma 3.12 gives
∥ M Ψ , ∣ ν ^ 1 ∣ r f ∥ L 2 ( ∣ ν ^ 2 ∣ ) ≲ ∥ f ∥ L 2 ( ∣ ν ^ 1 ∣ ) . (278) \|M^r_{\Psi,|\widehat{\nu}_1|}f\|_{L^2(|\widehat{\nu}_2|)}\lesssim\|f\|_{L^2(|\widehat{\nu}_1|)}. \tag*{(278)} ∥ M Ψ , ∣ ν 1 ∣ r f ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) . (278) Let x ∈ X x\in X x ∈ X ρ ≥ ε \rho\ge\varepsilon ρ ≥ ε ρ x : = max ( Ψ ( x ) , ρ ) \rho_x:=\max(\Psi(x),\rho) ρ x := max ( Ψ ( x ) , ρ ) Lemma 3.13 , applied on { y : ρ < d ( x , y ) < ρ x } \{y:\rho<d(x,y)<\rho_x\} { y : ρ < d ( x , y ) < ρ x } { y : d ( x , y ) = ρ x } \{y:d(x,y)=\rho_x\} { y : d ( x , y ) = ρ x } { y : d ( x , y ) > ρ x } \{y:d(x,y)>\rho_x\} { y : d ( x , y ) > ρ x }
∣ T Ψ , ∣ ν ^ 1 ∣ , ρ f ( x ) − T ~ Ψ , ∣ ν ^ 1 ∣ , ρ f ( x ) ∣ ≲ M Ψ , ∣ ν ^ 1 ∣ r f ( x ) . (279) |T_{\Psi,|\widehat{\nu}_1|,\rho}f(x)-\widetilde{T}_{\Psi,|\widehat{\nu}_1|,\rho}f(x)|\lesssim M^r_{\Psi,|\widehat{\nu}_1|}f(x). \tag*{(279)} ∣ T Ψ , ∣ ν 1 ∣ , ρ f ( x ) − T Ψ , ∣ ν 1 ∣ , ρ f ( x ) ∣ ≲ M Ψ , ∣ ν 1 ∣ r f ( x ) . (279) Applying (279) with ρ = ε \rho=\varepsilon ρ = ε (278) , (272) , and (273) , we obtain
∥ T Ψ , ∣ ν ^ 1 ∣ , ε ∥ ≲ 1 + ∥ T Ψ , ∣ ν ^ 1 ∣ , ε ∥ ≲ 1 + c φ − 1 ∥ T ~ Ψ , ∣ ν ^ 1 ∣ , ♯ , ε ∥ . (280) \|T_{\Psi,|\widehat{\nu}_1|,\varepsilon}\|\lesssim1+\|T_{\Psi,|\widehat{\nu}_1|,\varepsilon}\|\lesssim1+c_{\varphi}^{-1}\|\widetilde{T}_{\Psi,|\widehat{\nu}_1|,\sharp,\varepsilon}\|. \tag*{(280)} ∥ T Ψ , ∣ ν 1 ∣ , ε ∥ ≲ 1 + ∥ T Ψ , ∣ ν 1 ∣ , ε ∥ ≲ 1 + c φ − 1 ∥ T Ψ , ∣ ν 1 ∣ , ♯ , ε ∥. (280) Moreover, since Ξ w ≥ Φ \Xi_w\ge\Phi Ξ w ≥ Φ M Ξ w , ∣ ν ^ 1 ∣ r f ≤ M Φ , ∣ ν ^ 1 ∣ r f M^r_{\Xi_w,|\widehat{\nu}_1|}f\le M^r_{\Phi,|\widehat{\nu}_1|}f M Ξ w , ∣ ν 1 ∣ r f ≤ M Φ , ∣ ν 1 ∣ r f (279) with Ψ = Ξ w \Psi=\Xi_w Ψ = Ξ w w ∈ G w\in\mathcal{G} w ∈ G
∣ T ~ ∣ ν ^ 1 ∣ , ρ f − T ~ ∣ ν ^ 1 ∣ , ρ f ∣ ≲ M Φ , ∣ ν ^ 1 ∣ r f , for ρ ≥ ε , |\widetilde{T}_{|\widehat{\nu}_1|,\rho}f-\widetilde{T}_{|\widehat{\nu}_1|,\rho}f|\lesssim M^r_{\Phi,|\widehat{\nu}_1|}f,\qquad\text{for }\rho\ge\varepsilon, ∣ T ∣ ν 1 ∣ , ρ f − T ∣ ν 1 ∣ , ρ f ∣ ≲ M Φ , ∣ ν 1 ∣ r f , for ρ ≥ ε , and (278) yields ∥ T ~ ∣ ν ^ 1 ∣ , ♯ , ε ∥ ≲ 1 + ∥ T ~ ∣ ν ^ 1 ∣ , ♯ , ε ∥ \|\widetilde{T}_{|\widehat{\nu}_1|,\sharp,\varepsilon}\|\lesssim1+\|\widetilde{T}_{|\widehat{\nu}_1|,\sharp,\varepsilon}\| ∥ T ∣ ν 1 ∣ , ♯ , ε ∥ ≲ 1 + ∥ T ∣ ν 1 ∣ , ♯ , ε ∥ (280) , the lemma follows from
∥ T ~ ∣ ν ^ 1 ∣ , ♯ , ε ∥ ≲ 1 + sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ . (281) \|\widetilde{T}_{|\widehat{\nu}_1|,\sharp,\varepsilon}\|\lesssim1+\sup_{\rho\ge\varepsilon}\|\widetilde{T}_{|\widehat{\nu}_1|,\rho}\|. \tag*{(281)} ∥ T ∣ ν 1 ∣ , ♯ , ε ∥ ≲ 1 + ρ ≥ ε sup ∥ T ∣ ν 1 ∣ , ρ ∥. (281) To prove (281) , we argue as in § 5 , with Φ \Phi Φ Ξ w \Xi_w Ξ w K ~ \widetilde{K} K K Ξ w K_{\Xi_w} K Ξ w Lemma 5.3 , the function Φ \Phi Φ Ξ w \Xi_w Ξ w (154) , (155) , and (156) hold with Φ \Phi Φ Ξ w \Xi_w Ξ w Ξ w ≥ Φ \Xi_w\ge\Phi Ξ w ≥ Φ (269) and Lemma 3.13 imply that K ~ \widetilde{K} K
∣ K ~ ( x , y ) ∣ ≲ min { d ( x , y ) − n , Φ ( x ) − n , Φ ( y ) − n } . |\widetilde{K}(x,y)|\lesssim\min\{d(x,y)^{-n},\Phi(x)^{-n},\Phi(y)^{-n}\}. ∣ K ( x , y ) ∣ ≲ min { d ( x , y ) − n , Φ ( x ) − n , Φ ( y ) − n } . With the measures ∣ ν i ∣ |\nu_i| ∣ ν i ∣ r 0 r_0 r 0 χ i \chi_i χ i ψ \psi ψ φ \varphi φ § 5 , we define
L ~ ε ( x , y ) : = χ 2 ( x ) χ 1 ( y ) ψ ( d ( x , y ) r 0 ) φ ( d ( x , y ) ε ) K ~ ( x , y ) , \widetilde{L}_{\varepsilon}(x,y):=\chi_2(x)\chi_1(y)\psi\left(\frac{d(x,y)}{r_0}\right)\varphi\left(\frac{d(x,y)}{\varepsilon}\right)\widetilde{K}(x,y), L ε ( x , y ) := χ 2 ( x ) χ 1 ( y ) ψ ( r 0 d ( x , y ) ) φ ( ε d ( x , y ) ) K ( x , y ) , and we denote by S ~ ε \widetilde{S}_{\varepsilon} S ε S ~ ε , ρ \widetilde{S}_{\varepsilon,\rho} S ε , ρ ∣ ν 1 ∣ |\nu_1| ∣ ν 1 ∣ L ~ ε \widetilde{L}_{\varepsilon} L ε L ~ ε χ { d ( x , y ) > ρ } \widetilde{L}_{\varepsilon}\chi_{\{d(x,y)>\rho\}} L ε χ { d ( x , y ) > ρ } (158) and (159) hold for L ~ ε \widetilde{L}_{\varepsilon} L ε Φ \Phi Φ Ξ w \Xi_w Ξ w ε \varepsilon ε
Let f ∈ L 2 ( ∣ ν ^ 1 ∣ ) f\in L^2(|\widehat{\nu}_1|) f ∈ L 2 ( ∣ ν 1 ∣ ) B 1 B_1 B 1 x ∈ B 2 x\in B_2 x ∈ B 2 ρ ≥ ε \rho\ge\varepsilon ρ ≥ ε
T ~ ∣ ν ^ 1 ∣ , ρ f ( x ) = S ~ ε , ρ f ( x ) + ∫ d ( x , y ) > ρ ( 1 − φ ( d ( x , y ) ε ) ) K ~ ( x , y ) f ( y ) d ∣ ν ^ 1 ∣ ( y ) + ∫ d ( x , y ) > ρ D ( x , y ) f ( y ) d ∣ ν ^ 1 ∣ ( y ) = : S ~ ε , ρ f ( x ) + J ~ 1 ( x ) + J ~ 2 ( x ) , (282) \begin{aligned}
\widetilde{T}_{|\widehat{\nu}_1|,\rho}f(x)
&=\widetilde{S}_{\varepsilon,\rho}f(x)
+\int_{d(x,y)>\rho}\left(1-\varphi\left(\frac{d(x,y)}{\varepsilon}\right)\right)\widetilde{K}(x,y)f(y)\,d|\widehat{\nu}_1|(y)\\
&\quad+\int_{d(x,y)>\rho}\mathcal{D}(x,y)f(y)\,d|\widehat{\nu}_1|(y)
=:\widetilde{S}_{\varepsilon,\rho}f(x)+\widetilde{J}_1(x)+\widetilde{J}_2(x), \tag*{(282)}
\end{aligned} T ∣ ν 1 ∣ , ρ f ( x ) = S ε , ρ f ( x ) + ∫ d ( x , y ) > ρ ( 1 − φ ( ε d ( x , y ) ) ) K ( x , y ) f ( y ) d ∣ ν 1 ∣ ( y ) + ∫ d ( x , y ) > ρ D ( x , y ) f ( y ) d ∣ ν 1 ∣ ( y ) =: S ε , ρ f ( x ) + J 1 ( x ) + J 2 ( x ) , (282) where
D ( x , y ) : = ( 1 − χ 2 ( x ) χ 1 ( y ) ψ ( d ( x , y ) r 0 ) ) φ ( d ( x , y ) ε ) K ~ ( x , y ) . \mathcal{D}(x,y):=\left(1-\chi_2(x)\chi_1(y)\psi\left(\frac{d(x,y)}{r_0}\right)\right)\varphi\left(\frac{d(x,y)}{\varepsilon}\right)\widetilde{K}(x,y). D ( x , y ) := ( 1 − χ 2 ( x ) χ 1 ( y ) ψ ( r 0 d ( x , y ) ) ) φ ( ε d ( x , y ) ) K ( x , y ) . Since φ ( t / ε ) = 1 \varphi(t/\varepsilon)=1 φ ( t / ε ) = 1 t ≥ 2 ε t\ge2\varepsilon t ≥ 2 ε K ~ \widetilde{K} K
∣ J ~ 1 ( x ) ∣ ≲ max ( ε , Φ ( x ) ) − n ∫ B ( x , 2 ε ) ∣ f ∣ d ∣ ν ^ 1 ∣ ≲ M Φ , ∣ ν ^ 1 ∣ r f ( x ) . |\widetilde{J}_1(x)|\lesssim\max(\varepsilon,\Phi(x))^{-n}\int_{B(x,2\varepsilon)}|f|\,d|\widehat{\nu}_1|\lesssim M^r_{\Phi,|\widehat{\nu}_1|}f(x). ∣ J 1 ( x ) ∣ ≲ max ( ε , Φ ( x ) ) − n ∫ B ( x , 2 ε ) ∣ f ∣ d ∣ ν 1 ∣ ≲ M Φ , ∣ ν 1 ∣ r f ( x ) . As in (161) and (162) , we have ∣ D ( x , y ) ∣ ≲ H Φ ( x , y ) : = ( r 0 + d ( x , y ) + Φ ( x ) + Φ ( y ) ) − n |\mathcal{D}(x,y)| \lesssim H_{\Phi}(x,y) := (r_{0}+d(x,y)+\Phi(x)+\Phi(y))^{-n} ∣ D ( x , y ) ∣ ≲ H Φ ( x , y ) := ( r 0 + d ( x , y ) + Φ ( x ) + Φ ( y ) ) − n
∬ B 2 × B 1 H Φ ( x , y ) 2 d ∣ ν ^ 1 ∣ ( y ) d ∣ ν ^ 2 ∣ ( x ) ≲ 1. \iint_{B_{2}\times B_{1}} H_{\Phi}(x,y)^{2}\,d|\widehat{\nu}_{1}|(y)\,d|\widehat{\nu}_{2}|(x) \lesssim1. ∬ B 2 × B 1 H Φ ( x , y ) 2 d ∣ ν 1 ∣ ( y ) d ∣ ν 2 ∣ ( x ) ≲ 1. Hence, ∣ J ~ 2 ( x ) ∣ ≲ ∫ H Φ ( x , y ) ∣ f ( y ) ∣ d ∣ ν ^ 1 ∣ ( y ) |\widetilde{\mathcal{J}}_{2}(x)| \lesssim\int H_{\Phi}(x,y)|f(y)|\,d|\widehat{\nu}_{1}|(y) ∣ J 2 ( x ) ∣ ≲ ∫ H Φ ( x , y ) ∣ f ( y ) ∣ d ∣ ν 1 ∣ ( y ) L 2 ( ∣ ν ^ 2 ∣ ) L^{2}(|\widehat{\nu}_{2}|) L 2 ( ∣ ν 2 ∣ ) ∥ f ∥ L 2 ( ∣ ν ^ 1 ∣ ) \|f\|_{L^{2}(|\widehat{\nu}_{1}|)} ∥ f ∥ L 2 ( ∣ ν 1 ∣ )
To estimate S ~ ε , ρ \widetilde{S}_{\varepsilon,\rho} S ε , ρ φ ( t ) = ∫ 1 2 φ ′ ( u ) χ { t > u } d u \varphi(t)=\int_{1}^{2}\varphi'(u)\chi_{\{t>u\}}\,du φ ( t ) = ∫ 1 2 φ ′ ( u ) χ { t > u } d u t ≥ 0 t\geq0 t ≥ 0
∫ φ ( d ( x , y ) ε ) K ~ ( x , y ) f ( y ) d ∣ ν ^ 1 ∣ ( y ) = ∫ 1 2 φ ′ ( u ) T ~ ∣ ν ^ 1 ∣ , u ε f ( x ) d u . \int\varphi\left(\frac{d(x,y)}{\varepsilon}\right)\widetilde{K}(x,y)f(y)\,d|\widehat{\nu}_{1}|(y)=\int_{1}^{2}\varphi'(u)\widetilde{T}_{|\widehat{\nu}_{1}|,u\varepsilon}f(x)\,du. ∫ φ ( ε d ( x , y ) ) K ( x , y ) f ( y ) d ∣ ν 1 ∣ ( y ) = ∫ 1 2 φ ′ ( u ) T ∣ ν 1 ∣ , u ε f ( x ) d u . Since φ \varphi φ ∫ 1 2 φ ′ ( u ) d u = 1 \int_{1}^{2}\varphi'(u)\,du=1 ∫ 1 2 φ ′ ( u ) d u = 1 sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ \sup_{\rho\geq\varepsilon}\|\widetilde{T}_{|\widehat{\nu}_{1}|,\rho}\| sup ρ ≥ ε ∥ T ∣ ν 1 ∣ , ρ ∥ L ~ ε \widetilde{L}_{\varepsilon} L ε D \mathcal{D} D B 2 × B 1 B_{2}\times B_{1} B 2 × B 1 L ~ ε \widetilde{L}_{\varepsilon} L ε B 2 × B 1 B_{2}\times B_{1} B 2 × B 1 H Φ H_{\Phi} H Φ
∥ S ~ ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≲ 1 + sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ . \|\widetilde{S}_{\varepsilon}\|_{L^{2}(|\nu_{1}|)\to L^{2}(|\nu_{2}|)}\lesssim1+\sup_{\rho\geq\varepsilon}\|\widetilde{T}_{|\widehat{\nu}_{1}|,\rho}\|. ∥ S ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≲ 1 + ρ ≥ ε sup ∥ T ∣ ν 1 ∣ , ρ ∥. Since L ~ ε ( x , y ) = 0 \widetilde{L}_{\varepsilon}(x,y)=0 L ε ( x , y ) = 0 d ( x , y ) ≤ ε d(x,y)\leq\varepsilon d ( x , y ) ≤ ε Lemma 3.16 , applied as after (159) with Φ + ε \Phi+\varepsilon Φ + ε Ξ \Xi Ξ m 1 = ∣ ν 1 ∣ m_{1}=|\nu_{1}| m 1 = ∣ ν 1 ∣ m 2 = m 3 = ∣ ν 2 ∣ m_{2}=m_{3}=|\nu_{2}| m 2 = m 3 = ∣ ν 2 ∣
∥ sup ρ > 0 ∣ S ~ ε , ρ f ∣ ∥ L 2 ( ∣ ν 2 ∣ ) ≲ ( 1 + ∥ S ~ ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ( 1 + sup ρ ≥ ε ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) . \left\|\sup_{\rho>0}|\widetilde{S}_{\varepsilon,\rho}f|\right\|_{L^{2}(|\nu_{2}|)}
\lesssim\left(1+\|\widetilde{S}_{\varepsilon}\|_{L^{2}(|\nu_{1}|)\to L^{2}(|\nu_{2}|)}\right)\|f\|_{L^{2}(|\nu_{1}|)}
\lesssim\left(1+\sup_{\rho\geq\varepsilon}\|\widetilde{T}_{|\widehat{\nu}_{1}|,\rho}\|\right)\|f\|_{L^{2}(|\nu_{1}|)}. ρ > 0 sup ∣ S ε , ρ f ∣ L 2 ( ∣ ν 2 ∣ ) ≲ ( 1 + ∥ S ε ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) ≲ ( 1 + ρ ≥ ε sup ∥ T ∣ ν 1 ∣ , ρ ∥ ) ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) . Gathering the bounds for the terms in (282) , taking the supremum over ρ ≥ ε \rho\geq\varepsilon ρ ≥ ε (278) with Ψ = Φ \Psi=\Phi Ψ = Φ (281) . □
We now prove the uniform L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|) L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) T ~ \widetilde{T} T T Φ T_{\Phi} T Φ T Ξ w T_{\Xi_{w}} T Ξ w
Lemma 7.3. There exists c > 0 c>0 c > 0 δ > 0 \delta>0 δ > 0
∥ T ~ ∣ ν ^ 1 ∣ , δ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) + ∥ T Φ , ∣ ν ^ 1 ∣ , δ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) + sup w ∈ G ∥ T Ξ w , ∣ ν ^ 1 ∣ , δ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) ≤ c , \|\widetilde{T}_{|\widehat{\nu}_{1}|,\delta}\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}
+\|T_{\Phi,|\widehat{\nu}_{1}|,\delta}\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}
+\sup_{w\in\mathcal{G}}\|T_{\Xi_{w},|\widehat{\nu}_{1}|,\delta}\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}
\leq c, ∥ T ∣ ν 1 ∣ , δ ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) + ∥ T Φ , ∣ ν 1 ∣ , δ ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) + w ∈ G sup ∥ T Ξ w , ∣ ν 1 ∣ , δ ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) ≤ c , where c c c δ \delta δ R 1 R_{1} R 1 R 2 R_{2} R 2
Proof. For δ > 0 \delta>0 δ > 0
A δ : = sup ρ ≥ δ ∥ T ~ ∣ ν ^ 1 ∣ , ρ ∥ L 2 ( ∣ ν ^ 1 ∣ ) → L 2 ( ∣ ν ^ 2 ∣ ) . A_{\delta}:=\sup_{\rho\geq\delta}\|\widetilde{T}_{|\widehat{\nu}_{1}|,\rho}\|_{L^{2}(|\widehat{\nu}_{1}|)\to L^{2}(|\widehat{\nu}_{2}|)}. A δ := ρ ≥ δ sup ∥ T ∣ ν 1 ∣ , ρ ∥ L 2 ( ∣ ν 1 ∣ ) → L 2 ( ∣ ν 2 ∣ ) . Since ∣ K ~ ( x , y ) ∣ ≲ d ( x , y ) − n |\widetilde{K}(x,y)|\lesssim d(x,y)^{-n} ∣ K ( x , y ) ∣ ≲ d ( x , y ) − n
A δ ≲ δ − n ∣ ν ^ 1 ∣ ( B 1 ) 1 / 2 ∣ ν ^ 2 ∣ ( B 2 ) 1 / 2 < ∞ . A_{\delta}\lesssim\delta^{-n}|\widehat{\nu}_{1}|(B_{1})^{1/2}|\widehat{\nu}_{2}|(B_{2})^{1/2}<\infty. A δ ≲ δ − n ∣ ν 1 ∣ ( B 1 ) 1/2 ∣ ν 2 ∣ ( B 2 ) 1/2 < ∞. Let C ′ > 0 C'>0 C ′ > 0 (277) , which, by Lemma 7.2 , is independent of m m m ε ≥ δ \varepsilon\geq\delta ε ≥ δ N ε N_{\varepsilon} N ε (182) satisfies
N ε ≤ C ′ ( 1 + A δ ) . (283) N_{\varepsilon}\leq C'(1+A_{\delta}). \tag*{(283)} N ε ≤ C ′ ( 1 + A δ ) . (283) Let ε ≥ δ \varepsilon\geq\delta ε ≥ δ f ∈ L 2 ( ∣ ν ^ 1 ∣ ) f\in L^{2}(|\widehat{\nu}_{1}|) f ∈ L 2 ( ∣ ν 1 ∣ ) g ∈ L 2 ( ∣ ν ^ 2 ∣ ) g\in L^{2}(|\widehat{\nu}_{2}|) g ∈ L 2 ( ∣ ν 2 ∣ ) B 1 B_{1} B 1 B 2 B_{2} B 2 ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) = ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) = 1 \|f\|_{L^{2}(|\nu_{1}|)}=\|g\|_{L^{2}(|\nu_{2}|)}=1 ∥ f ∥ L 2 ( ∣ ν 1 ∣ ) = ∥ g ∥ L 2 ( ∣ ν 2 ∣ ) = 1
Let w ∈ G w\in\mathcal{G} w ∈ G T w : = T Ξ w , ∣ ν ^ 1 ∣ , ε T_{w}:=T_{\Xi_{w},|\widehat{\nu}_{1}|,\varepsilon} T w := T Ξ w , ∣ ν 1 ∣ , ε f f f g g g Q 2 ( w 1 ) Q_{2}(w_{1}) Q 2 ( w 1 ) Q 2 ( w 2 ) Q_{2}(w_{2}) Q 2 ( w 2 ) (177) gives f = F g o o d + F b a d f=F_{\mathrm{good}}+F_{\mathrm{bad}} f = F good + F bad g = G g o o d + G b a d g=G_{\mathrm{good}}+G_{\mathrm{bad}} g = G good + G bad
F g o o d : = E 1 f + P G 1 ( w ) f , F b a d : = P B 1 ( w ) f , F_{\mathrm{good}}:=E_{1}f+\mathcal{P}_{\mathcal{G}}^{1}(w)f,\qquad F_{\mathrm{bad}}:=\mathcal{P}_{\mathcal{B}}^{1}(w)f, F good := E 1 f + P G 1 ( w ) f , F bad := P B 1 ( w ) f , G g o o d : = E 2 g + P G 2 ( w ) g , G b a d : = P B 2 ( w ) g , G_{\mathrm{good}}:=E_{2}g+\mathcal{P}_{\mathcal{G}}^{2}(w)g,\qquad G_{\mathrm{bad}}:=\mathcal{P}_{\mathcal{B}}^{2}(w)g, G good := E 2 g + P G 2 ( w ) g , G bad := P B 2 ( w ) g , where we recall that E 1 E_1 E 1 E 2 E_2 E 2 (174) , and projections are introduced in Definition 6.4.
Note that the functions above need not be supported in the testing balls, so we set f g o o d : = χ B 1 F g o o d f_{\mathrm{good}} := \chi_{B_1}F_{\mathrm{good}} f good := χ B 1 F good f b a d : = χ B 1 F b a d f_{\mathrm{bad}} := \chi_{B_1}F_{\mathrm{bad}} f bad := χ B 1 F bad g g o o d g_{\mathrm{good}} g good g b a d g_{\mathrm{bad}} g bad f = f g o o d + f b a d f = f_{\mathrm{good}} + f_{\mathrm{bad}} f = f good + f bad g = g g o o d + g b a d g = g_{\mathrm{good}} + g_{\mathrm{bad}} g = g good + g bad
⟨ T w f , g ⟩ ∣ ν 2 ∣ = ⟨ T w f g o o d , g g o o d ⟩ ∣ ν 2 ∣ + ⟨ T w f g o o d , g b a d ⟩ ∣ ν 2 ∣ + ⟨ T w f b a d , g ⟩ ∣ ν 2 ∣ = J ~ 1 ( w ) + J ~ 2 ( w ) + J ~ 3 ( w ) . (284) \langle T_w f,g\rangle_{|\nu_2|}=\langle T_w f_{\mathrm{good}},g_{\mathrm{good}}\rangle_{|\nu_2|}+\langle T_w f_{\mathrm{good}},g_{\mathrm{bad}}\rangle_{|\nu_2|}+\langle T_w f_{\mathrm{bad}},g\rangle_{|\nu_2|}
= \widetilde{J}_1(w)+\widetilde{J}_2(w)+\widetilde{J}_3(w). \tag*{(284)} ⟨ T w f , g ⟩ ∣ ν 2 ∣ = ⟨ T w f good , g good ⟩ ∣ ν 2 ∣ + ⟨ T w f good , g bad ⟩ ∣ ν 2 ∣ + ⟨ T w f bad , g ⟩ ∣ ν 2 ∣ = J 1 ( w ) + J 2 ( w ) + J 3 ( w ) . (284) Since ∥ χ B i h ∥ L 2 ( ∣ ν i ∣ ) ≤ ∥ h ∥ L 2 ( ∣ ν i ∣ ) \|\chi_{B_i}h\|_{L^2(|\nu_i|)}\leq\|h\|_{L^2(|\nu_i|)} ∥ χ B i h ∥ L 2 ( ∣ ν i ∣ ) ≤ ∥ h ∥ L 2 ( ∣ ν i ∣ )
∣ J ~ 2 ( w ) ∣ + ∣ J ~ 3 ( w ) ∣ ≤ N ε ( ∥ F g o o d ∥ L 2 ( ∣ ν 1 ∣ ) ∥ P B 2 ( w ) g ∥ L 2 ( ∣ ν 2 ∣ ) + ∥ P B 1 ( w ) f ∥ L 2 ( ∣ ν 1 ∣ ) ) . |\widetilde{J}_2(w)|+|\widetilde{J}_3(w)|\leq N_{\varepsilon}\left(\|F_{\mathrm{good}}\|_{L^2(|\nu_1|)}\|\mathcal{P}^2_B(w)g\|_{L^2(|\nu_2|)}+\|\mathcal{P}^1_B(w)f\|_{L^2(|\nu_1|)}\right). ∣ J 2 ( w ) ∣ + ∣ J 3 ( w ) ∣ ≤ N ε ( ∥ F good ∥ L 2 ( ∣ ν 1 ∣ ) ∥ P B 2 ( w ) g ∥ L 2 ( ∣ ν 2 ∣ ) + ∥ P B 1 ( w ) f ∥ L 2 ( ∣ ν 1 ∣ ) ) . Hence, by (178) , (181) , (180) , and P Ω 2 ( G ) > 1 − κ > 1 / 2 \mathbb{P}^{\Omega^2}(\mathcal{G})>1-\kappa>1/2 P Ω 2 ( G ) > 1 − κ > 1/2
E ~ w ( ∣ J ~ 2 ( w ) ∣ + ∣ J ~ 3 ( w ) ∣ ) ≤ C b a d E B ( m ) N ε , (285) \widetilde{\mathbb{E}}_w\bigl(|\widetilde{J}_2(w)|+|\widetilde{J}_3(w)|\bigr)\leq C_{\mathrm{bad}}\mathcal{E}\mathcal{B}(m)N_{\varepsilon}, \tag*{(285)} E w ( ∣ J 2 ( w ) ∣ + ∣ J 3 ( w ) ∣ ) ≤ C bad E B ( m ) N ε , (285) where C b a d > 0 C_{\mathrm{bad}}>0 C bad > 0 m m m
To bound J ~ 1 ( w ) \widetilde{J}_1(w) J 1 ( w )
F g o o d − E 1 F g o o d = P G 1 ( w ) f and G g o o d − E 2 G g o o d = P G 2 ( w ) g . F_{\mathrm{good}}-E_1F_{\mathrm{good}}=\mathcal{P}^1_{\mathcal{G}}(w)f
\qquad\text{and}\qquad
G_{\mathrm{good}}-E_2G_{\mathrm{good}}=\mathcal{P}^2_{\mathcal{G}}(w)g. F good − E 1 F good = P G 1 ( w ) f and G good − E 2 G good = P G 2 ( w ) g . Hence, (185) , applied twice, the splitting (190) with F g o o d F_{\mathrm{good}} F good G g o o d G_{\mathrm{good}} G good f w f_w f w g w g_w g w (192) , (194) , and (195) for its first three terms give
∣ J ~ 1 ( w ) − ⟨ T w ( P G 1 ( w ) f ) , P G 2 ( w ) g ⟩ ∣ ν 2 ∣ ∣ ≲ 1 , \left|\widetilde{J}_1(w)-\langle T_w(\mathcal{P}^1_{\mathcal{G}}(w)f),\mathcal{P}^2_{\mathcal{G}}(w)g\rangle_{|\nu_2|}\right|\lesssim1, J 1 ( w ) − ⟨ T w ( P G 1 ( w ) f ) , P G 2 ( w ) g ⟩ ∣ ν 2 ∣ ≲ 1 , where we also used (178) and (181) . Thus, the Key Lemma 6.7 gives
E ~ w ∣ J ~ 1 ( w ) ∣ ≤ C m , λ , υ ( 1 + a − η ) + β m ( λ , υ , a ) ( 1 + N ε ) , (286) \widetilde{\mathbb{E}}_w|\widetilde{J}_1(w)|\leq C_{m,\lambda,\upsilon}(1+a^{-\eta})+\beta_m(\lambda,\upsilon,a)(1+N_{\varepsilon}), \tag*{(286)} E w ∣ J 1 ( w ) ∣ ≤ C m , λ , υ ( 1 + a − η ) + β m ( λ , υ , a ) ( 1 + N ε ) , (286) where, increasing C m , λ , υ C_{m,\lambda,\upsilon} C m , λ , υ
β m ( λ , υ , a ) : = C m ( ( 2 m + 1 ) λ η + υ P Ω 2 ( G ) ) 1 / 2 + C m , υ ( M Λ a ) 1 / 2 . \beta_m(\lambda,\upsilon,a):=C_m\left(\frac{(2m+1)\lambda^\eta+\upsilon}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\right)^{1/2}+C_{m,\upsilon}(M_{\Lambda}a)^{1/2}. β m ( λ , υ , a ) := C m ( P Ω 2 ( G ) ( 2 m + 1 ) λ η + υ ) 1/2 + C m , υ ( M Λ a ) 1/2 . We now fix the parameters, independently of δ \delta δ ε \varepsilon ε C ′ C' C ′ C b a d C_{\mathrm{bad}} C bad m m m m m m (172) holds and C ′ C b a d E B ( m ) ≤ 1 / 4 C'C_{\mathrm{bad}}\mathcal{E}\mathcal{B}(m)\leq1/4 C ′ C bad E B ( m ) ≤ 1/4 C m C_m C m λ \lambda λ υ \upsilon υ λ \lambda λ υ \upsilon υ
C ′ C m ( ( 2 m + 1 ) λ η + υ P Ω 2 ( G ) ) 1 / 2 ≤ 1 8 . C'C_m\left(\frac{(2m+1)\lambda^\eta+\upsilon}{\mathbb{P}^{\Omega^2}(\mathcal{G})}\right)^{1/2}\leq\frac{1}{8}. C ′ C m ( P Ω 2 ( G ) ( 2 m + 1 ) λ η + υ ) 1/2 ≤ 8 1 . Finally, since M Λ M_{\Lambda} M Λ C m , υ C_{m,\upsilon} C m , υ a ∈ ( 0 , 1 / 4 ) a\in(0,1/4) a ∈ ( 0 , 1/4 ) C ′ C m , υ ( M Λ a ) 1 / 2 ≤ 1 / 8 C'C_{m,\upsilon}(M_{\Lambda}a)^{1/2}\leq1/8 C ′ C m , υ ( M Λ a ) 1/2 ≤ 1/8
By (268) and Fubini’s theorem, we have
⟨ T ~ ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ = E ~ w ⟨ T w f , g ⟩ ∣ ν 2 ∣ . \langle\widetilde{T}_{|\nu_1|,\varepsilon}f,g\rangle_{|\nu_2|}=\widetilde{\mathbb{E}}_w\langle T_wf,g\rangle_{|\nu_2|}. ⟨ T ∣ ν 1 ∣ , ε f , g ⟩ ∣ ν 2 ∣ = E w ⟨ T w f , g ⟩ ∣ ν 2 ∣ . Hence, (284) , (285) , (286) , and (283) give, after taking the supremum over f f f g g g ε ≥ δ \varepsilon\geq\delta ε ≥ δ
A δ ≤ C ∗ + C ′ ( β m ( λ , υ , a ) + C b a d E B ( m ) ) A δ ≤ C ∗ + 1 2 A δ , (287) A_\delta\leq C_*+C'\bigl(\beta_m(\lambda,\upsilon,a)+C_{\mathrm{bad}}\mathcal{E}\mathcal{B}(m)\bigr)A_\delta\leq C_*+\frac{1}{2}A_\delta, \tag*{(287)} A δ ≤ C ∗ + C ′ ( β m ( λ , υ , a ) + C bad E B ( m ) ) A δ ≤ C ∗ + 2 1 A δ , (287) where C ∗ C_* C ∗ m m m λ \lambda λ υ \upsilon υ a a a δ \delta δ A δ < ∞ A_\delta<\infty A δ < ∞ A δ ≤ 2 C ∗ A_\delta\leq2C_* A δ ≤ 2 C ∗ (276) and (277) . □
The good-lambda argument
We use the dyadic system of Lemma 2.5 and a good-lambda inequality to prove the following result.
Proposition 7.4. Let μ \mu μ X X X n n n C 0 \mathcal{C}_0 C 0 θ 0 ∈ ( 0 , 1 ] \theta_0 \in(0,1] θ 0 ∈ ( 0 , 1 ] T T T n n n γ \gamma γ C C C s > 0 s>0 s > 0 Q ∈ D μ d b , ( s ) Q \in\mathcal{D}_{\mu}^{\mathrm{db},(s)} Q ∈ D μ db , ( s ) G Q ⊂ Q G_Q \subset Q G Q ⊂ Q μ ( G Q ) ≥ θ 0 μ ( Q ) \mu(G_Q) \ge\theta_0\mu(Q) μ ( G Q ) ≥ θ 0 μ ( Q ) T μ ∣ G Q T_{\mu|G_Q} T μ ∣ G Q L 2 ( μ ∣ G Q ) L^2(\mu|G_Q) L 2 ( μ ∣ G Q ) L 2 ( μ ∣ G Q ) L^2(\mu|G_Q) L 2 ( μ ∣ G Q ) Q Q Q s s s
Then, if the parameter K 0 K_0 K 0 D D D f ∈ L 2 ( μ ) f \in L^2(\mu) f ∈ L 2 ( μ )
sup ε > 0 ∥ T μ , ε f ∥ L 2 ( μ ) ≲ ∥ f ∥ L 2 ( μ ) , (288) \sup_{\varepsilon>0}\|T_{\mu,\varepsilon}f\|_{L^2(\mu)} \lesssim\|f\|_{L^2(\mu)}, \tag*{(288)} ε > 0 sup ∥ T μ , ε f ∥ L 2 ( μ ) ≲ ∥ f ∥ L 2 ( μ ) , (288) where the implicit constant depends only on D D D n n n C 0 \mathcal{C}_0 C 0 C C C γ \gamma γ θ 0 \theta_0 θ 0 K 0 K_0 K 0 A 0 A_0 A 0 T μ ∣ G Q T_{\mu|G_Q} T μ ∣ G Q
By density and Fatou’s lemma, it suffices to prove (288) for f ∈ L 2 ( μ ) f \in L^2(\mu) f ∈ L 2 ( μ ) h > 0 h>0 h > 0
T μ , ♯ , h f ( x ) : = sup ρ ≥ h ∣ T μ , ρ f ( x ) ∣ . T_{\mu,\sharp,h}f(x):=\sup_{\rho\ge h}|T_{\mu,\rho}f(x)|. T μ , ♯ , h f ( x ) := ρ ≥ h sup ∣ T μ , ρ f ( x ) ∣. If supp f ⊂ B ( x 0 , R ) \operatorname{supp} f \subset B(x_0,R) supp f ⊂ B ( x 0 , R ) T μ , ♯ , h f ≤ C h − n ∥ f ∥ L 1 ( μ ) T_{\mu,\sharp,h}f \le Ch^{-n}\|f\|_{L^1(\mu)} T μ , ♯ , h f ≤ C h − n ∥ f ∥ L 1 ( μ )
∫ X ∖ B ( x 0 , 2 R ) ( T μ , ♯ , h f ) 2 d μ ≲ ∥ f ∥ L 1 ( μ ) 2 ∑ j > 1 ( 2 j R ) − 2 n μ ( B ( x 0 , 2 j + 1 R ) ) < ∞ . \int_{X\setminus B(x_0,2R)}(T_{\mu,\sharp,h}f)^2\,d\mu\lesssim\|f\|_{L^1(\mu)}^2\sum_{j>1}(2^jR)^{-2n}\mu(B(x_0,2^{j+1}R))<\infty. ∫ X ∖ B ( x 0 , 2 R ) ( T μ , ♯ , h f ) 2 d μ ≲ ∥ f ∥ L 1 ( μ ) 2 j > 1 ∑ ( 2 j R ) − 2 n μ ( B ( x 0 , 2 j + 1 R )) < ∞. Since μ ( B ( x 0 , 2 R ) ) < ∞ \mu(B(x_0,2R))<\infty μ ( B ( x 0 , 2 R )) < ∞ T μ , ♯ , h f ∈ L 2 ( μ ) T_{\mu,\sharp,h}f\in L^2(\mu) T μ , ♯ , h f ∈ L 2 ( μ ) h h h 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 ε ′ > 0 \varepsilon'>0 ε ′ > 0 λ > 0 \lambda>0 λ > 0
μ ( { x : T μ , ♯ , h f ( x ) > ( 1 + ε ) λ , M ~ μ f ( x ) ≤ ε ′ λ } ) ≤ ( 1 − θ 0 4 ) μ ( { x : T μ , ♯ , h f ( x ) > λ } ) . (289) \mu\bigl(\{x:T_{\mu,\sharp,h}f(x)>(1+\varepsilon)\lambda,\ \widetilde{M}_{\mu}f(x)\le\varepsilon'\lambda\}\bigr)
\le\left(1-\frac{\theta_0}{4}\right)\mu\bigl(\{x:T_{\mu,\sharp,h}f(x)>\lambda\}\bigr). \tag*{(289)} μ ( { x : T μ , ♯ , h f ( x ) > ( 1 + ε ) λ , M μ f ( x ) ≤ ε ′ λ } ) ≤ ( 1 − 4 θ 0 ) μ ( { x : T μ , ♯ , h f ( x ) > λ } ) . (289) We follow the scheme of the proof of [MT20, Section 9], using the cells of D μ ( s ) \mathcal{D}_{\mu}^{(s)} D μ ( s )
For s > 0 s>0 s > 0 Q ∈ D μ d b , ( s ) Q\in\mathcal{D}_{\mu}^{\mathrm{db},(s)} Q ∈ D μ db , ( s ) σ Q : = μ ∣ G Q \sigma_Q:=\mu|_{G_Q} σ Q := μ ∣ G Q σ Q \sigma_Q σ Q n n n [40 Corollary 1.7(ii)] , applied to the L 2 ( σ Q ) L^2(\sigma_Q) L 2 ( σ Q ) T T T
μ ( { x ∈ G Q : T ♯ ω ( x ) > t } ) ≤ C t ∥ ω ∥ , t > 0 , (290) \mu(\{x\in G_Q:T^\sharp\omega(x)>t\})\le\frac{C}{t}\|\omega\|,\qquad t>0, \tag*{(290)} μ ({ x ∈ G Q : T ♯ ω ( x ) > t }) ≤ t C ∥ ω ∥ , t > 0 , (290) for every finite complex Borel measure ω \omega ω X X X Q Q Q s s s ω \omega ω G Q G_Q G Q
We first prove the following two lemmas.
Lemma 7.5. Let ( X , d ) (X,d) ( X , d ) μ \mu μ n n n C 0 \mathcal{C}_0 C 0 T T T C > 0 C>0 C > 0 γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) f ∈ L 2 ( μ ) f\in L^2(\mu) f ∈ L 2 ( μ ) h , λ > 0 h,\lambda>0 h , λ > 0
Ω λ : = { x ∈ supp ( μ ) : T μ , ♯ , h f ( x ) > λ } . \Omega_\lambda:=\{x\in\operatorname{supp}(\mu):T_{\mu,\sharp,h}f(x)>\lambda\}. Ω λ := { x ∈ supp ( μ ) : T μ , ♯ , h f ( x ) > λ } . For every λ > 0 \lambda>0 λ > 0 Ω λ \Omega_\lambda Ω λ μ ( Ω λ ) < ∞ \mu(\Omega_\lambda)<\infty μ ( Ω λ ) < ∞ Ω λ \Omega_\lambda Ω λ supp ( μ ) \operatorname{supp}(\mu) supp ( μ )
Proof. By the assumptions on f f f x 0 ∈ X x_0 \in X x 0 ∈ X R > 0 R > 0 R > 0 f ≡ 0 f \equiv0 f ≡ 0 B ( x 0 , R ) B(x_0,R) B ( x 0 , R ) f ∈ L 1 ( μ ) f \in L^1(\mu) f ∈ L 1 ( μ ) x ∈ X ∖ B ( x 0 , R ) ‾ x \in X \setminus\overline{B(x_0,R)} x ∈ X ∖ B ( x 0 , R ) ρ > 0 \rho> 0 ρ > 0
∣ T μ , ρ f ( x ) ∣ ≤ C dist ( x , B ( x 0 , R ) ‾ ) n ∫ ∣ f ∣ d μ , \lvert T_{\mu,\rho}f(x)\rvert\le\frac{C}{\operatorname{dist}(x,\overline{B(x_0,R)})^n}\int\lvert f\rvert\,d\mu, ∣ T μ , ρ f ( x )∣ ≤ dist ( x , B ( x 0 , R ) ) n C ∫ ∣ f ∣ d μ , which implies that Ω λ ⊂ B ( x 0 , R λ ) \Omega_\lambda\subset B(x_0,R_\lambda) Ω λ ⊂ B ( x 0 , R λ ) R λ > R R_\lambda> R R λ > R f f f λ \lambda λ Ω λ \Omega_\lambda Ω λ
μ ( Ω λ ) ≤ μ ( B ( x 0 , R λ ) ) ≤ C 0 R λ n < ∞ . \mu(\Omega_\lambda) \le\mu(B(x_0,R_\lambda)) \le C_0 R_\lambda^n < \infty. μ ( Ω λ ) ≤ μ ( B ( x 0 , R λ )) ≤ C 0 R λ n < ∞. It remains to prove that Ω λ \Omega_\lambda Ω λ x ∈ Ω λ x \in\Omega_\lambda x ∈ Ω λ ρ ≥ h \rho\ge h ρ ≥ h ∣ T μ , ρ f ( x ) ∣ > λ \lvert T_{\mu,\rho}f(x)\rvert> \lambda ∣ T μ , ρ f ( x )∣ > λ T μ , r f ( x ) → T μ , ρ f ( x ) T_{\mu,r}f(x) \to T_{\mu,\rho}f(x) T μ , r f ( x ) → T μ , ρ f ( x ) r ↓ ρ r \downarrow\rho r ↓ ρ { y : d ( x , y ) = r } \{y:d(x,y)=r\} { y : d ( x , y ) = r } ∣ f ∣ μ \lvert f\rvert\mu ∣ f ∣ μ r > 0 r > 0 r > 0 r > ρ r > \rho r > ρ ∣ T μ , r f ( x ) ∣ > λ \lvert T_{\mu,r}f(x)\rvert> \lambda ∣ T μ , r f ( x )∣ > λ
∫ d ( x , y ) = r ∣ f ( y ) ∣ d μ ( y ) = 0. \int_{d(x,y)=r} \lvert f(y)\rvert\,d\mu(y)=0. ∫ d ( x , y ) = r ∣ f ( y )∣ d μ ( y ) = 0. If z → x z \to x z → x K ( z , y ) χ { d ( z , y ) > r } f ( y ) → K ( x , y ) χ { d ( x , y ) > r } f ( y ) K(z,y)\chi_{\{d(z,y)>r\}}f(y) \to K(x,y)\chi_{\{d(x,y)>r\}}f(y) K ( z , y ) χ { d ( z , y ) > r } f ( y ) → K ( x , y ) χ { d ( x , y ) > r } f ( y ) ( ∣ f ∣ μ ) (\lvert f\rvert\mu) (∣ f ∣ μ ) y y y C r − n ∣ f ( y ) ∣ Cr^{-n}\lvert f(y)\rvert C r − n ∣ f ( y )∣ T μ , r f ( z ) → T μ , r f ( x ) T_{\mu,r}f(z) \to T_{\mu,r}f(x) T μ , r f ( z ) → T μ , r f ( x ) U U U x x x ∣ T μ , r f ( z ) ∣ > λ \lvert T_{\mu,r}f(z)\rvert> \lambda ∣ T μ , r f ( z )∣ > λ z ∈ U z \in U z ∈ U U ∩ supp ( μ ) ⊂ Ω λ U \cap\operatorname{supp}(\mu) \subset\Omega_\lambda U ∩ supp ( μ ) ⊂ Ω λ
If Ω λ = ∅ \Omega_\lambda= \varnothing Ω λ = ∅ (289) is immediate. Suppose first that ∅ ≠ Ω λ ⊊ supp ( μ ) \varnothing\ne\Omega_\lambda\subsetneq\operatorname{supp}(\mu) ∅ = Ω λ ⊊ supp ( μ ) Ω λ \Omega_\lambda Ω λ x ˉ ∈ supp ( μ ) ∖ Ω λ \bar{x} \in\operatorname{supp}(\mu) \setminus\Omega_\lambda x ˉ ∈ supp ( μ ) ∖ Ω λ
s : = 2 sup x ∈ Ω λ d ( x , x ˉ ) > 0. (291) s := 2\sup_{x\in\Omega_\lambda}d(x,\bar{x})>0. \tag*{(291)} s := 2 x ∈ Ω λ sup d ( x , x ˉ ) > 0. (291) If a top-generation cell Q ∈ D μ , 0 ( s ) Q \in\mathcal{D}^{(s)}_{\mu,0} Q ∈ D μ , 0 ( s ) Ω λ \Omega_\lambda Ω λ z Q z_Q z Q Ω λ \Omega_\lambda Ω λ
d ( z Q , x ˉ ) ≤ s / 2 < s ≤ r ( s ) ( Q ) , d(z_Q,\bar{x})\le s/2<s\le r^{(s)}(Q), d ( z Q , x ˉ ) ≤ s /2 < s ≤ r ( s ) ( Q ) , and (26) would give x ˉ ∈ supp ( μ ) ∩ B ( s ) ( Q ) ⊂ Q ⊂ Ω λ \bar{x} \in\operatorname{supp}(\mu)\cap B^{(s)}(Q)\subset Q\subset\Omega_\lambda x ˉ ∈ supp ( μ ) ∩ B ( s ) ( Q ) ⊂ Q ⊂ Ω λ Ω λ \Omega_\lambda Ω λ Ω λ = supp ( μ ) \Omega_\lambda=\operatorname{supp}(\mu) Ω λ = supp ( μ )
Lemma 7.6. Let Ω ⊂ supp ( μ ) \Omega\subset\operatorname{supp}(\mu) Ω ⊂ supp ( μ ) non-empty bounded set, relatively open in supp ( μ ) \operatorname{supp}(\mu) supp ( μ ) s > 0 s>0 s > 0 D μ , 0 ( s ) \mathcal{D}^{(s)}_{\mu,0} D μ , 0 ( s ) Ω \Omega Ω W ( s ) ⊂ D μ ( s ) \mathcal{W}^{(s)}\subset\mathcal{D}^{(s)}_\mu W ( s ) ⊂ D μ ( s )
Ω = ⋃ Q ∈ W ( s ) Q , \Omega=\bigcup_{Q\in\mathcal{W}^{(s)}}Q, Ω = Q ∈ W ( s ) ⋃ Q , and such that, for some constants T 0 > 10 4 T_0>10^4 T 0 > 1 0 4 A 0 A_0 A 0 K 0 K_0 K 0 N 0 ≥ 1 N_0\ge1 N 0 ≥ 1 D D D A 0 A_0 A 0 K 0 K_0 K 0
Every Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) Q ^ \widehat{Q} Q D μ ( s ) \mathcal{D}^{(s)}_\mu D μ ( s ) Q ^ ⊂ Ω \widehat{Q}\subset\Omega Q ⊂ Ω
For every Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) supp ( μ ) ∩ 10 4 B ( s ) ( Q ) ⊂ Ω \operatorname{supp}(\mu)\cap10^4B^{(s)}(Q)\subset\Omega supp ( μ ) ∩ 1 0 4 B ( s ) ( Q ) ⊂ Ω
For every Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) y Q ∈ supp ( μ ) ∖ Ω y_Q\in\operatorname{supp}(\mu)\setminus\Omega y Q ∈ supp ( μ ) ∖ Ω
d ( x , y Q ) ≤ T 0 r ( s ) ( Q ) , for every x ∈ Q . d(x,y_Q)\le T_0r^{(s)}(Q),\qquad\text{for every }x\in Q. d ( x , y Q ) ≤ T 0 r ( s ) ( Q ) , for every x ∈ Q . For every Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) N 0 N_0 N 0 P ∈ W ( s ) P\in\mathcal{W}^{(s)} P ∈ W ( s ) 10 4 B ( s ) ( Q ) ∩ 10 4 B ( s ) ( P ) ≠ ∅ 10^4B^{(s)}(Q)\cap10^4B^{(s)}(P)\ne\varnothing 1 0 4 B ( s ) ( Q ) ∩ 1 0 4 B ( s ) ( P ) = ∅ Q , P Q,P Q , P
( 4 A 0 ) − 1 ℓ ( s ) ( P ) ≤ ℓ ( s ) ( Q ) ≤ 4 A 0 ℓ ( s ) ( P ) . (292) (4A_0)^{-1}\ell^{(s)}(P)\le\ell^{(s)}(Q)\le4A_0\ell^{(s)}(P). \tag*{(292)} ( 4 A 0 ) − 1 ℓ ( s ) ( P ) ≤ ℓ ( s ) ( Q ) ≤ 4 A 0 ℓ ( s ) ( P ) . (292) (5) The family
W d b ( s ) : = W ( s ) ∩ D μ d b , ( s ) \mathcal{W}_{\mathrm{db}}^{(s)} := \mathcal{W}^{(s)} \cap\mathcal{D}_{\mu}^{\mathrm{db},(s)} W db ( s ) := W ( s ) ∩ D μ db , ( s ) satisfies
μ ( ⋃ Q ∈ W d b ( s ) Q ) ≥ 1 2 μ ( Ω ) , (293) \mu\left(\bigcup_{Q\in\mathcal{W}_{\mathrm{db}}^{(s)}} Q\right) \ge\frac{1}{2}\mu(\Omega), \tag*{(293)} μ Q ∈ W db ( s ) ⋃ Q ≥ 2 1 μ ( Ω ) , (293) assuming the parameter K 0 K_0 K 0 D D D
Proof. Fix a constant 0 < t < 1 0 < \mathfrak{t} < 1 0 < t < 1 t ≤ min { ( 2 A 0 ) − 1 , 14 ⋅ 10 − 4 } \mathfrak{t} \le\min\{(2A_0)^{-1},14\cdot10^{-4}\} t ≤ min {( 2 A 0 ) − 1 , 14 ⋅ 1 0 − 4 } Q ∈ D μ ( s ) Q\in\mathcal{D}_{\mu}^{(s)} Q ∈ D μ ( s ) t \mathfrak{t} t
ℓ ( s ) ( Q ) ≤ t dist ( Q , supp ( μ ) ∖ Ω ) . \ell^{(s)}(Q)\le\mathfrak{t}\operatorname{dist}(Q,\operatorname{supp}(\mu)\setminus\Omega). ℓ ( s ) ( Q ) ≤ t dist ( Q , supp ( μ ) ∖ Ω ) . In particular, every t \mathfrak{t} t Ω \Omega Ω Q ⊂ 28 B ( s ) ( Q ) Q\subset28B^{(s)}(Q) Q ⊂ 28 B ( s ) ( Q ) diam ( Q ) ≤ 56 r ( s ) ( Q ) ≤ ℓ ( s ) ( Q ) \operatorname{diam}(Q)\le56r^{(s)}(Q)\le\ell^{(s)}(Q) diam ( Q ) ≤ 56 r ( s ) ( Q ) ≤ ℓ ( s ) ( Q ) Q ∈ D μ ( s ) Q\in\mathcal{D}_{\mu}^{(s)} Q ∈ D μ ( s )
For k ≥ 0 k\ge0 k ≥ 0 x ∈ Ω x\in\Omega x ∈ Ω Q k ( x ) Q_k(x) Q k ( x ) D μ , k ( s ) \mathcal{D}_{\mu,k}^{(s)} D μ , k ( s ) x x x diam ( Q k ( x ) ) ≤ ℓ ( s ) ( Q k ( x ) ) \operatorname{diam}(Q_k(x))\le\ell^{(s)}(Q_k(x)) diam ( Q k ( x )) ≤ ℓ ( s ) ( Q k ( x ))
dist ( Q k ( x ) , supp ( μ ) ∖ Ω ) ≥ dist ( x , supp ( μ ) ∖ Ω ) − ℓ ( s ) ( Q k ( x ) ) . \operatorname{dist}(Q_k(x),\operatorname{supp}(\mu)\setminus\Omega)\ge\operatorname{dist}(x,\operatorname{supp}(\mu)\setminus\Omega)-\ell^{(s)}(Q_k(x)). dist ( Q k ( x ) , supp ( μ ) ∖ Ω ) ≥ dist ( x , supp ( μ ) ∖ Ω ) − ℓ ( s ) ( Q k ( x )) . Since Ω \Omega Ω supp ( μ ) \operatorname{supp}(\mu) supp ( μ ) dist ( x , supp ( μ ) ∖ Ω ) > 0 \operatorname{dist}(x,\operatorname{supp}(\mu)\setminus\Omega)>0 dist ( x , supp ( μ ) ∖ Ω ) > 0 ℓ ( s ) ( Q k ( x ) ) → 0 \ell^{(s)}(Q_k(x))\to0 ℓ ( s ) ( Q k ( x )) → 0 k → ∞ k\to\infty k → ∞ k k k
ℓ ( s ) ( Q k ( x ) ) ≤ t ( 1 + t ) − 1 dist ( x , supp ( μ ) ∖ Ω ) . \ell^{(s)}(Q_k(x))\le\mathfrak{t}(1+\mathfrak{t})^{-1}\operatorname{dist}(x,\operatorname{supp}(\mu)\setminus\Omega). ℓ ( s ) ( Q k ( x )) ≤ t ( 1 + t ) − 1 dist ( x , supp ( μ ) ∖ Ω ) . Combining the two inequalities gives
ℓ ( s ) ( Q k ( x ) ) ≤ t dist ( Q k ( x ) , supp ( μ ) ∖ Ω ) . \ell^{(s)}(Q_k(x))\le\mathfrak{t}\operatorname{dist}(Q_k(x),\operatorname{supp}(\mu)\setminus\Omega). ℓ ( s ) ( Q k ( x )) ≤ t dist ( Q k ( x ) , supp ( μ ) ∖ Ω ) . Thus, for k k k Q k ( x ) Q_k(x) Q k ( x ) t \mathfrak{t} t W ( s ) \mathcal{W}^{(s)} W ( s ) t \mathfrak{t} t t \mathfrak{t} t X X X W ( s ) \mathcal{W}^{(s)} W ( s ) t \mathfrak{t} t Ω \Omega Ω Ω \Omega Ω
By assumption, no cell from D μ , 0 ( s ) \mathcal{D}_{\mu,0}^{(s)} D μ , 0 ( s ) Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) Q ^ \widehat{Q} Q
diam ( Q ^ ) ≤ A 0 ℓ ( s ) ( Q ) ≤ A 0 t dist ( Q , supp ( μ ) ∖ Ω ) < dist ( Q , supp ( μ ) ∖ Ω ) . \operatorname{diam}(\widehat{Q})\le A_0\ell^{(s)}(Q)\le A_0\mathfrak{t}\operatorname{dist}(Q,\operatorname{supp}(\mu)\setminus\Omega)<\operatorname{dist}(Q,\operatorname{supp}(\mu)\setminus\Omega). diam ( Q ) ≤ A 0 ℓ ( s ) ( Q ) ≤ A 0 t dist ( Q , supp ( μ ) ∖ Ω ) < dist ( Q , supp ( μ ) ∖ Ω ) . Since Q ⊂ Q ^ Q\subset\widehat{Q} Q ⊂ Q Q ^ ⊂ Ω \widehat{Q}\subset\Omega Q ⊂ Ω
To prove (2), let Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) z ∈ supp ( μ ) ∩ 10 4 B ( s ) ( Q ) z\in\operatorname{supp}(\mu)\cap10^4B^{(s)}(Q) z ∈ supp ( μ ) ∩ 1 0 4 B ( s ) ( Q ) r ( s ) ( Q ) ≤ ℓ ( s ) ( Q ) / 56 r^{(s)}(Q)\le\ell^{(s)}(Q)/56 r ( s ) ( Q ) ≤ ℓ ( s ) ( Q ) /56 t ≤ 14 ⋅ 10 − 4 \mathfrak{t}\le14\cdot10^{-4} t ≤ 14 ⋅ 1 0 − 4 Q Q Q
d ( z , z Q ) < 10 4 r ( s ) ( Q ) ≤ 10 4 56 ℓ ( s ) ( Q ) ≤ 10 4 56 t dist ( Q , supp ( μ ) ∖ Ω ) ≤ 1 4 dist ( Q , supp ( μ ) ∖ Ω ) ≤ 1 4 dist ( z Q , supp ( μ ) ∖ Ω ) , \begin{aligned}
d(z,z_Q)<10^4r^{(s)}(Q)\le\frac{10^4}{56}\ell^{(s)}(Q)\le\frac{10^4}{56}\mathfrak{t}\operatorname{dist}(Q,\operatorname{supp}(\mu)\setminus\Omega) \\
&\le\frac14\operatorname{dist}(Q,\operatorname{supp}(\mu)\setminus\Omega)\le\frac14\operatorname{dist}(z_Q,\operatorname{supp}(\mu)\setminus\Omega),
\end{aligned} d ( z , z Q ) < 1 0 4 r ( s ) ( Q ) ≤ 56 1 0 4 ℓ ( s ) ( Q ) ≤ 56 1 0 4 t dist ( Q , supp ( μ ) ∖ Ω ) ≤ 4 1 dist ( Q , supp ( μ ) ∖ Ω ) ≤ 4 1 dist ( z Q , supp ( μ ) ∖ Ω ) , which proves that z ∈ Ω z\in\Omega z ∈ Ω
Now, we prove (3). Let Q ∈ W ( s ) Q\in\mathcal{W}^{(s)} Q ∈ W ( s ) Q ^ \widehat{Q} Q D μ ( s ) \mathcal{D}_{\mu}^{(s)} D μ ( s ) Q ^ \widehat{Q} Q y ′ ∈ Q ^ y'\in\widehat{Q} y ′ ∈ Q y Q ∈ supp ( μ ) ∖ Ω y_Q\in\operatorname{supp}(\mu)\setminus\Omega y Q ∈ supp ( μ ) ∖ Ω
d ( y Q , y ′ ) < ℓ ( s ) ( Q ^ ) t = A 0 ℓ ( s ) ( Q ) t . d(y_Q,y')<\frac{\ell^{(s)}(\widehat{Q})}{\mathfrak{t}}=A_0\frac{\ell^{(s)}(Q)}{\mathfrak{t}}. d ( y Q , y ′ ) < t ℓ ( s ) ( Q ) = A 0 t ℓ ( s ) ( Q ) . Thus, by the triangle inequality, for every x ∈ Q x \in Q x ∈ Q
d ( x , y Q ) ≤ diam ( Q ^ ) + d ( y Q , y ′ ) < A 0 ℓ ( s ) ( Q ) + A 0 ℓ ( s ) ( Q ) t d(x,y_Q) \le\operatorname{diam}(\widehat{Q}) + d(y_Q,y') < A_0\ell^{(s)}(Q) + \frac{A_0\ell^{(s)}(Q)}{\mathfrak{t}} d ( x , y Q ) ≤ diam ( Q ) + d ( y Q , y ′ ) < A 0 ℓ ( s ) ( Q ) + t A 0 ℓ ( s ) ( Q ) ≤ 56 A 0 K 0 ( 1 + t − 1 ) r ( s ) ( Q ) = : T 0 r ( s ) ( Q ) . \le56A_0K_0(1+\mathfrak{t}^{-1})r^{(s)}(Q) =: T_0r^{(s)}(Q). ≤ 56 A 0 K 0 ( 1 + t − 1 ) r ( s ) ( Q ) =: T 0 r ( s ) ( Q ) . To prove (4), let Q , P ∈ W ( s ) Q,P \in\mathcal{W}^{(s)} Q , P ∈ W ( s ) 10 4 B ( s ) ( Q ) ∩ 10 4 B ( s ) ( P ) ≠ ∅ 10^4B^{(s)}(Q) \cap10^4B^{(s)}(P) \ne\varnothing 1 0 4 B ( s ) ( Q ) ∩ 1 0 4 B ( s ) ( P ) = ∅ Q , P Q,P Q , P W ( s ) \mathcal{W}^{(s)} W ( s ) t \mathfrak{t} t
ℓ ( s ) ( Q ) ≤ t dist ( z Q , supp ( μ ) ∖ Ω ) ≤ t ( d ( z Q , z P ) + dist ( z P , supp ( μ ) ∖ Ω ) ) ≤ t ( 10 4 56 ( ℓ ( s ) ( Q ) + ℓ ( s ) ( P ) ) + A 0 ( 1 + t − 1 ) ℓ ( s ) ( P ) ) ≤ 1 4 ℓ ( s ) ( Q ) + 1 4 ℓ ( s ) ( P ) + 2 A 0 ℓ ( s ) ( P ) , \begin{aligned}
\ell^{(s)}(Q) \le\mathfrak{t}\operatorname{dist}(z_Q,\operatorname{supp}(\mu)\setminus\Omega) \\
&\le\mathfrak{t}\bigl(d(z_Q,z_P)+\operatorname{dist}(z_P,\operatorname{supp}(\mu)\setminus\Omega)\bigr) \\
&\le\mathfrak{t}\left(\frac{10^4}{56}\bigl(\ell^{(s)}(Q)+\ell^{(s)}(P)\bigr)+A_0(1+\mathfrak{t}^{-1})\ell^{(s)}(P)\right) \\
&\le\frac{1}{4}\ell^{(s)}(Q)+\frac{1}{4}\ell^{(s)}(P)+2A_0\ell^{(s)}(P),
\end{aligned} ℓ ( s ) ( Q ) ≤ t dist ( z Q , supp ( μ ) ∖ Ω ) ≤ t ( d ( z Q , z P ) + dist ( z P , supp ( μ ) ∖ Ω ) ) ≤ t ( 56 1 0 4 ( ℓ ( s ) ( Q ) + ℓ ( s ) ( P ) ) + A 0 ( 1 + t − 1 ) ℓ ( s ) ( P ) ) ≤ 4 1 ℓ ( s ) ( Q ) + 4 1 ℓ ( s ) ( P ) + 2 A 0 ℓ ( s ) ( P ) , which implies that ℓ ( s ) ( Q ) ≤ 4 A 0 ℓ ( s ) ( P ) \ell^{(s)}(Q) \le4A_0\ell^{(s)}(P) ℓ ( s ) ( Q ) ≤ 4 A 0 ℓ ( s ) ( P ) Q Q Q P P P ℓ ( s ) ( P ) ≤ 4 A 0 ℓ ( s ) ( Q ) \ell^{(s)}(P) \le4A_0\ell^{(s)}(Q) ℓ ( s ) ( P ) ≤ 4 A 0 ℓ ( s ) ( Q ) (292) . In particular, (292) implies that the generations of Q Q Q P P P D μ ( s ) D_\mu^{(s)} D μ ( s ) X X X
We now prove (5). We define W n d ( s ) : = W ( s ) ∖ W d b ( s ) \mathcal{W}_{nd}^{(s)} := \mathcal{W}^{(s)} \setminus\mathcal{W}_{db}^{(s)} W n d ( s ) := W ( s ) ∖ W d b ( s ) K 0 ≥ 100 K_0 \ge100 K 0 ≥ 100 ℓ = 1 \ell=1 ℓ = 1 Q ∈ W n d ( s ) Q \in\mathcal{W}_{nd}^{(s)} Q ∈ W n d ( s )
(7.27)
μ ( Q ) ≤ μ ( 100 B ( s ) ( Q ) ) ≤ K 0 − 1 μ ( 10 4 B ( s ) ( Q ) ) . (294) \mu(Q) \le\mu(100B^{(s)}(Q)) \le K_0^{-1}\mu(10^4B^{(s)}(Q)). \tag*{(294)} μ ( Q ) ≤ μ ( 100 B ( s ) ( Q )) ≤ K 0 − 1 μ ( 1 0 4 B ( s ) ( Q )) . (294) We claim that there exists C D > 0 C_D>0 C D > 0 D D D
(7.28)
∑ Q ∈ W n d ( s ) χ 10 4 B ( s ) ( Q ) ≤ C D . (295) \sum_{Q\in\mathcal{W}_{nd}^{(s)}}\chi_{10^4B^{(s)}(Q)} \le C_D. \tag*{(295)} Q ∈ W n d ( s ) ∑ χ 1 0 4 B ( s ) ( Q ) ≤ C D . (295) Note that item (2), (294) and (295) readily imply (293) . Indeed,
μ ( ⋃ Q ∈ W d b ( s ) Q ) = μ ( Ω ) − ∑ Q ∈ W n d ( s ) μ ( Q ) ≥ μ ( Ω ) − 1 K 0 ∑ Q ∈ W n d ( s ) μ ( 10 4 B ( s ) ( Q ) ) ≥ μ ( Ω ) − 1 K 0 ∫ Ω ∑ Q ∈ W n d ( s ) χ 10 4 B ( s ) ( Q ) d μ ≥ ( 1 − C D K 0 ) μ ( Ω ) > 1 2 μ ( Ω ) , \begin{aligned}
\mu\left(\bigcup_{Q\in\mathcal{W}_{db}^{(s)}}Q\right)
= \mu(\Omega)-\sum_{Q\in\mathcal{W}_{nd}^{(s)}}\mu(Q)
\ge\mu(\Omega)-\frac{1}{K_0}\sum_{Q\in\mathcal{W}_{nd}^{(s)}}\mu(10^4B^{(s)}(Q)) \\
&\ge\mu(\Omega)-\frac{1}{K_0}\int_{\Omega}\sum_{Q\in\mathcal{W}_{nd}^{(s)}}\chi_{10^4B^{(s)}(Q)}\,d\mu
\ge\left(1-\frac{C_D}{K_0}\right)\mu(\Omega)>\frac{1}{2}\mu(\Omega),
\end{aligned} μ Q ∈ W d b ( s ) ⋃ Q = μ ( Ω ) − Q ∈ W n d ( s ) ∑ μ ( Q ) ≥ μ ( Ω ) − K 0 1 Q ∈ W n d ( s ) ∑ μ ( 1 0 4 B ( s ) ( Q )) ≥ μ ( Ω ) − K 0 1 ∫ Ω Q ∈ W n d ( s ) ∑ χ 1 0 4 B ( s ) ( Q ) d μ ≥ ( 1 − K 0 C D ) μ ( Ω ) > 2 1 μ ( Ω ) , provided that C D / K 0 < 1 / 2 C_D/K_0<1/2 C D / K 0 < 1/2
Hence, we are left only with proving (295) . Fix x ∈ X x\in X x ∈ X Q ∈ W n d ( s ) Q\in\mathcal{W}_{nd}^{(s)} Q ∈ W n d ( s ) k k k x ∈ 10 4 B ( s ) ( Q ) x\in10^4B^{(s)}(Q) x ∈ 1 0 4 B ( s ) ( Q ) r ( s ) ( Q ) = s A 0 − k r^{(s)}(Q)=sA_0^{-k} r ( s ) ( Q ) = s A 0 − k B ( x , 10 4 s A 0 − k ) B(x,10^4sA_0^{-k}) B ( x , 1 0 4 s A 0 − k ) Q ′ Q' Q ′ W n d ( s ) \mathcal{W}_{nd}^{(s)} W n d ( s ) k k k x ∈ 10 4 B ( s ) ( Q ′ ) x\in10^4B^{(s)}(Q') x ∈ 1 0 4 B ( s ) ( Q ′ ) 5 B ( s ) ( Q ) ∩ 5 B ( s ) ( Q ′ ) = ∅ 5B^{(s)}(Q)\cap5B^{(s)}(Q')=\varnothing 5 B ( s ) ( Q ) ∩ 5 B ( s ) ( Q ′ ) = ∅ X X X D ⌈ log 2 ( 10 4 ) ⌉ D^{\lceil\log_2(10^4)\rceil} D ⌈ l o g 2 ( 1 0 4 )⌉ (295) is nonzero.
The claim (295) follows with C D : = 2 D ⌈ log 2 ( 10 4 ) ⌉ C_D:=2D^{\lceil\log_2(10^4)\rceil} C D := 2 D ⌈ l o g 2 ( 1 0 4 )⌉ P , P ′ ∈ W n d ( s ) P,P'\in\mathcal{W}_{nd}^{(s)} P , P ′ ∈ W n d ( s ) x ∈ 10 4 B ( s ) ( P ) ∩ 10 4 B ( s ) ( P ′ ) x\in10^4B^{(s)}(P)\cap10^4B^{(s)}(P') x ∈ 1 0 4 B ( s ) ( P ) ∩ 1 0 4 B ( s ) ( P ′ )
We now prove (289) , which completes the proof of Proposition 7.4 . We denote by E λ E_\lambda E λ ( 1 − θ 0 / 4 ) μ ( Ω λ ) (1-\theta_0/4)\mu(\Omega_\lambda) ( 1 − θ 0 /4 ) μ ( Ω λ )
We first assume that ∅ ≠ Ω λ ⊊ supp ( μ ) \varnothing\ne\Omega_\lambda\subsetneq\operatorname{supp}(\mu) ∅ = Ω λ ⊊ supp ( μ ) s s s (291) , and we denote by W ( s ) \mathcal{W}^{(s)} W ( s ) Lemma 7.6 with Ω = Ω λ \Omega=\Omega_\lambda Ω = Ω λ Q ∈ W d b ( s ) Q\in\mathcal{W}^{(s)}_{db} Q ∈ W d b ( s ) r : = r ( s ) ( Q ) r:=r^{(s)}(Q) r := r ( s ) ( Q ) Q ⊂ 28 B ( s ) ( Q ) Q\subset28B^{(s)}(Q) Q ⊂ 28 B ( s ) ( Q ) N D N_D N D V j : = B ( c j , r ) V_j:=B(c_j,r) V j := B ( c j , r ) c j ∈ Q c_j\in Q c j ∈ Q N D N_D N D D D D Q Q Q
Let j j j V j ∩ E λ ∩ G Q ≠ ∅ V_j\cap E_\lambda\cap G_Q\ne\varnothing V j ∩ E λ ∩ G Q = ∅ x j ∈ V j ∩ E λ ∩ G Q x_j\in V_j\cap E_\lambda\cap G_Q x j ∈ V j ∩ E λ ∩ G Q f j : = f χ B ( c j , 4 r ) f_j:=f\chi_{B(c_j,4r)} f j := f χ B ( c j , 4 r ) x j ∈ B ( c j , 4 r ) x_j\in B(c_j,4r) x j ∈ B ( c j , 4 r ) M ~ μ f ( x j ) ≤ ε ′ λ \widetilde{M}_{\mu}f(x_j)\le\varepsilon'\lambda M μ f ( x j ) ≤ ε ′ λ B ( c j , 20 r ) ⊂ 100 B ( s ) ( Q ) B(c_j,20r)\subset100B^{(s)}(Q) B ( c j , 20 r ) ⊂ 100 B ( s ) ( Q ) Q Q Q
∥ f j ∥ L 1 ( μ ) ≤ ε ′ λ μ ( B ( c j , 20 r ) ) ≤ K 0 ε ′ λ μ ( Q ) . (296) \lVert f_j\rVert_{L^1(\mu)}\le\varepsilon'\lambda\mu(B(c_j,20r))\le K_0\varepsilon'\lambda\mu(Q). \tag*{(296)} ∥ f j ∥ L 1 ( μ ) ≤ ε ′ λ μ ( B ( c j , 20 r )) ≤ K 0 ε ′ λ μ ( Q ) . (296) Let y Q y_Q y Q Lemma 7.6 , let x ∈ Q ∩ V j x\in Q\cap V_j x ∈ Q ∩ V j ρ ≥ h \rho\ge h ρ ≥ h u : = max { ρ , 4 ( T 0 + 2 ) r } u:=\max\{\rho,4(T_0+2)r\} u := max { ρ , 4 ( T 0 + 2 ) r } B j : = B ( x j , 4 ( T 0 + 2 ) r ) B_j:=B(x_j,4(T_0+2)r) B j := B ( x j , 4 ( T 0 + 2 ) r ) B ( c j , 4 r ) ⊂ B j B(c_j,4r)\subset B_j B ( c j , 4 r ) ⊂ B j
T μ , ρ ( f − f j ) ( x ) − T μ , u f ( y Q ) = ∫ B j ( K ( x , z ) χ A ( z ) − K ( y Q , z ) χ { d ( y Q , z ) > u } ( z ) ) f ( z ) d μ ( z ) + ∫ X ∖ B j ( K ( x , z ) − K ( y Q , z ) ) χ { d ( y Q , z ) > u } ( z ) f ( z ) d μ ( z ) + ∫ X ∖ B j K ( x , z ) ( χ { d ( x , z ) > ρ } ( z ) − χ { d ( y Q , z ) > u } ( z ) ) f ( z ) d μ ( z ) = : J ~ 1 + J ~ 2 + J ~ 3 , (297) \begin{aligned}
T_{\mu,\rho}(f-f_j)(x)-T_{\mu,u}f(y_Q)
={}&\int_{B_j}\left(K(x,z)\chi_A(z)-K(y_Q,z)\chi_{\{d(y_Q,z)>u\}}(z)\right)f(z)\,d\mu(z)\\
&+\int_{X\setminus B_j}\left(K(x,z)-K(y_Q,z)\right)\chi_{\{d(y_Q,z)>u\}}(z)f(z)\,d\mu(z)\\
&+\int_{X\setminus B_j}K(x,z)\left(\chi_{\{d(x,z)>\rho\}}(z)-\chi_{\{d(y_Q,z)>u\}}(z)\right)f(z)\,d\mu(z)\\
=:{}&\widetilde{J}_1+\widetilde{J}_2+\widetilde{J}_3, \tag*{(297)}
\end{aligned} T μ , ρ ( f − f j ) ( x ) − T μ , u f ( y Q ) = =: ∫ B j ( K ( x , z ) χ A ( z ) − K ( y Q , z ) χ { d ( y Q , z ) > u } ( z ) ) f ( z ) d μ ( z ) + ∫ X ∖ B j ( K ( x , z ) − K ( y Q , z ) ) χ { d ( y Q , z ) > u } ( z ) f ( z ) d μ ( z ) + ∫ X ∖ B j K ( x , z ) ( χ { d ( x , z ) > ρ } ( z ) − χ { d ( y Q , z ) > u } ( z ) ) f ( z ) d μ ( z ) J 1 + J 2 + J 3 , (297) where A : = { z : d ( x , z ) > ρ } ∖ B ( c j , 4 r ) A:=\{z:d(x,z)>\rho\}\setminus B(c_j,4r) A := { z : d ( x , z ) > ρ } ∖ B ( c j , 4 r ) d ( x , x j ) < 2 r d(x,x_j)<2r d ( x , x j ) < 2 r d ( x , z ) ≥ 3 r d(x,z)\ge3r d ( x , z ) ≥ 3 r z ∉ B ( c j , 4 r ) z\notin B(c_j,4r) z ∈ / B ( c j , 4 r ) d ( x , y Q ) ≤ T 0 r d(x,y_Q)\le T_0r d ( x , y Q ) ≤ T 0 r M ~ μ \widetilde{M}_\mu M μ n n n μ \mu μ
∫ B ( x j , t ) ∣ f ∣ d μ ≤ M ~ μ f ( x j ) μ ( B ( x j , 5 t ) ) ≲ t n M ~ μ f ( x j ) , t > 0. (298) \int_{B(x_j,t)}|f|\,d\mu\le\widetilde{M}_\mu f(x_j)\mu(B(x_j,5t))\lesssim t^n\widetilde{M}_\mu f(x_j),\qquad t>0. \tag*{(298)} ∫ B ( x j , t ) ∣ f ∣ d μ ≤ M μ f ( x j ) μ ( B ( x j , 5 t )) ≲ t n M μ f ( x j ) , t > 0. (298) Since the kernels in J ~ 1 \widetilde{J}_1 J 1 C r − n Cr^{-n} C r − n (298) gives ∣ J ~ 1 ∣ ≲ M ~ μ f ( x j ) |\widetilde{J}_1|\lesssim\widetilde{M}_\mu f(x_j) ∣ J 1 ∣ ≲ M μ f ( x j )
If z ∉ B j z\notin B_j z ∈ / B j d ( x , z ) ≥ d ( x j , z ) / 2 ≥ 2 T 0 r ≥ 2 d ( x , y Q ) d(x,z)\ge d(x_j,z)/2\ge2T_0r\ge2d(x,y_Q) d ( x , z ) ≥ d ( x j , z ) /2 ≥ 2 T 0 r ≥ 2 d ( x , y Q ) (298) , applied on the annuli B ( x j , 2 k + 1 t j ) ∖ B ( x j , 2 k t j ) B(x_j,2^{k+1}t_j)\setminus B(x_j,2^kt_j) B ( x j , 2 k + 1 t j ) ∖ B ( x j , 2 k t j ) t j : = 4 ( T 0 + 2 ) r t_j:=4(T_0+2)r t j := 4 ( T 0 + 2 ) r k ≥ 0 k\ge0 k ≥ 0
∣ J ~ 2 ∣ ≲ ∫ X ∖ B j r γ ∣ f ( z ) ∣ d ( x j , z ) n + γ d μ ( z ) ≲ ∑ k ≥ 0 2 − k γ M ~ μ f ( x j ) ≲ M ~ μ f ( x j ) . |\widetilde{J}_2|\lesssim\int_{X\setminus B_j}\frac{r^\gamma|f(z)|}{d(x_j,z)^{n+\gamma}}\,d\mu(z)\lesssim\sum_{k\ge0}2^{-k\gamma}\widetilde{M}_\mu f(x_j)\lesssim\widetilde{M}_\mu f(x_j). ∣ J 2 ∣ ≲ ∫ X ∖ B j d ( x j , z ) n + γ r γ ∣ f ( z ) ∣ d μ ( z ) ≲ k ≥ 0 ∑ 2 − k γ M μ f ( x j ) ≲ M μ f ( x j ) . Moreover ∣ d ( x , z ) − d ( y Q , z ) ∣ ≤ T 0 r ≤ u / 4 |d(x,z)-d(y_Q,z)|\le T_0r\le u/4 ∣ d ( x , z ) − d ( y Q , z ) ∣ ≤ T 0 r ≤ u /4 d ( x , z ) ≥ 2 ( T 0 + 2 ) r d(x,z)\ge2(T_0+2)r d ( x , z ) ≥ 2 ( T 0 + 2 ) r z ∉ B j z\notin B_j z ∈ / B j J ~ 3 \widetilde{J}_3 J 3 z ∈ B ( x j , 2 u ) z\in B(x_j,2u) z ∈ B ( x j , 2 u ) d ( x , z ) ≥ u / 2 d(x,z)\ge u/2 d ( x , z ) ≥ u /2 (298) gives
∣ J ~ 3 ∣ ≲ u − n ∫ B ( x j , 2 u ) ∣ f ∣ d μ ≲ M ~ μ f ( x j ) . |\widetilde{J}_3|\lesssim u^{-n}\int_{B(x_j,2u)}|f|\,d\mu\lesssim\widetilde{M}_\mu f(x_j). ∣ J 3 ∣ ≲ u − n ∫ B ( x j , 2 u ) ∣ f ∣ d μ ≲ M μ f ( x j ) . Gathering the bounds for the terms in (297) , we obtain
∣ T μ , ρ ( f − f j ) ( x ) − T μ , u f ( y Q ) ∣ ≤ C M ~ μ f ( x j ) , (299) \left|T_{\mu,\rho}(f-f_j)(x)-T_{\mu,u}f(y_Q)\right|\le C\widetilde{M}_\mu f(x_j), \tag*{(299)} ∣ T μ , ρ ( f − f j ) ( x ) − T μ , u f ( y Q ) ∣ ≤ C M μ f ( x j ) , (299) where C C C r r r h h h ρ \rho ρ
Since u ≥ h u\ge h u ≥ h y Q ∈ supp ( μ ) ∖ Ω λ y_Q\in\operatorname{supp}(\mu)\setminus\Omega_\lambda y Q ∈ supp ( μ ) ∖ Ω λ M ~ μ f ( x j ) ≤ ε ′ λ \widetilde{M}_\mu f(x_j)\le\varepsilon'\lambda M μ f ( x j ) ≤ ε ′ λ (299) implies
T μ , ♯ , h ( f − f j ) ( x ) ≤ ( 1 + C ε ′ ) λ . T_{\mu,\sharp,h}(f-f_j)(x)\le(1+C\varepsilon')\lambda. T μ , ♯ , h ( f − f j ) ( x ) ≤ ( 1 + C ε ′ ) λ . We choose ε ′ \varepsilon' ε ′ C ε ′ ≤ ε / 2 C\varepsilon'\le\varepsilon/2 C ε ′ ≤ ε /2 x ∈ E λ ∩ G Q ∩ V j x\in E_\lambda\cap G_Q\cap V_j x ∈ E λ ∩ G Q ∩ V j
T ♯ ( f j μ ) ( x ) ≥ T μ , ♯ , h f ( x ) − T μ , ♯ , h ( f − f j ) ( x ) > ε 2 λ . T_{\sharp}(f_j\mu)(x)\ge T_{\mu,\sharp,h}f(x)-T_{\mu,\sharp,h}(f-f_j)(x)>\frac{\varepsilon}{2}\lambda. T ♯ ( f j μ ) ( x ) ≥ T μ , ♯ , h f ( x ) − T μ , ♯ , h ( f − f j ) ( x ) > 2 ε λ . Hence, (290) with ω = f j μ \omega=f_j\mu ω = f j μ (296) give
μ ( E λ ∩ G Q ∩ V j ) ≤ 2 C ε λ ∥ f j ∥ L 1 ( μ ) ≤ 2 C K 0 ε ′ ε μ ( Q ) . \mu(E_\lambda\cap G_Q\cap V_j)\leq\frac{2C}{\varepsilon\lambda}\|f_j\|_{L^1(\mu)}\leq2CK_0\frac{\varepsilon'}{\varepsilon}\mu(Q). μ ( E λ ∩ G Q ∩ V j ) ≤ ε λ 2 C ∥ f j ∥ L 1 ( μ ) ≤ 2 C K 0 ε ε ′ μ ( Q ) . Summing over j j j ε ′ \varepsilon' ε ′ μ ( E λ ∩ G Q ) ≤ θ 0 μ ( Q ) / 2 \mu(E_\lambda\cap G_Q)\leq\theta_0\mu(Q)/2 μ ( E λ ∩ G Q ) ≤ θ 0 μ ( Q ) /2 μ ( G Q ) ≥ θ 0 μ ( Q ) \mu(G_Q)\geq\theta_0\mu(Q) μ ( G Q ) ≥ θ 0 μ ( Q ) μ ( E λ ∩ Q ) ≤ ( 1 − θ 0 / 2 ) μ ( Q ) \mu(E_\lambda\cap Q)\leq(1-\theta_0/2)\mu(Q) μ ( E λ ∩ Q ) ≤ ( 1 − θ 0 /2 ) μ ( Q ) E λ ∩ supp ( μ ) ⊂ Ω λ E_\lambda\cap\operatorname{supp}(\mu)\subset\Omega_\lambda E λ ∩ supp ( μ ) ⊂ Ω λ (293) gives
μ ( E λ ) ≤ μ ( Ω λ ) − θ 0 2 ∑ Q ∈ W μ , 0 ( s ) μ ( Q ) ≤ ( 1 − θ 0 4 ) μ ( Ω λ ) . \mu(E_\lambda)\leq\mu(\Omega_\lambda)-\frac{\theta_0}{2}\sum_{Q\in\mathcal{W}^{(s)}_{\mu,0}}\mu(Q)\leq\left(1-\frac{\theta_0}{4}\right)\mu(\Omega_\lambda). μ ( E λ ) ≤ μ ( Ω λ ) − 2 θ 0 Q ∈ W μ , 0 ( s ) ∑ μ ( Q ) ≤ ( 1 − 4 θ 0 ) μ ( Ω λ ) . We now assume that Ω λ = supp ( μ ) \Omega_\lambda=\operatorname{supp}(\mu) Ω λ = supp ( μ ) (289) is trivial, that E λ ≠ ∅ E_\lambda\neq\varnothing E λ = ∅ supp ( μ ) \operatorname{supp}(\mu) supp ( μ ) μ ( X ) < ∞ \mu(X)<\infty μ ( X ) < ∞ s > diam ( supp ( μ ) ) s>\operatorname{diam}(\operatorname{supp}(\mu)) s > diam ( supp ( μ )) 5 B ( s ) ( Q ) 5B^{(s)}(Q) 5 B ( s ) ( Q ) Q ∈ D μ , 0 ( s ) Q\in\mathcal{D}^{(s)}_{\mu,0} Q ∈ D μ , 0 ( s ) supp ( μ ) \operatorname{supp}(\mu) supp ( μ ) diam ( supp ( μ ) ) \operatorname{diam}(\operatorname{supp}(\mu)) diam ( supp ( μ )) Q = supp ( μ ) Q=\operatorname{supp}(\mu) Q = supp ( μ ) supp ( μ ) ⊂ B ( s ) ( Q ) \operatorname{supp}(\mu)\subset B^{(s)}(Q) supp ( μ ) ⊂ B ( s ) ( Q ) Q Q Q x ∈ E λ x\in E_\lambda x ∈ E λ M ~ μ f ( x ) ≤ ε ′ λ \widetilde{M}_\mu f(x)\leq\varepsilon'\lambda M μ f ( x ) ≤ ε ′ λ x x x supp ( μ ) \operatorname{supp}(\mu) supp ( μ ) ∥ f ∥ L 1 ( μ ) ≤ ε ′ λ μ ( X ) \|f\|_{L^1(\mu)}\leq\varepsilon'\lambda\mu(X) ∥ f ∥ L 1 ( μ ) ≤ ε ′ λ μ ( X ) (290) with ω = f μ \omega=f\mu ω = f μ μ ( E λ ∩ G Q ) ≤ C ε ′ μ ( X ) \mu(E_\lambda\cap G_Q)\leq C\varepsilon'\mu(X) μ ( E λ ∩ G Q ) ≤ C ε ′ μ ( X ) ε ′ \varepsilon' ε ′
μ ( E λ ) ≤ μ ( X ∖ G Q ) + θ 0 2 μ ( X ) ≤ ( 1 − θ 0 2 ) μ ( Ω λ ) , \mu(E_\lambda)\leq\mu(X\setminus G_Q)+\frac{\theta_0}{2}\mu(X)\leq\left(1-\frac{\theta_0}{2}\right)\mu(\Omega_\lambda), μ ( E λ ) ≤ μ ( X ∖ G Q ) + 2 θ 0 μ ( X ) ≤ ( 1 − 2 θ 0 ) μ ( Ω λ ) , which proves (289) also in this case.
Finally, we choose ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) ( 1 + ε ) 2 ( 1 − θ 0 / 4 ) < 1 (1+\varepsilon)^2(1-\theta_0/4)<1 ( 1 + ε ) 2 ( 1 − θ 0 /4 ) < 1 ε ′ \varepsilon' ε ′
{ T μ , h ♯ f > ( 1 + ε ) λ } ⊂ E λ ∪ { M ~ μ f > ε ′ λ } , \{T_{\mu,h}^{\sharp}f>(1+\varepsilon)\lambda\}\subset E_\lambda\cup\{\widetilde{M}_\mu f>\varepsilon'\lambda\}, { T μ , h ♯ f > ( 1 + ε ) λ } ⊂ E λ ∪ { M μ f > ε ′ λ } , integrating (289) with respect to 2 λ d λ 2\lambda\,d\lambda 2 λ d λ
∥ T μ , h ♯ f ∥ L 2 ( μ ) 2 ≤ ( 1 + ε ) 2 ( 1 − θ 0 4 ) ∥ T μ , h ♯ f ∥ L 2 ( μ ) 2 + ( 1 + ε ) 2 ( ε ′ ) 2 ∥ M ~ μ f ∥ L 2 ( μ ) 2 . \|T_{\mu,h}^{\sharp}f\|_{L^2(\mu)}^2\leq(1+\varepsilon)^2\left(1-\frac{\theta_0}{4}\right)\|T_{\mu,h}^{\sharp}f\|_{L^2(\mu)}^2+\frac{(1+\varepsilon)^2}{(\varepsilon')^2}\|\widetilde{M}_\mu f\|_{L^2(\mu)}^2. ∥ T μ , h ♯ f ∥ L 2 ( μ ) 2 ≤ ( 1 + ε ) 2 ( 1 − 4 θ 0 ) ∥ T μ , h ♯ f ∥ L 2 ( μ ) 2 + ( ε ′ ) 2 ( 1 + ε ) 2 ∥ M μ f ∥ L 2 ( μ ) 2 . Since T μ , h f ∈ L 2 ( μ ) T_{\mu,h}f\in L^2(\mu) T μ , h f ∈ L 2 ( μ ) M ~ μ \widetilde{M}_\mu M μ L 2 ( μ ) L^2(\mu) L 2 ( μ ) ∥ T μ , h ♯ f ∥ L 2 ( μ ) ≲ ∥ f ∥ L 2 ( μ ) \|T_{\mu,h}^{\sharp}f\|_{L^2(\mu)}\lesssim\|f\|_{L^2(\mu)} ∥ T μ , h ♯ f ∥ L 2 ( μ ) ≲ ∥ f ∥ L 2 ( μ ) h h h ∣ T μ , h f ∣ ≤ T μ , h ♯ f |T_{\mu,h}f|\leq T_{\mu,h}^{\sharp}f ∣ T μ , h f ∣ ≤ T μ , h ♯ f h > 0 h>0 h > 0 (288) , which completes the proof of Proposition 7.4.
Completion of the proof
We now prove Proposition 3.1 by verifying the hypotheses of Proposition 7.4, and we obtain the required big pieces by applying the construction of Sections 3 and 4, together with the bounds of Section 7.1, to the testing measures μ ∣ B \mu|_B μ ∣ B
Proof of Proposition 3.1. Let B 0 ∈ B B_0\in\mathcal{B} B 0 ∈ B B 0 B_0 B 0
sup ε > 0 sup R ∈ Top ( B 0 ) ∥ T Φ ~ R , μ ∣ 84 B ( R ) ε ∥ L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≲ 1. (300) \sup_{\varepsilon>0}\sup_{R\in\operatorname{Top}(B_0)}
\left\|T_{\widetilde{\Phi}_R,\mu|_{84B(R)} }^\varepsilon\right\|_{L^2(\mu|_{84B(R)})\to L^2(\mu|_{84B(R)})}\lesssim1. \tag*{(300)} ε > 0 sup R ∈ Top ( B 0 ) sup T Φ R , μ ∣ 84 B ( R ) ε L 2 ( μ ∣ 84 B ( R ) ) → L 2 ( μ ∣ 84 B ( R ) ) ≲ 1. (300) so the assumption (146) of Corollary 4.13 holds with C T C_T C T ε \varepsilon ε T T T d ( x , y ) ≤ h d(x,y)\leq h d ( x , y ) ≤ h (36) , the bound (300) also holds for the whole operator T Φ ~ R , μ ∣ 84 B ( R ) T_{\widetilde{\Phi}_R,\mu|_{84B(R)}} T Φ R , μ ∣ 84 B ( R )
Let g ∈ L ∞ ( μ ) g\in L^\infty(\mu) g ∈ L ∞ ( μ ) B 0 B_0 B 0 (123) , (141) , Lemma 4.8, and the proofs of Lemmas 4.9, 4.10, and 4.11, there exists a non-negative function A g A_g A g
∫ B 0 A g d μ ≲ ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) \int_{B_0}A_g\,d\mu\lesssim\|g\|_{L^\infty(\mu)}\mu(B_0) ∫ B 0 A g d μ ≲ ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) and, after taking the supremum over the truncation parameter in (127) , we get
T μ , ♯ g ( x ) ≲ A g ( x ) + ∑ R ∈ Top ~ ( B 0 ) χ R ( x ) T Φ ~ R , ♯ ( χ 84 B ( R ) g μ ) ( x ) , for μ -a.e. x ∈ B 0 . T_{\mu,\sharp}g(x) \lesssim A_g(x) + \sum_{R\in\widetilde{\operatorname{Top}}(B_0)}\chi_R(x)T_{\widetilde{\Phi}_R,\sharp}(\chi_{84B(R)}g\mu)(x), \qquad\text{for }\mu\text{-a.e. }x\in B_0. T μ , ♯ g ( x ) ≲ A g ( x ) + R ∈ Top ( B 0 ) ∑ χ R ( x ) T Φ R , ♯ ( χ 84 B ( R ) g μ ) ( x ) , for μ -a.e. x ∈ B 0 . For R ∈ Top ~ ( B 0 ) R\in\widetilde{\operatorname{Top}}(B_0) R ∈ Top ( B 0 ) m 1 = m 2 = m 3 = μ ∣ 84 B ( R ) m_1=m_2=m_3=\mu|_{84B(R)} m 1 = m 2 = m 3 = μ ∣ 84 B ( R ) Ξ ≡ 0 \Xi\equiv0 Ξ ≡ 0 δ 0 = h \delta_0=h δ 0 = h (300) for the whole operator, the Cauchy–Schwarz inequality, and μ ( 84 B ( R ) ) ≤ K 0 μ ( R ) \mu(84B(R))\le K_0\mu(R) μ ( 84 B ( R )) ≤ K 0 μ ( R )
∫ R T Φ ~ R , ♯ ( χ 84 B ( R ) g μ ) d μ ≲ μ ( R ) 1 / 2 ∥ g ∥ L 2 ( μ ∣ 84 B ( R ) ) ≲ ∥ g ∥ L ∞ ( μ ) μ ( R ) . \int_R T_{\widetilde{\Phi}_R,\sharp}(\chi_{84B(R)}g\mu)\,d\mu\lesssim\mu(R)^{1/2}\|g\|_{L^2(\mu|_{84B(R)})}\lesssim\|g\|_{L^\infty(\mu)}\mu(R). ∫ R T Φ R , ♯ ( χ 84 B ( R ) g μ ) d μ ≲ μ ( R ) 1/2 ∥ g ∥ L 2 ( μ ∣ 84 B ( R ) ) ≲ ∥ g ∥ L ∞ ( μ ) μ ( R ) . Hence, summing over R ∈ Top ~ ( B 0 ) R\in\widetilde{\operatorname{Top}}(B_0) R ∈ Top ( B 0 ) T ∗ T^* T ∗ C > 0 C>0 C > 0 B 0 B_0 B 0 g g g h h h L L L
∫ B 0 ( T μ , ♯ g + ( T ∗ ) μ , ♯ g ) d μ ≤ C ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) . (301) \int_{B_0}\left(T_{\mu,\sharp}g+(T^*)_{\mu,\sharp}g\right)\,d\mu\le C\|g\|_{L^\infty(\mu)}\mu(B_0). \tag*{(301)} ∫ B 0 ( T μ , ♯ g + ( T ∗ ) μ , ♯ g ) d μ ≤ C ∥ g ∥ L ∞ ( μ ) μ ( B 0 ) . (301) We may assume that μ ≠ 0 \mu\ne0 μ = 0 W : = supp ( μ ) W:=\operatorname{supp}(\mu) W := supp ( μ ) s > 0 s>0 s > 0 D μ ( s ) \mathcal{D}^{(s)}_\mu D μ ( s ) μ \mu μ K 0 K_0 K 0 R ∈ D μ d b , ( s ) R\in\mathcal{D}^{\mathrm{db},(s)}_\mu R ∈ D μ db , ( s ) B : = B ~ ( s ) ( R ) B:=\widetilde{B}^{(s)}(R) B := B ( s ) ( R ) B B B W W W B ∩ W ⊂ R B\cap W\subset R B ∩ W ⊂ R μ ( R ) ≤ β μ ( B ) \mu(R)\le\beta\mu(B) μ ( R ) ≤ β μ ( B ) μ ( Λ B ) ≤ β μ ( δ B ) \mu(\Lambda B)\le\beta\mu(\delta B) μ ( Λ B ) ≤ β μ ( δ B )
Case I. We first assume that r ( B ) ≤ diam W r(B)\le\operatorname{diam}W r ( B ) ≤ diam W C 3 ≥ 40 C_3\ge40 C 3 ≥ 40 B B B B \mathcal{B} B B 1 = B 2 = B B_1=B_2=B B 1 = B 2 = B ν ^ B = ν ^ B ∗ = μ ∣ B \widehat{\nu}_B=\widehat{\nu}_B^*=\mu|_B ν B = ν B ∗ = μ ∣ B 1 1 1 μ ∣ B + μ ∣ X ∖ B = μ \mu|_B+\mu|_{X\setminus B}=\mu μ ∣ B + μ ∣ X ∖ B = μ 1 / 200 1/200 1/200
By (301) , with B B B B 0 B_0 B 0 g = χ B g=\chi_B g = χ B
U B : = { x ∈ B : T ♯ ( μ ∣ B ) ( x ) > 200 C } U_B:=\{x\in B:T_{\sharp}(\mu|_B)(x)>200C\} U B := { x ∈ B : T ♯ ( μ ∣ B ) ( x ) > 200 C } and the analogous set U B ∗ U_B^* U B ∗ T ∗ T^* T ∗ μ \mu μ μ ( B ) / 200 \mu(B)/200 μ ( B ) /200 C 2 = 200 C C_2=200C C 2 = 200 C
Since the augmented measures equal μ \mu μ S 1 \mathcal{S}_1 S 1 S 2 \mathcal{S}_2 S 2 1 1 1 μ \mu μ T T T T ∗ T^* T ∗ B ∩ E T ⊂ U B B\cap\mathcal{E}_T\subset U_B B ∩ E T ⊂ U B B ∩ E T ∗ ⊂ U B ∗ B\cap\mathcal{E}_{T^*}\subset U_B^* B ∩ E T ∗ ⊂ U B ∗
Let θ \theta θ (64) gives μ ( B ∖ ( 1 − θ ) B ) ≤ μ ( B ) / 400 \mu(B\setminus(1-\theta)B)\le\mu(B)/400 μ ( B ∖ ( 1 − θ ) B ) ≤ μ ( B ) /400 S 1 \mathcal{S}_1 S 1 S 2 \mathcal{S}_2 S 2 (61) of the two systems are E T ∪ U B \mathcal{E}_T\cup U_B E T ∪ U B E T ∗ ∪ U B ∗ \mathcal{E}_{T^*}\cup U_B^* E T ∗ ∪ U B ∗ B B B U B U_B U B U B ∗ U_B^* U B ∗ B ∩ W ⊂ R B\cap W\subset R B ∩ W ⊂ R μ ( U B ∪ U B ∗ ) ≤ μ ( B ) / 100 \mu(U_B\cup U_B^*)\le\mu(B)/100 μ ( U B ∪ U B ∗ ) ≤ μ ( B ) /100
F ⊂ ( ( 1 − θ ) B ∩ R ) ∖ ( U B ∪ U B ∗ ) with μ ( F ) ≥ μ ( B ) / 2. F\subset((1-\theta)B\cap R)\setminus(U_B\cup U_B^*)\qquad\text{with}\qquad\mu(F)\ge\mu(B)/2. F ⊂ (( 1 − θ ) B ∩ R ) ∖ ( U B ∪ U B ∗ ) with μ ( F ) ≥ μ ( B ) /2. Since F F F F 1 ( w 1 ) = F 2 ( w 2 ) = F F_1(w_1)=F_2(w_2)=F F 1 ( w 1 ) = F 2 ( w 2 ) = F w 1 , w 2 ∈ Ω w_1,w_2\in\Omega w 1 , w 2 ∈ Ω Ψ : = dist ( ⋅ , F ) \Psi:=\operatorname{dist}(\cdot,F) Ψ := dist ( ⋅ , F ) Ξ w = Ψ \Xi_w=\Psi Ξ w = Ψ w ∈ Ω 2 w\in\Omega^2 w ∈ Ω 2
Moreover, Lemma 3.15 gives the suppressed testing estimates for μ ∣ B \mu|_B μ ∣ B F F F E T ∪ E T ∗ \mathcal{E}_T\cup\mathcal{E}_{T^*} E T ∪ E T ∗ ∣ ν i ∣ ( Q ) = μ ( Q ) |\nu_i|(Q)=\mu(Q) ∣ ν i ∣ ( Q ) = μ ( Q ) Q Q Q i = 1 , 2 i=1,2 i = 1 , 2 ∣ ν i ∣ = μ |\nu_i|=\mu ∣ ν i ∣ = μ C 2 = 200 C C_2=200C C 2 = 200 C E 2 \mathcal{E}_2 E 2 B 1 = B 2 = B B_1=B_2=B B 1 = B 2 = B Ξ w = Ψ \Xi_w=\Psi Ξ w = Ψ T Ψ , μ ∣ B , ε T_{\Psi,\mu|B,\varepsilon} T Ψ , μ ∣ B , ε L 2 ( μ ∣ B ) L^2(\mu|B) L 2 ( μ ∣ B ) ε > 0 \varepsilon>0 ε > 0 K Ψ = K K_{\Psi}=K K Ψ = K F × F F\times F F × F Lemma 3.13 , the set G R : = F ⊂ R G_R:=F\subset R G R := F ⊂ R
sup ε > 0 ∥ T μ ∣ G R , ε ∥ L 2 ( μ ∣ G R ) → L 2 ( μ ∣ G R ) ≲ 1 and μ ( G R ) ≥ 1 2 μ ( B ) ≥ ( 2 β ) − 1 μ ( R ) , (302) \sup_{\varepsilon>0}\|T_{\mu|G_R,\varepsilon}\|_{L^2(\mu|G_R)\to L^2(\mu|G_R)}\lesssim1
\qquad\text{and}\qquad
\mu(G_R)\ge\frac{1}{2}\mu(B)\ge(2\beta)^{-1}\mu(R), \tag*{(302)} ε > 0 sup ∥ T μ ∣ G R , ε ∥ L 2 ( μ ∣ G R ) → L 2 ( μ ∣ G R ) ≲ 1 and μ ( G R ) ≥ 2 1 μ ( B ) ≥ ( 2 β ) − 1 μ ( R ) , (302) with constants independent of R R R s s s
Case II. It remains to consider the case r ( B ) > diam W r(B)>\operatorname{diam}W r ( B ) > diam W W W W a : = diam W a:=\operatorname{diam}W a := diam W μ \mu μ
r ( s ) ( R ) ≥ 2 Λ c ∗ r ( B ) > a . r^{(s)}(R)\ge\frac{2\Lambda}{c_*}r(B)>a. r ( s ) ( R ) ≥ c ∗ 2Λ r ( B ) > a . Since the center of R R R W W W W ⊂ B ( s ) ( R ) W\subset B^{(s)}(R) W ⊂ B ( s ) ( R ) R = W R=W R = W (26) . We set s 0 : = 3 a / 2 s_0:=3a/2 s 0 := 3 a /2 Proposition 7.4 , the lattice D μ ( s 0 ) \mathcal{D}^{(s_0)}_\mu D μ ( s 0 ) W W W s 0 ≤ 2 a s_0\le2a s 0 ≤ 2 a W W W B \mathcal{B} B Lemma A.1 . Hence, the construction above, with s 0 s_0 s 0 s s s G W ⊂ W G_W\subset W G W ⊂ W (302) with R = W R=W R = W G R : = G W G_R:=G_W G R := G W
Therefore, the hypotheses of Proposition 7.4 hold for every s > 0 s>0 s > 0 θ 0 = ( 2 β ) − 1 \theta_0=(2\beta)^{-1} θ 0 = ( 2 β ) − 1 Proposition 7.4 gives the L 2 ( μ ) L^2(\mu) L 2 ( μ ) T T T h h h L L L
Since the bound in Proposition 3.1 is independent of h h h L L L § 3.1 and prove the main result of the paper.
Proof of Theorem 1.3 . The theorem follows from Proposition 3.1 and the reduction to the auxiliary kernel in Lemma 3.2 .
Applications to caloric measure
In this section we prove Theorem 1.9 , and we use the notation of § 1.2 . We set
B + ( X , r ) : = B ( X , r ) ∩ { ( Y , s ) : s > t } , (303) B^+(X,r):=B(X,r)\cap\{(Y,s):s>t\}, \tag*{(303)} B + ( X , r ) := B ( X , r ) ∩ {( Y , s ) : s > t } , (303) and
B − ( X , r ) : = B ( X , r ) ∩ { ( Y , s ) : s < t } (304) B^-(X,r):=B(X,r)\cap\{(Y,s):s<t\} \tag*{(304)} B − ( X , r ) := B ( X , r ) ∩ {( Y , s ) : s < t } (304) to denote the time-forward and time-backward half-balls, respectively. For a ball B = B ( X , r ) B=B(X,r) B = B ( X , r ) B ± : = B ± ( X , r ) B^\pm:=B^\pm(X,r) B ± := B ± ( X , r )
As in the introduction, for a finite measure ν \nu ν P n \mathbb{P}^n P n T ε ν T_\varepsilon\nu T ε ν T ε ∗ ν T_\varepsilon^*\nu T ε ∗ ν K ( X , Y ) = ∇ X Γ ( X , Y ) K(X,Y)=\nabla_X\Gamma(X,Y) K ( X , Y ) = ∇ X Γ ( X , Y ) K ( Y , X ) = ∇ Y Γ ( Y , X ) K(Y,X)=\nabla_Y\Gamma(Y,X) K ( Y , X ) = ∇ Y Γ ( Y , X )
For f ∈ L 1 ( σ ) f\in L^1(\sigma) f ∈ L 1 ( σ ) T σ , ε f : = T ε ( f σ ) T_{\sigma,\varepsilon}f:=T_\varepsilon(f\sigma) T σ , ε f := T ε ( f σ ) T σ , ε ∗ f : = T ε ∗ ( f σ ) T_{\sigma,\varepsilon}^*f:=T_\varepsilon^*(f\sigma) T σ , ε ∗ f := T ε ∗ ( f σ ) T σ , ♯ f T_{\sigma,\sharp}f T σ , ♯ f T σ , ♯ ∗ f T_{\sigma,\sharp}^*f T σ , ♯ ∗ f T ♯ ν T_\sharp\nu T ♯ ν T ♯ ∗ ν T_\sharp^*\nu T ♯ ∗ ν ν \nu ν
Throughout this section, Ω ⊂ P n \Omega\subset\mathbb{P}^n Ω ⊂ P n n ≥ 1 n\ge1 n ≥ 1 σ \sigma σ ∂ Ω \partial\Omega ∂ Ω
σ ( B ( X , r ) ) ≤ C 0 r n + 1 for every X ∈ P n , r > 0. \sigma(B(X,r))\le C_0r^{n+1}\qquad\text{for every }X\in\mathbb{P}^n,\ r>0. σ ( B ( X , r )) ≤ C 0 r n + 1 for every X ∈ P n , r > 0. For a pole P ∈ Ω P\in\Omega P ∈ Ω Ω \Omega Ω ω P \omega^P ω P ω ∗ P \omega_*^P ω ∗ P ∂ e Ω , ∂ e ∗ Ω ⊂ ∂ Ω \partial_e\Omega,\partial_e^*\Omega\subset\partial\Omega ∂ e Ω , ∂ e ∗ Ω ⊂ ∂ Ω ∂ Ω \partial\Omega ∂ Ω ω P ( ∂ Ω ) ≤ 1 \omega^{P}(\partial\Omega)\le1 ω P ( ∂ Ω ) ≤ 1 ω ∗ P ( ∂ Ω ) ≤ 1 \omega_{*}^{P}(\partial\Omega)\le1 ω ∗ P ( ∂ Ω ) ≤ 1 [66 Sections 2 and 3] and [Wal2, Chapter 8].
In Section 8.4, we associate with each ball of the local T b Tb T b
Lemma 8.1. Let σ \sigma σ P n \mathbb{P}^{n} P n ( n + 1 ) (n+1) ( n + 1 ) E ⊂ P n E\subset\mathbb{P}^{n} E ⊂ P n 0 < σ ( E ) < ∞ 0<\sigma(E)<\infty 0 < σ ( E ) < ∞ X E = ( X E , t E ) ∈ E X_E=(X_E,t_E)\in E X E = ( X E , t E ) ∈ E
σ ( { ( Y , s ) ∈ E : s > t E } ) = σ ( { ( Y , s ) ∈ E : s < t E } ) = 1 2 σ ( E ) . (305) \sigma\bigl(\{(Y,s)\in E:s>t_E\}\bigr)=\sigma\bigl(\{(Y,s)\in E:s<t_E\}\bigr)=\frac{1}{2}\sigma(E). \tag*{(305)} σ ( {( Y , s ) ∈ E : s > t E } ) = σ ( {( Y , s ) ∈ E : s < t E } ) = 2 1 σ ( E ) . (305) Proof. First, we observe that, for every t 0 ∈ R t_0\in\mathbb{R} t 0 ∈ R E ∩ { ( Y , s ) : s = t 0 } E\cap\{(Y,s):s=t_0\} E ∩ {( Y , s ) : s = t 0 } n n n ( n + 1 ) (n+1) ( n + 1 ) σ \sigma σ
σ ( E ∩ { ( Y , s ) : s = t 0 } ) = 0 , for every t 0 ∈ R . \sigma\bigl(E\cap\{(Y,s):s=t_0\}\bigr)=0,\qquad\text{for every }t_0\in\mathbb{R}. σ ( E ∩ {( Y , s ) : s = t 0 } ) = 0 , for every t 0 ∈ R . Hence, if we denote by π t ( X , t ) : = t \pi_t(X,t):=t π t ( X , t ) := t ν : = ( π t ) # ( σ ∣ E ) \nu:=(\pi_t)_{\#}(\sigma|_E) ν := ( π t ) # ( σ ∣ E ) ν \nu ν
We define the function F ν ( t ) : = ν ( ( − ∞ , t ] ) F_\nu(t):=\nu((-\infty,t]) F ν ( t ) := ν (( − ∞ , t ]) t ∈ R t\in\mathbb{R} t ∈ R
lim t → − ∞ F ν ( t ) = 0 and lim t → ∞ F ν ( t ) = σ ( E ) . \lim_{t\to-\infty}F_\nu(t)=0\qquad\text{and}\qquad\lim_{t\to\infty}F_\nu(t)=\sigma(E). t → − ∞ lim F ν ( t ) = 0 and t → ∞ lim F ν ( t ) = σ ( E ) . Thus, there exists t E ∈ R t_E\in\mathbb{R} t E ∈ R t E = min { t ∈ R : F ν ( t ) ≥ σ ( E ) / 2 } t_E=\min\{t\in\mathbb{R}:F_\nu(t)\ge\sigma(E)/2\} t E = min { t ∈ R : F ν ( t ) ≥ σ ( E ) /2 } ν ( ( t E − ε , t E + ε ) ) > 0 \nu((t_E-\varepsilon,t_E+\varepsilon))>0 ν (( t E − ε , t E + ε )) > 0 ε > 0 \varepsilon>0 ε > 0 t E ∈ supp ( ν ) t_E\in\operatorname{supp}(\nu) t E ∈ supp ( ν ) E E E supp ( ν ) ⊂ π t ( E ) \operatorname{supp}(\nu)\subset\pi_t(E) supp ( ν ) ⊂ π t ( E ) ( X E , t E ) ∈ E (X_E,t_E)\in E ( X E , t E ) ∈ E (305) holds, because ν \nu ν
Elements of heat potential theory
In this subsection, we collect some auxiliary potential-theoretic results.
Lemma 8.2. Let Ω \Omega Ω Theorem 1.9 , let Z ∈ P n Z\in\mathbb{P}^{n} Z ∈ P n r > 0 r>0 r > 0 E ⊂ ∂ Ω ∩ B ( Z , r ) E\subset\partial\Omega\cap B(Z,r) E ⊂ ∂ Ω ∩ B ( Z , r )
v ( X ) : = ∫ E Γ ( X , Y ) d σ ( Y ) v(X):=\int_E\Gamma(X,Y)\,d\sigma(Y) v ( X ) := ∫ E Γ ( X , Y ) d σ ( Y ) is continuous on P n \mathbb{P}^{n} P n Ω \Omega Ω 0 ≤ v ≲ E 0 r 0\le v\lesssim\mathcal{E}_0r 0 ≤ v ≲ E 0 r
Proof. Standard kernel estimates and the upper growth of σ \sigma σ X ∈ P n X\in\mathbb{P}^{n} X ∈ P n a > 0 a>0 a > 0
∫ E ∩ B ( X , a ) Γ ( X , Y ) d σ ( Y ) ≲ ∑ j ≥ 0 ( 2 − j a ) − n σ ( B ( X , 2 − j a ) ) ≲ E 0 a . (306) \int_{E\cap B(X,a)}\Gamma(X,Y)\,d\sigma(Y)\lesssim\sum_{j\ge0}(2^{-j}a)^{-n}\sigma(B(X,2^{-j}a))\lesssim\mathcal{E}_0a. \tag*{(306)} ∫ E ∩ B ( X , a ) Γ ( X , Y ) d σ ( Y ) ≲ j ≥ 0 ∑ ( 2 − j a ) − n σ ( B ( X , 2 − j a )) ≲ E 0 a . (306) If X ∈ B ( Z , 2 r ) X\in B(Z,2r) X ∈ B ( Z , 2 r ) E ⊂ B ( X , 3 r ) E\subset B(X,3r) E ⊂ B ( X , 3 r ) (306) with a = 3 r a=3r a = 3 r v ( X ) ≲ E 0 r v(X)\lesssim\mathcal{E}_0r v ( X ) ≲ E 0 r X ∈ P n ∖ B ( Z , 2 r ) X\in\mathbb{P}^{n}\setminus B(Z,2r) X ∈ P n ∖ B ( Z , 2 r )
v ( X ) ≲ r − n σ ( E ) ≲ E 0 r . v(X)\lesssim r^{-n}\sigma(E)\lesssim\mathcal{E}_0r. v ( X ) ≲ r − n σ ( E ) ≲ E 0 r . The estimate (306) with a → 0 a\to0 a → 0 X X X v v v E ⊂ ∂ Ω E\subset\partial\Omega E ⊂ ∂ Ω v v v Ω \Omega Ω σ ( E ) \sigma(E) σ ( E )
Extended by zero at infinity, the function v v v P n \mathbb{P}^n P n Lemma 8.2 , we obtain the identity
v ( P ) = ∫ v ( W ) d ω P ( W ) , for P ∈ Ω , (307) v(\mathbf{P})=\int v(\mathbf{W})\,d\omega^{\mathbf{P}}(\mathbf{W}), \qquad\text{for } \mathbf{P}\in\Omega, \tag*{(307)} v ( P ) = ∫ v ( W ) d ω P ( W ) , for P ∈ Ω , (307) and the corresponding statements hold for the adjoint caloric potential v ∗ ( X ) : = ∫ E Γ ( Y , X ) d σ ( Y ) v^*(\mathbf{X}):=\int_E\Gamma(\mathbf{Y},\mathbf{X})\,d\sigma(\mathbf{Y}) v ∗ ( X ) := ∫ E Γ ( Y , X ) d σ ( Y ) ω ∗ P \omega_*^{\mathbf{P}} ω ∗ P v ( X ) v(\mathbf{X}) v ( X ) ω P \omega^{\mathbf{P}} ω P
The next lemma is a parabolic version of Bourgain’s lemma, see also for instance [25 ] . Here H p , ∞ n + s \mathcal{H}^{n+s}_{p,\infty} H p , ∞ n + s ( n + s ) (n+s) ( n + s ) [66 Section 2] .
Lemma 8.3. Let X 0 = ( X 0 , t 0 ) ∈ ∂ Ω \mathbf{X}_0=(X_0,t_0)\in\partial\Omega X 0 = ( X 0 , t 0 ) ∈ ∂ Ω r > 0 r>0 r > 0 A ≥ 1 A\ge1 A ≥ 1 ϑ ∈ ( 0 , 1 ) \vartheta\in(0,1) ϑ ∈ ( 0 , 1 ) P + ∈ Ω \mathbf{P}^+\in\Omega P + ∈ Ω ∣ X P + − X 0 ∣ ≤ A r |X_{P^+}-X_0|\le Ar ∣ X P + − X 0 ∣ ≤ A r
t 0 + ϑ r 2 ≤ t P + ≤ t 0 + ϑ − 1 r 2 . t_0+\vartheta r^2\le t_{P^+}\le t_0+\vartheta^{-1}r^2. t 0 + ϑ r 2 ≤ t P + ≤ t 0 + ϑ − 1 r 2 . Then there are constants M 0 = M 0 ( n , A , ϑ ) > 1 M_0=M_0(n,A,\vartheta)>1 M 0 = M 0 ( n , A , ϑ ) > 1 c = c ( n , A , ϑ ) > 0 c=c(n,A,\vartheta)>0 c = c ( n , A , ϑ ) > 0 M ≥ M 0 M\ge M_0 M ≥ M 0
ω P + ( B ( X 0 , M r ) ) ≥ c σ ( B − ( X 0 , r ) ) E 0 r n + 1 . (308) \omega^{\mathbf{P}^+}(B(\mathbf{X}_0,Mr))\ge c\frac{\sigma(B^-(\mathbf{X}_0,r))}{\mathcal{E}_0r^{n+1}}. \tag*{(308)} ω P + ( B ( X 0 , M r )) ≥ c E 0 r n + 1 σ ( B − ( X 0 , r )) . (308) More generally, for 0 < s ≤ 2 0<s\le2 0 < s ≤ 2
ω P + ( B ( X 0 , M r ) ) ≳ n , s , A , ϑ H p , ∞ n + s ( B − ( X 0 , r ) ∩ ∂ Ω ) r n + s . (309) \omega^{\mathbf{P}^+}(B(\mathbf{X}_0,Mr))\gtrsim_{n,s,A,\vartheta}\frac{\mathcal{H}^{n+s}_{p,\infty}(B^-(\mathbf{X}_0,r)\cap\partial\Omega)}{r^{n+s}}. \tag*{(309)} ω P + ( B ( X 0 , M r )) ≳ n , s , A , ϑ r n + s H p , ∞ n + s ( B − ( X 0 , r ) ∩ ∂ Ω ) . (309) Proof. Take a compact set E ⊂ B − ( X 0 , r ) ∩ ∂ Ω E\subset B^-(\mathbf{X}_0,r)\cap\partial\Omega E ⊂ B − ( X 0 , r ) ∩ ∂ Ω v v v Lemma 8.2 . The time and spatial separation give v ( P + ) ≥ c 1 r − n σ ( E ) v(\mathbf{P}^+)\ge c_1r^{-n}\sigma(E) v ( P + ) ≥ c 1 r − n σ ( E ) M ≥ M 0 M\ge M_0 M ≥ M 0
sup W ∉ B ( X 0 , M r ) v ( W ) ≤ C ( M − 1 ) − n r − n σ ( E ) ≤ 1 2 c 1 r − n σ ( E ) . \sup_{\mathbf{W}\notin B(\mathbf{X}_0,Mr)}v(\mathbf{W})\le C(M-1)^{-n}r^{-n}\sigma(E)\le\frac{1}{2}c_1r^{-n}\sigma(E). W ∈ / B ( X 0 , M r ) sup v ( W ) ≤ C ( M − 1 ) − n r − n σ ( E ) ≤ 2 1 c 1 r − n σ ( E ) . Since ω P + ( ∂ Ω ) ≤ 1 \omega^{\mathbf{P}^+}(\partial\Omega)\le1 ω P + ( ∂ Ω ) ≤ 1 (307) and Lemma 8.2 imply
c 1 r − n σ ( E ) ≤ C E 0 r ω P + ( B ( X 0 , M r ) ) + 1 2 c 1 r − n σ ( E ) , c_1r^{-n}\sigma(E)\le C\mathcal{E}_0r\,\omega^{\mathbf{P}^+}(B(\mathbf{X}_0,Mr))+\frac{1}{2}c_1r^{-n}\sigma(E), c 1 r − n σ ( E ) ≤ C E 0 r ω P + ( B ( X 0 , M r )) + 2 1 c 1 r − n σ ( E ) , which readily yields (308) by the inner regularity of σ \sigma σ (309) follows from Frostman’s lemma. □
A time reflection of Lemma 8.3 implies that, if P − ∈ Ω \mathbf{P}^-\in\Omega P − ∈ Ω ∣ X P − − X 0 ∣ ≤ A r |X_{P^-}-X_0|\le Ar ∣ X P − − X 0 ∣ ≤ A r t 0 − ϑ − 1 r 2 ≤ t P − ≤ t 0 − ϑ r 2 t_0-\vartheta^{-1}r^2\le t_{P^-}\le t_0-\vartheta r^2 t 0 − ϑ − 1 r 2 ≤ t P − ≤ t 0 − ϑ r 2
ω ∗ P − ( B ( X 0 , M r ) ) ≥ c σ ( B + ( X 0 , r ) ) E 0 r n + 1 , for M ≥ M 0 . \omega_*^{\mathbf{P}^-}(B(\mathbf{X}_0,Mr))\ge c\frac{\sigma(B^+(\mathbf{X}_0,r))}{\mathcal{E}_0r^{n+1}},\qquad\text{for } M\ge M_0. ω ∗ P − ( B ( X 0 , M r )) ≥ c E 0 r n + 1 σ ( B + ( X 0 , r )) , for M ≥ M 0 . In particular, in our application, we take δ ≤ M 0 ( n , 1 , ϑ ) − 1 \delta\le M_0(n,1,\vartheta)^{-1} δ ≤ M 0 ( n , 1 , ϑ ) − 1 B B B Definition 1.8 such that σ ( ( δ B ) ± ) ≳ σ ( B ) \sigma((\delta B)^\pm)\gtrsim\sigma(B) σ (( δ B ) ± ) ≳ σ ( B ) Lemma 8.3 and its time reflection with δ r ( B ) \delta r(B) δ r ( B ) r r r A = 1 A=1 A = 1 M = δ − 1 ≥ M 0 M=\delta^{-1}\ge M_0 M = δ − 1 ≥ M 0
ω B P + ( B ) ≳ σ ( ( δ B ) − ) E 0 ( δ r ( B ) ) n + 1 ≳ σ ( B ) E 0 r ( B ) n + 1 , and ω ∗ B P − ( B ) ≳ σ ( B ) E 0 r ( B ) n + 1 , (310) \omega^{\mathbf{P}^+}_B(B)\gtrsim\frac{\sigma((\delta B)^-)}{\mathcal{E}_0(\delta r(B))^{n+1}}\gtrsim\frac{\sigma(B)}{\mathcal{E}_0r(B)^{n+1}},\qquad\text{and}\qquad\omega^{\mathbf{P}^-}_{*B}(B)\gtrsim\frac{\sigma(B)}{\mathcal{E}_0r(B)^{n+1}}, \tag*{(310)} ω B P + ( B ) ≳ E 0 ( δ r ( B ) ) n + 1 σ (( δ B ) − ) ≳ E 0 r ( B ) n + 1 σ ( B ) , and ω ∗ B P − ( B ) ≳ E 0 r ( B ) n + 1 σ ( B ) , (310) where the implicit constants depend on n n n ϑ \vartheta ϑ δ \delta δ B B B Lemma 8.9 .
A local estimate for the cutoff potential
We first recall a consequence of the interior gradient estimate for solutions to the adjoint heat equation and Harnack’s inequality.
Lemma 8.4. Suppose that u ≥ 0 u \ge0 u ≥ 0 is adjoint caloric in B ( Z , 10 r ) B(Z,10r) B ( Z , 10 r ) where Z = ( Z , t Z ) Z=(Z,t_Z) Z = ( Z , t Z ) and put Z + = ( Z , t Z + 4 r 2 ) Z^+=(Z,t_Z+4r^2) Z + = ( Z , t Z + 4 r 2 ) Then
∣ ∇ X u ( Z + ) ∣ ≤ C r u ( Y ) , for every Y ∈ B ( Z , r ) . \lvert\nabla_X u(Z^+)\rvert\le\frac{C}{r}u(Y), \qquad\text{for every }Y\in B(Z,r). ∣ ∇ X u ( Z + )∣ ≤ r C u ( Y ) , for every Y ∈ B ( Z , r ) . For a non-negative caloric function, the analogous estimate holds at Z − = ( Z , t Z − 4 r 2 ) Z^-=(Z,t_Z-4r^2) Z − = ( Z , t Z − 4 r 2 )
Proof. The interior gradient estimate gives ∣ ∇ X u ( Z + ) ∣ ≲ r − 1 sup B ( Z + , r ) u \lvert\nabla_Xu(Z^+)\rvert\lesssim r^{-1}\sup_{B(Z^+,r)}u ∣ ∇ X u ( Z + )∣ ≲ r − 1 sup B ( Z + , r ) u B ( Z + , r ) B(Z^+,r) B ( Z + , r ) ( t Z + 3 r 2 , t Z + 5 r 2 ) (t_Z+3r^2,t_Z+5r^2) ( t Z + 3 r 2 , t Z + 5 r 2 ) Y ∈ B ( Z , r ) Y\in B(Z,r) Y ∈ B ( Z , r ) ( t Z − r 2 , t Z + r 2 ) (t_Z-r^2,t_Z+r^2) ( t Z − r 2 , t Z + r 2 ) B ( Z , 10 r ) B(Z,10r) B ( Z , 10 r ) C u ( Y ) Cu(Y) C u ( Y ) Y Y Y
Lemma 8.5. Let Z ∈ supp σ Z\in\operatorname{supp}\sigma Z ∈ supp σ r > 0 r>0 r > 0 P + ∈ Ω ∖ B ( Z , 100 r ) P^+\in\Omega\setminus B(Z,100r) P + ∈ Ω ∖ B ( Z , 100 r ) 0 ≤ φ Z , r ≤ 1 0\le\varphi_{Z,r}\le1 0 ≤ φ Z , r ≤ 1 B ( Z , 10 r ) B(Z,10r) B ( Z , 10 r ) 1 1 1 B ( Z , 20 r ) B(Z,20r) B ( Z , 20 r )
u + ( Y ) : = Γ ( P + , Y ) − ∫ Γ ( W , Y ) φ Z , r ( W ) d ω ∗ P + ( W ) . (311) u^+(Y):=\Gamma(P^+,Y)-\int\Gamma(W,Y)\varphi_{Z,r}(W)\,d\omega_{*}^{P^+}(W). \tag*{(311)} u + ( Y ) := Γ ( P + , Y ) − ∫ Γ ( W , Y ) φ Z , r ( W ) d ω ∗ P + ( W ) . (311) Then u + ≥ 0 u^+\ge0 u + ≥ 0 B ( Z , 10 r ) B(Z,10r) B ( Z , 10 r ) Z + : = ( Z , t Z + 4 r 2 ) Z^+:=(Z,t_Z+4r^2) Z + := ( Z , t Z + 4 r 2 )
∣ ∇ X u + ( Z + ) ∣ ≲ E 0 ω P + ( B ( Z , 20 r ) ) σ ( B ( Z , r ) ) . (312) \lvert\nabla_Xu^+(Z^+)\rvert\lesssim\mathcal{E}_0\frac{\omega^{P^+}(B(Z,20r))}{\sigma(B(Z,r))}. \tag*{(312)} ∣ ∇ X u + ( Z + )∣ ≲ E 0 σ ( B ( Z , r )) ω P + ( B ( Z , 20 r )) . (312) Proof. We first claim that, for P ∈ Ω P\in\Omega P ∈ Ω P ≠ Y P\ne Y P = Y
∫ Γ ( W , Y ) d ω P ( W ) ≤ Γ ( P , Y ) . (313) \int\Gamma(W,Y)\,d\omega^P(W)\le\Gamma(P,Y). \tag*{(313)} ∫ Γ ( W , Y ) d ω P ( W ) ≤ Γ ( P , Y ) . (313) Note that, since 0 ≤ φ Z , r ≤ 1 0\le\varphi_{Z,r}\le1 0 ≤ φ Z , r ≤ 1 (313) proves u + ≥ 0 u^+\ge0 u + ≥ 0
To prove the claim, it suffices to approximate a unit point mass at Y Y Y (313) ; at the diagonal we assign the kernel the value zero.
The adjoint caloricity of u + u^+ u + B ( Z , 10 r ) B(Z,10r) B ( Z , 10 r ) φ Z , r ω P + \varphi_{Z,r}\omega^{P^+} φ Z , r ω P +
We are left with proving (312) . Let E ⊂ supp σ ∩ B ( Z , r ) E\subset\operatorname{supp}\sigma\cap B(Z,r) E ⊂ supp σ ∩ B ( Z , r ) σ ( E ) ≥ σ ( B ( Z , r ) ) / 2 \sigma(E)\ge\sigma(B(Z,r))/2 σ ( E ) ≥ σ ( B ( Z , r )) /2 v v v (307) , we have
∫ E u + d σ = v ( P + ) − ∫ v φ Z , r d ω P + = ∫ v ( 1 − φ Z , r ) d ω P + ≲ E 0 r ω P + ( B ( Z , 20 r ) ) . (314) \int_E u^+\,d\sigma=v(P^+)-\int v\varphi_{Z,r}\,d\omega^{P^+}
=\int v(1-\varphi_{Z,r})\,d\omega^{P^+}
\lesssim\mathcal{E}_0r\,\omega^{P^+}(B(Z,20r)). \tag*{(314)} ∫ E u + d σ = v ( P + ) − ∫ v φ Z , r d ω P + = ∫ v ( 1 − φ Z , r ) d ω P + ≲ E 0 r ω P + ( B ( Z , 20 r )) . (314) On the other hand, Lemma 8.4 and the inequality u + ≥ 0 u^+\ge0 u + ≥ 0
∣ ∇ X u + ( Z + ) ∣ ≲ 1 r σ ( E ) ∫ E u + d σ . (315) \lvert\nabla_Xu^+(Z^+)\rvert\lesssim\frac{1}{r\sigma(E)}\int_Eu^+\,d\sigma. \tag*{(315)} ∣ ∇ X u + ( Z + )∣ ≲ r σ ( E ) 1 ∫ E u + d σ . (315) Thus, gathering the last two estimates proves (312) . □
A time reflection yields an estimate analogous to (312) for the function
u − ( Y ) : = Γ ( Y , P − ) − ∫ Γ ( Y , W ) φ Z , r ( W ) d ω ∗ P − ( W ) , u^-(Y):=\Gamma(Y,P^-)-\int\Gamma(Y,W)\varphi_{Z,r}(W)\,d\omega_*^{P^-}(W), u − ( Y ) := Γ ( Y , P − ) − ∫ Γ ( Y , W ) φ Z , r ( W ) d ω ∗ P − ( W ) , at the time-lagged point Z − : = ( Z , t Z − 4 r 2 ) Z^-:=(Z,t_Z-4r^2) Z − := ( Z , t Z − 4 r 2 )
The maximal truncations on a testing ball
Let B B B Definition 1.8 such that
σ ( ( δ B ) − ) ≳ σ ( B ) and σ ( ( δ B ) + ) ≳ σ ( B ) , (316) \sigma((\delta B)^-) \gtrsim\sigma(B) \qquad\text{and}\qquad\sigma((\delta B)^+) \gtrsim\sigma(B), \tag*{(316)} σ (( δ B ) − ) ≳ σ ( B ) and σ (( δ B ) + ) ≳ σ ( B ) , (316) with fixed constants. In particular, this holds for the balls of Lemma 8.9 . To apply the local T b Tb T b
ν B ∗ : = σ ( B ) ω B + ( B ) ω B + ∣ B , ν B : = σ ( B ) ω B − ( B ) ω B − ∣ B , (317) \nu_B^\ast:= \frac{\sigma(B)}{\omega_B^{+}(B)}\omega_B^{+}|_B,\qquad\nu_B := \frac{\sigma(B)}{\omega_B^{-}(B)}\omega_B^{-}|_B, \tag*{(317)} ν B ∗ := ω B + ( B ) σ ( B ) ω B + ∣ B , ν B := ω B − ( B ) σ ( B ) ω B − ∣ B , (317) and we remark that the denominators in (317) are positive by (310) . Both measures are non-negative and concentrated on B B B
ν B ( B ) = ν B ∗ ( B ) = σ ( B ) . (318) \nu_B(B)=\nu_B^\ast(B)=\sigma(B). \tag*{(318)} ν B ( B ) = ν B ∗ ( B ) = σ ( B ) . (318) Moreover, by Definition 1.8 , for every Borel set E ⊂ B E\subset B E ⊂ B
If σ ( E ) ≤ κ σ ( B ) , then ν B ( E ) ≤ κ ′ σ ( B ) and ν B ∗ ( E ) ≤ κ ′ σ ( B ) . (319) \text{If }\sigma(E)\leq\kappa\sigma(B),\qquad\text{then}\qquad\nu_B(E)\leq\kappa'\sigma(B)\text{ and }\nu_B^\ast(E)\leq\kappa'\sigma(B). \tag*{(319)} If σ ( E ) ≤ κσ ( B ) , then ν B ( E ) ≤ κ ′ σ ( B ) and ν B ∗ ( E ) ≤ κ ′ σ ( B ) . (319) To verify the alternative (5) of Definition 1.2 , we need pointwise bounds for the maximal truncations of T T T T ∗ T^\ast T ∗ (21) of a non-negative measure ν \nu ν
M ~ σ ν ( Z ) = sup B ′ ∋ Z , σ ( 5 B ′ ) > 0 ν ( B ′ ) σ ( 5 B ′ ) , for Z ∈ P n . \widetilde{\mathcal{M}}_{\sigma}\nu(Z)=\sup_{\substack{B'\ni Z,\ \sigma(5B')>0}}\frac{\nu(B')}{\sigma(5B')},\qquad\text{for }Z\in\mathbb{P}^n. M σ ν ( Z ) = B ′ ∋ Z , σ ( 5 B ′ ) > 0 sup σ ( 5 B ′ ) ν ( B ′ ) , for Z ∈ P n . Proposition 8.6. For τ , η ∈ ( 0 , 1 / 10 ) \tau,\eta\in(0,1/10) τ , η ∈ ( 0 , 1/10 ) (317) satisfy
T # ∗ ν B ∗ ( Z ) ≤ C τ , η ( 1 + M ~ σ ν B ∗ ( Z ) ) , Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ B ( P B + , η r ( B ) / 2 ) , (320) T^\ast_{\#}\nu_B^\ast(Z)\leq C_{\tau,\eta}\left(1+\widetilde{\mathcal{M}}_{\sigma}\nu_B^\ast(Z)\right),\qquad Z\in\bigl((1-\tau)B\cap\operatorname{supp}\sigma\bigr)\setminus B(P_B^+,\eta r(B)/2), \tag*{(320)} T # ∗ ν B ∗ ( Z ) ≤ C τ , η ( 1 + M σ ν B ∗ ( Z ) ) , Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ B ( P B + , η r ( B ) /2 ) , (320) and
T # ν B ( Z ) ≤ C τ , η ( 1 + M ~ σ ν B ( Z ) ) , Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ B ( P B − , η r ( B ) / 2 ) , (321) T_{\#}\nu_B(Z)\leq C_{\tau,\eta}\left(1+\widetilde{\mathcal{M}}_{\sigma}\nu_B(Z)\right),\qquad Z\in\bigl((1-\tau)B\cap\operatorname{supp}\sigma\bigr)\setminus B(P_B^-,\eta r(B)/2), \tag*{(321)} T # ν B ( Z ) ≤ C τ , η ( 1 + M σ ν B ( Z ) ) , Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ B ( P B − , η r ( B ) /2 ) , (321) where C τ , η : = C min { τ , η } − n − 1 C_{\tau,\eta}:=C\min\{\tau,\eta\}^{-n-1} C τ , η := C min { τ , η } − n − 1 C > 0 C>0 C > 0 n n n E 0 \mathcal{E}_0 E 0 ϑ \vartheta ϑ δ \delta δ (316) .
The proof of the proposition above relies on the following lemma.
Lemma 8.7. We write P + : = P B + P^+:=P_B^+ P + := P B + ω + , : = ω + \omega^{+,}:=\omega^+ ω + , := ω + a B : = σ ( B ) / ω + ( B ) a_B:=\sigma(B)/\omega^+(B) a B := σ ( B ) / ω + ( B ) Z Z Z C ~ : = M ~ σ ν B ∗ ( Z ) < ∞ \widetilde{C}:=\widetilde{\mathcal{M}}_{\sigma}\nu_B^\ast(Z)<\infty C := M σ ν B ∗ ( Z ) < ∞ r ∗ : = 10 − 4 min { τ , η } r ( B ) r_\ast:=10^{-4}\min\{\tau,\eta\}r(B) r ∗ := 1 0 − 4 min { τ , η } r ( B ) 0 < R ≤ r ∗ 0<R\leq r_\ast 0 < R ≤ r ∗
σ ( B ( Z , 100 R ) ) ≤ 100 n + 2 σ ( B ( Z , R ) ) , (322) \sigma(B(Z,100R))\leq100^{n+2}\sigma(B(Z,R)), \tag*{(322)} σ ( B ( Z , 100 R )) ≤ 10 0 n + 2 σ ( B ( Z , R )) , (322) then
∣ T R ∗ ν B ∗ ( Z ) ∣ ≤ C τ , η ( 1 + C ~ ) . (323) |T_R^\ast\nu_B^\ast(Z)|\leq C_{\tau,\eta}(1+\widetilde{C}). \tag*{(323)} ∣ T R ∗ ν B ∗ ( Z ) ∣ ≤ C τ , η ( 1 + C ) . (323) Proof. By the definition of C ~ \widetilde{C} C σ \sigma σ r > 0 r>0 r > 0
ν B ∗ ( B ( Z , r ) ) ≤ C ~ σ ( B ( Z , 5 r ) ) ≤ 5 n + 1 C 0 C ~ r n + 1 . (324) \nu_B^\ast(B(Z,r))\leq\widetilde{C}\sigma(B(Z,5r))\leq5^{n+1}\mathcal{C}_0\widetilde{C}r^{n+1}. \tag*{(324)} ν B ∗ ( B ( Z , r )) ≤ C σ ( B ( Z , 5 r )) ≤ 5 n + 1 C 0 C r n + 1 . (324) Let Z R : = ( Z , t Z + 4 R 2 ) Z_R:=(Z,t_Z+4R^2) Z R := ( Z , t Z + 4 R 2 ) R ≤ 10 − 4 min { τ , η } r ( B ) R\leq10^{-4}\min\{\tau,\eta\}r(B) R ≤ 1 0 − 4 min { τ , η } r ( B ) P + ∉ B ( Z , 100 R ) P^+\notin B(Z,100R) P + ∈ / B ( Z , 100 R ) B ( Z , 100 R ) ⊂ B B(Z,100R)\subset B B ( Z , 100 R ) ⊂ B ∥ P + − Z R ∥ ≥ η r ( B ) / 4 \lVert P^+-Z_R\rVert\geq\eta r(B)/4 ∥ P + − Z R ∥ ≥ η r ( B ) /4 Lemma 8.5 with r = R r=R r = R (311) with respect to the space variable, we obtain
∫ K ( W , Z R ) φ Z , R ( W ) d ω + ( W ) = ∇ X u + ( Z R ) + K ( P + , Z R ) . (325) \int K(W,Z_R)\varphi_{Z,R}(W)\,d\omega^+(W)=\nabla_Xu^+(Z_R)+K(P^+,Z_R). \tag*{(325)} ∫ K ( W , Z R ) φ Z , R ( W ) d ω + ( W ) = ∇ X u + ( Z R ) + K ( P + , Z R ) . (325) To prove the desired bound, we denote ν : = ν B ∗ = a B ω + ∣ B \nu:= \nu_B^* = a_B\omega^+|_B ν := ν B ∗ = a B ω + ∣ B
T R ∗ ν ( Z ) = a B ∇ X u + ( Z R ) + a B K ( P + , Z R ) + ∫ R < ∥ W − Z ∥ < 20 R K ( W , Z ) d ν ( W ) + ∫ ∥ W − Z ∥ ≥ 20 R ( K ( W , Z ) − K ( W , Z R ) ) d ν ( W ) − ∫ B ( Z , 20 R ) K ( W , Z R ) φ Z , R ( W ) d ν ( W ) − a B ∫ P n ∖ B K ( W , Z R ) φ Z , R ( W ) d ω + ( W ) = a B ∇ X u + ( Z R ) + a B K ( P + , Z R ) + I 1 + I 2 + I 3 + I 4 . (326) \begin{aligned}
T_R^*\nu(Z)={}&a_B\nabla_Xu^+(Z_R)+a_BK(\mathbb{P}^+,Z_R)+\int_{R<\lVert W-Z\rVert<20R}K(W,Z)\,d\nu(W)\\
&+\int_{\lVert W-Z\rVert\ge20R}\bigl(K(W,Z)-K(W,Z_R)\bigr)\,d\nu(W)\\
&-\int_{B(Z,20R)}K(W,Z_R)\varphi_{Z,R}(W)\,d\nu(W)-a_B\int_{\mathbb{P}^n\setminus B}K(W,Z_R)\varphi_{Z,R}(W)\,d\omega^+(W)\\
={}&a_B\nabla_Xu^+(Z_R)+a_BK(\mathbb{P}^+,Z_R)+\mathfrak{I}_1+\mathfrak{I}_2+\mathfrak{I}_3+\mathfrak{I}_4. \tag*{(326)}
\end{aligned} T R ∗ ν ( Z ) = = a B ∇ X u + ( Z R ) + a B K ( P + , Z R ) + ∫ R < ∥ W − Z ∥ < 20 R K ( W , Z ) d ν ( W ) + ∫ ∥ W − Z ∥ ≥ 20 R ( K ( W , Z ) − K ( W , Z R ) ) d ν ( W ) − ∫ B ( Z , 20 R ) K ( W , Z R ) φ Z , R ( W ) d ν ( W ) − a B ∫ P n ∖ B K ( W , Z R ) φ Z , R ( W ) d ω + ( W ) a B ∇ X u + ( Z R ) + a B K ( P + , Z R ) + I 1 + I 2 + I 3 + I 4 . (326) By {(8.10)}, the definition of C ~ \widetilde C C
a B ∣ ∇ X u + ( Z R ) ∣ ≲ ε 0 σ ( B ( Z , R ) ) ν ( B ( Z , 20 R ) ) ≤ ε 0 C ~ σ ( B ( Z , 100 R ) ) σ ( B ( Z , R ) ) ≲ ε 0 C ~ . (327) \begin{aligned}
a_B\left|\nabla_Xu^+(Z_R)\right|\lesssim\frac{\varepsilon_0}{\sigma(B(Z,R))}\nu(B(Z,20R)) \\
&\le\varepsilon_0\widetilde C\frac{\sigma(B(Z,100R))}{\sigma(B(Z,R))}\lesssim\varepsilon_0\widetilde C. \tag*{(327)}
\end{aligned} a B ∇ X u + ( Z R ) ≲ σ ( B ( Z , R )) ε 0 ν ( B ( Z , 20 R )) ≤ ε 0 C σ ( B ( Z , R )) σ ( B ( Z , 100 R )) ≲ ε 0 C . (327) Moreover, {(8.8)} and standard kernel estimates yield
a B ∣ K ( P + , Z R ) ∣ ≲ ε 0 r ( B ) n + 1 ( η r ( B ) ) n + 1 ≤ ε 0 η − n − 1 . (328) a_B\left|K(\mathbb{P}^+,Z_R)\right|\lesssim\frac{\varepsilon_0r(B)^{n+1}}{(\eta r(B))^{n+1}}\le\varepsilon_0\eta^{-n-1}. \tag*{(328)} a B K ( P + , Z R ) ≲ ( η r ( B ) ) n + 1 ε 0 r ( B ) n + 1 ≤ ε 0 η − n − 1 . (328) By the size estimate and {(8.22)}, we obtain
∣ I 1 ∣ + ∣ I 3 ∣ ≲ R − n − 1 ν ( B ( Z , 20 R ) ) ≲ ε 0 C ~ . |\mathfrak{I}_1|+|\mathfrak{I}_3|\lesssim R^{-n-1}\nu(B(Z,20R))\lesssim\varepsilon_0\widetilde C. ∣ I 1 ∣ + ∣ I 3 ∣ ≲ R − n − 1 ν ( B ( Z , 20 R )) ≲ ε 0 C . To bound the second integral, note that kernel regularity and {(8.22)} give
∣ I 2 ∣ ≲ ∫ ∥ W − Z ∥ ≥ 20 R R ∥ W − Z ∥ n + 2 d ν ( W ) ≲ ε 0 C ~ ∑ j ≥ 0 2 − j ≲ ε 0 C ~ . |\mathfrak{I}_2|\lesssim\int_{\lVert W-Z\rVert\ge20R}\frac{R}{\lVert W-Z\rVert^{n+2}}\,d\nu(W)\lesssim\varepsilon_0\widetilde C\sum_{j\ge0}2^{-j}\lesssim\varepsilon_0\widetilde C. ∣ I 2 ∣ ≲ ∫ ∥ W − Z ∥ ≥ 20 R ∥ W − Z ∥ n + 2 R d ν ( W ) ≲ ε 0 C j ≥ 0 ∑ 2 − j ≲ ε 0 C . Since ∥ Z R − Z ∥ = 2 R \lVert Z_R-Z\rVert=2R ∥ Z R − Z ∥ = 2 R Z R Z_R Z R P n ∖ B \mathbb{P}^n\setminus B P n ∖ B τ r ( B ) / 2 \tau r(B)/2 τ r ( B ) /2 ω + ( ∂ Ω ) ≤ 1 \omega^+(\partial\Omega)\le1 ω + ( ∂ Ω ) ≤ 1
∣ I 4 ∣ ≲ a B ∫ P n ∖ B ∣ K ( W , Z R ) ∣ φ Z , R ( W ) d ω + ( W ) ≲ a B ( τ r ( B ) ) n + 1 ≲ ε 0 τ − n − 1 , |\mathfrak{I}_4|\lesssim a_B\int_{\mathbb{P}^n\setminus B}|K(W,Z_R)|\varphi_{Z,R}(W)\,d\omega^+(W)\lesssim\frac{a_B}{(\tau r(B))^{n+1}}\lesssim\varepsilon_0\tau^{-n-1}, ∣ I 4 ∣ ≲ a B ∫ P n ∖ B ∣ K ( W , Z R ) ∣ φ Z , R ( W ) d ω + ( W ) ≲ ( τ r ( B ) ) n + 1 a B ≲ ε 0 τ − n − 1 , which, together with the previous bounds for the other terms in {(8.24)}, allows us to conclude {(8.21)}. □
The proof of Proposition 8.6 combines the lemma above with the standard kernel estimates.
Proof of Proposition 8.6. We prove {(8.18)} and note that {(8.19)} readily follows by time reversal. Moreover, we fix Z Z Z C ~ : = M σ ν B ∗ ( Z ) < ∞ \widetilde C:=\mathcal{M}_{\sigma}\nu_B^*(Z)<\infty C := M σ ν B ∗ ( Z ) < ∞ r ∗ : = 10 − 4 min { τ , η } r ( B ) r_*:=10^{-4}\min\{\tau,\eta\}r(B) r ∗ := 1 0 − 4 min { τ , η } r ( B ) ν : = ν B ∗ \nu:=\nu_B^* ν := ν B ∗
Let ε > 0 \varepsilon>0 ε > 0 ε ≥ r ∗ / 100 \varepsilon\ge r_*/100 ε ≥ r ∗ /100
∣ T ε ∗ ν ( Z ) ∣ ≲ ε − n − 1 σ ( B ) ≲ ε 0 min { τ , η } − n − 1 . (329) |T_\varepsilon^*\nu(Z)|\lesssim\varepsilon^{-n-1}\sigma(B)\lesssim\varepsilon_0\min\{\tau,\eta\}^{-n-1}. \tag*{(329)} ∣ T ε ∗ ν ( Z ) ∣ ≲ ε − n − 1 σ ( B ) ≲ ε 0 min { τ , η } − n − 1 . (329) Let us now assume that ε < r ∗ / 100 \varepsilon<r_*/100 ε < r ∗ /100 j ≥ 0 j\ge0 j ≥ 0 R j : = 100 j ε R_j:=100^j\varepsilon R j := 10 0 j ε k ≥ 0 k\ge0 k ≥ 0 R = R k R=R_k R = R k R k ≥ r ∗ / 100 R_k\ge r_*/100 R k ≥ r ∗ /100 R k < r ∗ R_k<r_* R k < r ∗
σ ( B ( Z , R j + 1 ) ) > 100 n + 2 σ ( B ( Z , R j ) ) , for 0 ≤ j < k . (330) \sigma(B(Z,R_{j+1}))>100^{n+2}\sigma(B(Z,R_j)),\qquad\text{for }0\le j<k. \tag*{(330)} σ ( B ( Z , R j + 1 )) > 10 0 n + 2 σ ( B ( Z , R j )) , for 0 ≤ j < k . (330) In either case, {(8.21)} or the preceding size estimate gives ∣ T R k ∗ ν ( Z ) ∣ ≤ C τ , η ( 1 + C ~ ) |T_{R_k}^*\nu(Z)|\le C_{\tau,\eta}(1+\widetilde C) ∣ T R k ∗ ν ( Z ) ∣ ≤ C τ , η ( 1 + C )
∣ T ε ∗ ν ( Z ) − T R k ∗ ν ( Z ) ∣ ≲ ∑ j = 0 k − 1 ν ( B ( Z , 2 R j + 1 ) ) R j n + 1 ≤ C ~ ∑ j = 0 k − 1 σ ( B ( Z , R j + 2 ) ) R j n + 1 . (331) \left|T_{\varepsilon}^{*}\nu(Z)-T_{R_k}^{*}\nu(Z)\right|\lesssim\sum_{j=0}^{k-1}\frac{\nu(B(Z,2R_{j+1}))}{R_j^{n+1}}\leq\widetilde C\sum_{j=0}^{k-1}\frac{\sigma(B(Z,R_{j+2}))}{R_j^{n+1}}. \tag*{(331)} T ε ∗ ν ( Z ) − T R k ∗ ν ( Z ) ≲ j = 0 ∑ k − 1 R j n + 1 ν ( B ( Z , 2 R j + 1 )) ≤ C j = 0 ∑ k − 1 R j n + 1 σ ( B ( Z , R j + 2 )) . (331) We assume without loss of generality that k ≥ 1 k\geq1 k ≥ 1 (331) is empty. Note that upper growth implies
σ ( B ( Z , R k + 1 ) ) R k n + 1 ≲ 100 2 ( n + 1 ) E 0 . (332) \frac{\sigma(B(Z,R_{k+1}))}{R_k^{n+1}}\lesssim100^{2(n+1)}\mathcal{E}_0. \tag*{(332)} R k n + 1 σ ( B ( Z , R k + 1 )) ≲ 10 0 2 ( n + 1 ) E 0 . (332) For j ≤ k − 2 j\leq k-2 j ≤ k − 2 (330) and again using upper growth yields
σ ( B ( Z , R j + 2 ) ) R j n + 1 ≤ 100 − ( n + 2 ) ( k − j − 2 ) σ ( B ( Z , R k ) ) R j n + 1 ≤ 100 2 ( n + 1 ) E 0 100 − ( k − j − 2 ) . (333) \frac{\sigma(B(Z,R_{j+2}))}{R_j^{n+1}}\leq100^{-(n+2)(k-j-2)}\frac{\sigma(B(Z,R_k))}{R_j^{n+1}}\leq100^{2(n+1)}\mathcal{E}_0\,100^{-(k-j-2)}. \tag*{(333)} R j n + 1 σ ( B ( Z , R j + 2 )) ≤ 10 0 − ( n + 2 ) ( k − j − 2 ) R j n + 1 σ ( B ( Z , R k )) ≤ 10 0 2 ( n + 1 ) E 0 10 0 − ( k − j − 2 ) . (333) Hence, we gather (331) , (332) , (333) , and obtain
∣ T ε ∗ ν ( Z ) − T R k ∗ ν ( Z ) ∣ ≲ E 0 C ~ , \left|T_{\varepsilon}^{*}\nu(Z)-T_{R_k}^{*}\nu(Z)\right|\lesssim\mathcal{E}_0\widetilde C, T ε ∗ ν ( Z ) − T R k ∗ ν ( Z ) ≲ E 0 C , which, combined with (323) and (329) , proves (320) . □
The exceptional sets on the selected balls and the proof of Theorem 1.9
Fix τ , η ∈ ( 0 , 1 / 10 ) \tau,\eta\in(0,1/10) τ , η ∈ ( 0 , 1/10 ) B B B (316) . We set
V B : = B ∩ B ( P B − , η r ( B ) / 2 ) , V B ∗ : = B ∩ B ( P B + , η r ( B ) / 2 ) , V_B:=B\cap B(P_B^{-},\eta r(B)/2),\qquad V_B^{*}:=B\cap B(P_B^{+},\eta r(B)/2), V B := B ∩ B ( P B − , η r ( B ) /2 ) , V B ∗ := B ∩ B ( P B + , η r ( B ) /2 ) , and, for A > 1 A>1 A > 1
U B : = V B ∪ { Z ∈ B : M σ ν B ( Z ) > A } , U B ∗ : = V B ∗ ∪ { Z ∈ B : M σ ν B ∗ ( Z ) > A } . (334) U_B:=V_B\cup\{Z\in B:\mathcal{M}_{\sigma\nu_B}(Z)>A\},\qquad U_B^{*}:=V_B^{*}\cup\{Z\in B:\mathcal{M}_{\sigma\nu_B^{*}}(Z)>A\}. \tag*{(334)} U B := V B ∪ { Z ∈ B : M σ ν B ( Z ) > A } , U B ∗ := V B ∗ ∪ { Z ∈ B : M σ ν B ∗ ( Z ) > A } . (334) Lemma 8.8. For every A > 1 A>1 A > 1
σ ( U B ) ≤ σ ( V B ) + A − 1 σ ( B ) and σ ( U B ∗ ) ≤ σ ( V B ∗ ) + A − 1 σ ( B ) . (335) \sigma(U_B)\leq\sigma(V_B)+A^{-1}\sigma(B)\qquad\text{and}\qquad\sigma(U_B^{*})\leq\sigma(V_B^{*})+A^{-1}\sigma(B). \tag*{(335)} σ ( U B ) ≤ σ ( V B ) + A − 1 σ ( B ) and σ ( U B ∗ ) ≤ σ ( V B ∗ ) + A − 1 σ ( B ) . (335) Moreover,
T ψ ν B ♯ ( Z ) ≤ C τ , η ( 1 + A ) , for Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ U B , (336) T_{\psi\nu_B}^{\sharp}(Z)\leq C_{\tau,\eta}(1+A),\qquad\text{for }Z\in((1-\tau)B\cap\operatorname{supp}\sigma)\setminus U_B, \tag*{(336)} T ψ ν B ♯ ( Z ) ≤ C τ , η ( 1 + A ) , for Z ∈ (( 1 − τ ) B ∩ supp σ ) ∖ U B , (336) and
T ψ ∗ ν B ∗ ( Z ) ≤ C τ , η ( 1 + A ) , for Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ U B ∗ , (337) T_{\psi^{*}\nu_B^{*}}(Z)\leq C_{\tau,\eta}(1+A),\qquad\text{for }Z\in((1-\tau)B\cap\operatorname{supp}\sigma)\setminus U_B^{*}, \tag*{(337)} T ψ ∗ ν B ∗ ( Z ) ≤ C τ , η ( 1 + A ) , for Z ∈ (( 1 − τ ) B ∩ supp σ ) ∖ U B ∗ , (337) where C τ , η C_{\tau,\eta} C τ , η
Proof. The estimates in (335) follow from (22) and (318) . If Z ∈ ( ( 1 − τ ) B ∩ supp σ ) ∖ U B Z\in((1-\tau)B\cap\operatorname{supp}\sigma)\setminus U_B Z ∈ (( 1 − τ ) B ∩ supp σ ) ∖ U B Z ∉ B ( P B − , η r ( B ) / 2 ) Z\notin B(P_B^{-},\eta r(B)/2) Z ∈ / B ( P B − , η r ( B ) /2 ) M σ ν B ( Z ) ≤ A \mathcal{M}_{\sigma\nu_B}(Z)\leq A M σ ν B ( Z ) ≤ A (321) gives (336) . The proof of (337) is analogous, with (320) in place of (321) . □
We prove Theorem 1.9 by applying Theorem 1.3 with ( n + 1 ) (n+1) ( n + 1 ) n n n μ : = σ \mu:=\sigma μ := σ δ 0 : = 1 / 2 \delta_0:=1/2 δ 0 := 1/2 Λ 0 : = 100 \Lambda_0:=100 Λ 0 := 100 δ \delta δ Λ \Lambda Λ β 0 : = β 0 ( D , n + 1 , δ 0 , Λ 0 ) \beta_0:=\beta_0(D,n+1,\delta_0,\Lambda_0) β 0 := β 0 ( D , n + 1 , δ 0 , Λ 0 ) β \beta β D D D P n \mathbb{P}^n P n B = B ( X 0 , r ) B=B(X_0,r) B = B ( X 0 , r ) B \mathcal{B} B X 0 ∈ supp σ X_0\in\operatorname{supp}\sigma X 0 ∈ supp σ σ ( Λ 0 B ) ≤ β 0 σ ( δ 0 B ) \sigma(\Lambda_0B)\leq\beta_0\sigma(\delta_0B) σ ( Λ 0 B ) ≤ β 0 σ ( δ 0 B ) δ \delta δ Λ \Lambda Λ
Lemma 8.9. There exist N = N ( n , δ , Λ ) ≥ 1 N=N(n,\delta,\Lambda)\ge1 N = N ( n , δ , Λ ) ≥ 1 B ′ = B ( X E , ρ ) B'=B(X_{\mathrm E},\rho) B ′ = B ( X E , ρ ) X E ∈ B / 2 ∩ supp σ X_{\mathrm E}\in B/2\cap\operatorname{supp}\sigma X E ∈ B /2 ∩ supp σ r / ( 8 Λ ) ≤ ρ ≤ r / ( 4 Λ ) r/(8\Lambda)\le\rho\le r/(4\Lambda) r / ( 8Λ ) ≤ ρ ≤ r / ( 4Λ )
Λ B ′ ⊂ B , σ ( ( δ B ′ ) ± ) ≥ σ ( B ) 8 β 0 N , and σ ( Λ B ′ ) ≤ 4 β 0 N σ ( δ B ′ ) . (338) \Lambda B'\subset B,\qquad\sigma\left((\delta B')^{\pm}\right)\ge\frac{\sigma(B)}{8\beta_0N},\qquad\text{and}\qquad\sigma(\Lambda B')\le4\beta_0N\sigma(\delta B'). \tag*{(338)} Λ B ′ ⊂ B , σ ( ( δ B ′ ) ± ) ≥ 8 β 0 N σ ( B ) , and σ ( Λ B ′ ) ≤ 4 β 0 N σ ( δ B ′ ) . (338) Proof. We put R : = r / ( 8 Λ ) R:=r/(8\Lambda) R := r / ( 8Λ ) σ ( B / 2 ) ≥ β 0 − 1 σ ( B ) \sigma(B/2)\ge\beta_0^{-1}\sigma(B) σ ( B /2 ) ≥ β 0 − 1 σ ( B ) F ⊂ B / 2 ∩ supp σ F\subset B/2\cap\operatorname{supp}\sigma F ⊂ B /2 ∩ supp σ σ ( F ) ≥ σ ( B ) / ( 2 β 0 ) \sigma(F)\ge\sigma(B)/(2\beta_0) σ ( F ) ≥ σ ( B ) / ( 2 β 0 ) F F F N N N δ R / 4 \delta R/4 δ R /4 N N N n n n δ \delta δ Λ \Lambda Λ E E E F F F σ ( E ) ≥ σ ( B ) / ( 4 β 0 N ) \sigma(E)\ge\sigma(B)/(4\beta_0N) σ ( E ) ≥ σ ( B ) / ( 4 β 0 N ) Lemma 8.1 , there exists X E = ( X E , t E ) ∈ E X_{\mathrm E}=(X_{\mathrm E},t_{\mathrm E})\in E X E = ( X E , t E ) ∈ E σ ∣ E \sigma|_E σ ∣ E E E E δ R / 4 \delta R/4 δ R /4 E ⊂ B ( X E , δ R / 2 ) E\subset B(X_{\mathrm E},\delta R/2) E ⊂ B ( X E , δ R /2 ) ε = 1 \varepsilon=1 ε = 1 R ≤ ρ ≤ 2 R R\le\rho\le2R R ≤ ρ ≤ 2 R B ′ : = B ( X E , ρ ) B':=B(X_{\mathrm E},\rho) B ′ := B ( X E , ρ )
The inclusion in (338) holds because X E ∈ B / 2 X_{\mathrm E}\in B/2 X E ∈ B /2 Λ ρ ≤ r / 4 \Lambda\rho\le r/4 Λ ρ ≤ r /4 E ⊂ δ B ′ E\subset\delta B' E ⊂ δ B ′ X E X_{\mathrm E} X E
σ ( ( δ B ′ ) ± ) ≥ 1 2 σ ( E ) ≥ σ ( B ) 8 β 0 N \sigma\left((\delta B')^{\pm}\right)\ge\frac{1}{2}\sigma(E)\ge\frac{\sigma(B)}{8\beta_0N} σ ( ( δ B ′ ) ± ) ≥ 2 1 σ ( E ) ≥ 8 β 0 N σ ( B ) and
σ ( Λ B ′ ) ≤ σ ( B ) ≤ 4 β 0 N σ ( E ) ≤ 4 β 0 N σ ( δ B ′ ) , \sigma(\Lambda B')\le\sigma(B)\le4\beta_0N\sigma(E)\le4\beta_0N\sigma(\delta B'), σ ( Λ B ′ ) ≤ σ ( B ) ≤ 4 β 0 N σ ( E ) ≤ 4 β 0 N σ ( δ B ′ ) , which concludes the proof. □
Let B ′ B' B ′ Lemma 8.9 . By construction and (338) , we have r ( B ′ ) ≈ r ( B ) r(B')\approx r(B) r ( B ′ ) ≈ r ( B ) σ ( B ′ ) ≈ σ ( B ) \sigma(B')\approx\sigma(B) σ ( B ′ ) ≈ σ ( B ) B ′ B' B ′ (316) . Moreover, B ′ B' B ′ Definition 1.8 , provided that the doubling constant β \beta β Definition 1.8 is at least 4 β 0 N 4\beta_0N 4 β 0 N B ′ B' B ′ supp σ \operatorname{supp}\sigma supp σ r ≤ diam ( supp σ ) r\le\operatorname{diam}(\operatorname{supp}\sigma) r ≤ diam ( supp σ ) C 3 C_3 C 3 C 3 ≥ 40 C_3\ge40 C 3 ≥ 40 σ ( Λ B ′ ) ≤ β σ ( δ B ′ ) \sigma(\Lambda B')\le\beta\sigma(\delta B') σ ( Λ B ′ ) ≤ β σ ( δ B ′ ) ν B ′ \nu_{B'} ν B ′ ν B ′ ∗ \nu^*_{B'} ν B ′ ∗ (317) , are defined, and we set
ν ^ B : = σ ( B ) σ ( B ′ ) ν B ′ and ν ^ B ∗ : = σ ( B ) σ ( B ′ ) ν B ′ ∗ . \widehat{\nu}_B:=\frac{\sigma(B)}{\sigma(B')}\nu_{B'}
\qquad\text{and}\qquad
\widehat{\nu}_B^*:=\frac{\sigma(B)}{\sigma(B')}\nu_{B'}^*. ν B := σ ( B ′ ) σ ( B ) ν B ′ and ν B ∗ := σ ( B ′ ) σ ( B ) ν B ′ ∗ . By (318) , these measures are non-negative, supported in B ′ ‾ ⊂ B \overline{B'}\subset B B ′ ⊂ B
ν ^ B ( B ) = ∥ ν ^ B ∥ = σ ( B ) and ν ^ B ∗ ( B ) = ∥ ν ^ B ∗ ∥ = σ ( B ) . (339) \widehat{\nu}_B(B)=\lVert\widehat{\nu}_B\rVert=\sigma(B)
\qquad\text{and}\qquad
\widehat{\nu}_B^*(B)=\lVert\widehat{\nu}_B^*\rVert=\sigma(B). \tag*{(339)} ν B ( B ) = ∥ ν B ∥ = σ ( B ) and ν B ∗ ( B ) = ∥ ν B ∗ ∥ = σ ( B ) . (339) Moreover, since σ ( B ) ≤ 4 β 0 N σ ( B ′ ) \sigma(B)\le4\beta_0N\sigma(B') σ ( B ) ≤ 4 β 0 N σ ( B ′ ) (338) , the condition (319) applied to B ′ B' B ′ A ⊂ B A\subset B A ⊂ B
if σ ( A ) ≤ κ 4 β 0 N σ ( B ) , then ν ^ B ( A ) ≤ κ ′ σ ( B ) and ν ^ B ∗ ( A ) ≤ κ ′ σ ( B ) . (340) \text{if }\sigma(A)\le\frac{\kappa}{4\beta_0N}\sigma(B),\qquad\text{then}\qquad
\widehat{\nu}_B(A)\le\kappa'\sigma(B)\quad\text{and}\quad
\widehat{\nu}_B^*(A)\le\kappa'\sigma(B). \tag*{(340)} if σ ( A ) ≤ 4 β 0 N κ σ ( B ) , then ν B ( A ) ≤ κ ′ σ ( B ) and ν B ∗ ( A ) ≤ κ ′ σ ( B ) . (340) Indeed, if σ ( A ) ≤ κ σ ( B ) / ( 4 β 0 N ) \sigma(A)\le\kappa\sigma(B)/(4\beta_0N) σ ( A ) ≤ κσ ( B ) / ( 4 β 0 N ) σ ( A ∩ B ′ ) ≤ κ σ ( B ′ ) \sigma(A\cap B')\le\kappa\sigma(B') σ ( A ∩ B ′ ) ≤ κσ ( B ′ )
ν ^ B ( A ) = σ ( B ) σ ( B ′ ) ν B ′ ( A ∩ B ′ ) ≤ κ ′ σ ( B ) , \widehat{\nu}_B(A)=\frac{\sigma(B)}{\sigma(B')}\nu_{B'}(A\cap B')\le\kappa'\sigma(B), ν B ( A ) = σ ( B ′ ) σ ( B ) ν B ′ ( A ∩ B ′ ) ≤ κ ′ σ ( B ) , and the same holds for ν ^ B ∗ \widehat{\nu}_B^* ν B ∗
Let ζ ∈ ( 0 , 1 ) \zeta\in(0,1) ζ ∈ ( 0 , 1 ) A : = 8 / ζ A:=8/\zeta A := 8/ ζ B ′ B' B ′ 5 B ′ ⊂ B 5B'\subset B 5 B ′ ⊂ B
σ ( ( 1 + h ) B ′ ∖ ( 1 − h ) B ′ ) ≤ 40 h σ ( B ) for h ∈ ( 0 , 1 ) . \sigma\left((1+h)B'\setminus(1-h)B'\right)\le40h\,\sigma(B)
\qquad\text{for }h\in(0,1). σ ( ( 1 + h ) B ′ ∖ ( 1 − h ) B ′ ) ≤ 40 h σ ( B ) for h ∈ ( 0 , 1 ) . Hence, we fix h ∈ ( 0 , 1 / 10 ) h\in(0,1/10) h ∈ ( 0 , 1/10 )
W B : = ( 1 + h ) B ′ ∖ ( 1 − h ) B ′ satisfies σ ( W B ) ≤ ζ 8 σ ( B ) . W_B:=(1+h)B'\setminus(1-h)B'
\qquad\text{satisfies}\qquad
\sigma(W_B)\le\frac{\zeta}{8}\sigma(B). W B := ( 1 + h ) B ′ ∖ ( 1 − h ) B ′ satisfies σ ( W B ) ≤ 8 ζ σ ( B ) . For η ∈ ( 0 , 1 / 10 ) \eta\in(0,1/10) η ∈ ( 0 , 1/10 ) V B ′ V_{B'} V B ′ U B ′ U_{B'} U B ′ V B ′ ∗ V_{B'}^* V B ′ ∗ U B ′ ∗ U_{B'}^* U B ′ ∗ B ′ B' B ′ A A A (334) . We define
V B : = V B ′ , U B : = U B ′ ∪ W B ∪ ( B ∖ supp σ ) , V B ∗ : = V B ′ ∗ , U B ∗ : = U B ′ ∗ ∪ W B ∪ ( B ∖ supp σ ) , V_B := V_{B'}, \qquad U_B := U_{B'} \cup W_B \cup(B \setminus\operatorname{supp}\sigma), \qquad V_B^* := V_{B'}^*, \qquad U_B^* := U_{B'}^* \cup W_B \cup(B \setminus\operatorname{supp}\sigma), V B := V B ′ , U B := U B ′ ∪ W B ∪ ( B ∖ supp σ ) , V B ∗ := V B ′ ∗ , U B ∗ := U B ′ ∗ ∪ W B ∪ ( B ∖ supp σ ) , and we prove that the alternative in the following lemma holds.
Lemma 8.10. There exists C 2 = C 2 ( η , ζ ) > 0 \mathfrak{C}_2=\mathfrak{C}_2(\eta,\zeta)>0 C 2 = C 2 ( η , ζ ) > 0 η \eta η ζ \zeta ζ Theorem 1.9 , such that the following holds.
If σ ( V B ) ≥ ζ σ ( B ) / 2 \sigma(V_B)\geq\zeta\sigma(B)/2 σ ( V B ) ≥ ζ σ ( B ) /2 V B ⊂ U B V_B\subset U_B V B ⊂ U B (4) .
If σ ( V B ) < ζ σ ( B ) / 2 \sigma(V_B)<\zeta\sigma(B)/2 σ ( V B ) < ζ σ ( B ) /2 σ ( U B ) < 3 ζ σ ( B ) / 4 \sigma(U_B)<3\zeta\sigma(B)/4 σ ( U B ) < 3 ζ σ ( B ) /4
T ♯ ν ^ B ( Z ) ≤ C 2 , Z ∈ B ∖ U B . (341) T_{\sharp\widehat{\nu}_B}(\mathbf{Z})\leq\mathfrak{C}_2,\qquad\mathbf{Z}\in B\setminus U_B. \tag*{(341)} T ♯ ν B ( Z ) ≤ C 2 , Z ∈ B ∖ U B . (341) The analogous statement holds for V B ∗ V_B^* V B ∗ U B ∗ U_B^* U B ∗ T ♯ ν ^ B ∗ T_{\sharp\widehat{\nu}_B^*} T ♯ ν B ∗
Proof. The first alternative is immediate because diam ( V B ) ≤ η r ( B ′ ) ≤ η r ( B ) \operatorname{diam}(V_B)\leq\eta r(B')\leq\eta r(B) diam ( V B ) ≤ η r ( B ′ ) ≤ η r ( B ) σ ( V B ) < ζ σ ( B ) / 2 \sigma(V_B)<\zeta\sigma(B)/2 σ ( V B ) < ζ σ ( B ) /2 (335) applied to B ′ B' B ′ A A A h h h σ ( B ∖ supp σ ) = 0 \sigma(B\setminus\operatorname{supp}\sigma)=0 σ ( B ∖ supp σ ) = 0
σ ( U B ) ≤ σ ( V B ′ ) + A − 1 σ ( B ′ ) + σ ( W B ) < ζ 2 σ ( B ) + ζ 8 σ ( B ) + ζ 8 σ ( B ) = 3 4 ζ σ ( B ) . \begin{aligned}
\sigma(U_B)&\leq\sigma(V_{B'})+A^{-1}\sigma(B')+\sigma(W_B)\\
&<\frac{\zeta}{2}\sigma(B)+\frac{\zeta}{8}\sigma(B)+\frac{\zeta}{8}\sigma(B)=\frac{3}{4}\zeta\sigma(B).
\end{aligned} σ ( U B ) ≤ σ ( V B ′ ) + A − 1 σ ( B ′ ) + σ ( W B ) < 2 ζ σ ( B ) + 8 ζ σ ( B ) + 8 ζ σ ( B ) = 4 3 ζ σ ( B ) . To prove (341) , we fix Z ∈ B ∖ U B \mathbf{Z}\in B\setminus U_B Z ∈ B ∖ U B W B ⊂ U B W_B\subset U_B W B ⊂ U B Z ∈ ( 1 − h ) B ′ \mathbf{Z}\in(1-h)B' Z ∈ ( 1 − h ) B ′ Z ∉ ( 1 + h ) B ′ \mathbf{Z}\notin(1+h)B' Z ∈ / ( 1 + h ) B ′ Z ∈ ( ( 1 − h ) B ′ ∩ supp σ ) ∖ U B ′ \mathbf{Z}\in((1-h)B'\cap\operatorname{supp}\sigma)\setminus U_{B'} Z ∈ (( 1 − h ) B ′ ∩ supp σ ) ∖ U B ′ (336) applied to B ′ B' B ′ τ = h \tau=h τ = h σ ( B ) ≤ 4 β 0 N σ ( B ′ ) \sigma(B)\leq4\beta_0N\sigma(B') σ ( B ) ≤ 4 β 0 N σ ( B ′ ) (338) , readily give
T ♯ ν ^ B ( Z ) = σ ( B ) σ ( B ′ ) T ♯ ν ^ B ′ ( Z ) ≤ 4 β 0 N C h , η ( 1 + A ) . T_{\sharp\widehat{\nu}_B}(\mathbf{Z})=\frac{\sigma(B)}{\sigma(B')}T_{\sharp\widehat{\nu}_{B'}}(\mathbf{Z})\leq4\beta_0N\mathcal{C}_{h,\eta}(1+A). T ♯ ν B ( Z ) = σ ( B ′ ) σ ( B ) T ♯ ν B ′ ( Z ) ≤ 4 β 0 N C h , η ( 1 + A ) . In the second case, the support of ν ^ B \widehat{\nu}_B ν B h r ( B ′ ) h r(B') h r ( B ′ ) Z \mathbf{Z} Z (339) , the ( n + 1 ) (n+1) ( n + 1 ) σ \sigma σ r ( B ′ ) ≥ r / ( 8 Λ ) r(B')\geq r/(8\Lambda) r ( B ′ ) ≥ r / ( 8Λ )
T ♯ ν ^ B ( Z ) ≲ σ ( B ) ( h r ( B ′ ) ) n + 1 ≲ C 0 ( 8 Λ h ) n + 1 . T_{\sharp\widehat{\nu}_B}(\mathbf{Z})\lesssim\frac{\sigma(B)}{(h r(B'))^{n+1}}\lesssim\mathfrak{C}_0\left(\frac{8\Lambda}{h}\right)^{n+1}. T ♯ ν B ( Z ) ≲ ( h r ( B ′ ) ) n + 1 σ ( B ) ≲ C 0 ( h 8Λ ) n + 1 . Since h h h A A A ζ \zeta ζ (341) . The proof for ν ^ B ∗ \widehat{\nu}_B^* ν B ∗ (337) in place of (336) . □
We finally conclude the section by gathering all the results above and applying our local T b Tb T b
Proof of Theorem 1.9 . We apply Theorem 1.3 , as described before Lemma 8.9 , to each component of T σ T_\sigma T σ ν ^ B \widehat{\nu}_B ν B ν ^ B ∗ \widehat{\nu}_B^* ν B ∗ B ∈ B B\in\mathcal{B} B ∈ B (339) , the measures ν ^ B \widehat{\nu}_B ν B ν ^ B ∗ \widehat{\nu}_B^* ν B ∗ (2) with C 1 : = 1 \mathfrak{C}_1:=1 C 1 := 1 κ ′ ≤ 1 / 200 \kappa'\leq1/200 κ ′ ≤ 1/200 (340) gives (3) with C 1 = 1 \mathfrak{C}_1=1 C 1 = 1 ε 0 : = κ / ( 4 β 0 N ) \varepsilon_0:=\kappa/(4\beta_0N) ε 0 := κ / ( 4 β 0 N ) Theorem 1.9 .
We fix τ ∈ ( 0 , τ 0 ) \tau\in(0,\tau_0) τ ∈ ( 0 , τ 0 ) ζ ∈ ( 0 , ζ 0 ) \zeta\in(0,\zeta_0) ζ ∈ ( 0 , ζ 0 ) τ 0 \tau_0 τ 0 ζ 0 \zeta_0 ζ 0 (8) . Since (341) holds on the whole B ∖ U B B\setminus U_B B ∖ U B Lemma 8.10 shows that, for every η ∈ ( 0 , 1 / 10 ) \eta\in(0,1/10) η ∈ ( 0 , 1/10 ) (4) and (5) holds with C 2 ( η ) : = C 2 ( η , ζ ) \mathfrak{C}_2(\eta):=\mathfrak{C}_2(\eta,\zeta) C 2 ( η ) := C 2 ( η , ζ ) ν ^ B \widehat{\nu}_B ν B T σ T_\sigma T σ ν ^ B ∗ \widehat{\nu}_B^* ν B ∗ T σ ∗ T_\sigma^* T σ ∗ T σ T_\sigma T σ T σ ∗ T_\sigma^* T σ ∗ T σ T_\sigma T σ T σ ∗ T_\sigma^* T σ ∗ { ν ^ B } B ∈ B \{\widehat{\nu}_B\}_{B\in\mathcal{B}} { ν B } B ∈ B { ν ^ B ∗ } B ∈ B \{\widehat{\nu}_B^*\}_{B\in\mathcal{B}} { ν B ∗ } B ∈ B T σ T_\sigma T σ Definition 1.2 . Finally, Theorem 1.3 applies to each component of T σ T_\sigma T σ Theorem 1.9 . □
Remark 8.11. The level C 2 ( η , ζ ) \mathcal{C}_2(\eta,\zeta) C 2 ( η , ζ ) Lemma 8.10 associated with the L 0 L^0 L 0 η \eta η C h , η C_{\mathrm{h},\eta} C h , η (328) . The poles P B ± \mathbf{P}^{\pm}_{B} P B ± ∂ Ω \partial\Omega ∂ Ω (4) in the testing dichotomy takes care of the balls on which σ \sigma σ Theorem 1.3 depends on the levels only through C 2 ( η ∗ ) C_2(\eta_*) C 2 ( η ∗ ) η ∗ \eta_* η ∗ n n n ζ \zeta ζ (101) . Thus, in Theorem 1.9 , the level is ultimately determined by ζ \zeta ζ Corollary 1.5 , where it does not depend on η \eta η
Appendix A. David-Mattila, random, and Whitney-type cubes in metric spaces
Throughout this appendix, ( X , d ) (X,d) ( X , d ) D D D
Construction of the David-Mattila testing balls
A.1. Construction of the David-Mattila testing balls. Let σ \sigma σ X X X supp σ \operatorname{supp}\sigma supp σ D σ \mathcal{D}_{\sigma} D σ Lemma 2.3 and, otherwise, use the lattice D σ ( s ) \mathcal{D}_{\sigma}^{(s)} D σ ( s ) Lemma 2.5 , for a fixed s > 0 s>0 s > 0 supp σ \operatorname{supp}\sigma supp σ D σ \mathcal{D}_{\sigma} D σ r Q r_Q r Q N j ∗ i n t ( Q ) N_{j_*}^{\mathrm{int}}(Q) N j ∗ int ( Q ) § 2.3.1 , with the same boundary-decay constant for both lattices.
Lemma A.1. For every Q ∈ D σ d b Q\in\mathcal{D}_{\sigma}^{\mathrm{db}} Q ∈ D σ db x Q ∈ Q ∖ N j ∗ i n t ( Q ) x_Q\in Q\setminus N_{j_*}^{\mathrm{int}}(Q) x Q ∈ Q ∖ N j ∗ int ( Q ) ρ Q > 0 \rho_Q>0 ρ Q > 0 B ~ ( Q ) : = B ( x Q , ρ Q ) \widetilde{B}(Q):=B(x_Q,\rho_Q) B ( Q ) := B ( x Q , ρ Q )
(1) We have that ρ Q ≈ r Q \rho_Q\approx r_Q ρ Q ≈ r Q
c ∗ 4 Λ r Q ≤ ρ Q ≤ c ∗ 2 Λ r Q . (342) \frac{c_*}{4\Lambda}r_Q\leq\rho_Q\leq\frac{c_*}{2\Lambda}r_Q. \tag*{(342)} 4Λ c ∗ r Q ≤ ρ Q ≤ 2Λ c ∗ r Q . (342) (2) We have the inclusion
Λ B ~ ( Q ) ∩ supp ( σ ) ⊂ Q ⊂ M ∗ B ~ ( Q ) . \Lambda\widetilde{B}(Q)\cap\operatorname{supp}(\sigma)\subset Q\subset M_*\widetilde{B}(Q). Λ B ( Q ) ∩ supp ( σ ) ⊂ Q ⊂ M ∗ B ( Q ) . (3) B ~ ( Q ) \widetilde{B}(Q) B ( Q )
σ ( Λ B ~ ( Q ) ) ≤ σ ( Q ) ≤ β ∗ σ ( δ B ~ ( Q ) ) . (343) \sigma(\Lambda\widetilde{B}(Q))\leq\sigma(Q)\leq\beta_*\sigma(\delta\widetilde{B}(Q)). \tag*{(343)} σ ( Λ B ( Q )) ≤ σ ( Q ) ≤ β ∗ σ ( δ B ( Q )) . (343) (4) B ~ ( Q ) \widetilde{B}(Q) B ( Q ) σ \sigma σ 0 < λ ≤ 1 0<\lambda\leq1 0 < λ ≤ 1
σ ( { x ∈ 5 B ~ ( Q ) : ∣ dist ( x , x Q ) − ρ Q ∣ ≤ λ ρ Q } ) ≤ 40 λ σ ( 5 B ~ ( Q ) ) . (344) \sigma\left(\left\{x\in5\widetilde{B}(Q):\left|\operatorname{dist}(x,x_Q)-\rho_Q\right|\leq\lambda\rho_Q\right\}\right)\leq40\lambda\sigma(5\widetilde{B}(Q)). \tag*{(344)} σ ( { x ∈ 5 B ( Q ) : ∣ dist ( x , x Q ) − ρ Q ∣ ≤ λ ρ Q } ) ≤ 40 λσ ( 5 B ( Q )) . (344) Proof. First, we define the center x Q x_Q x Q B ~ ( Q ) \widetilde{B}(Q) B ( Q ) F Q : = Q ∖ N j ∗ i n t ( Q ) F_Q:=Q\setminus N_{j_*}^{\mathrm{int}}(Q) F Q := Q ∖ N j ∗ int ( Q ) F Q ⊂ Q ⊂ 28 B ( Q ) F_Q\subset Q\subset28B(Q) F Q ⊂ Q ⊂ 28 B ( Q ) R Q : = c ∗ 4 Λ r Q R_Q:=\frac{c_*}{4\Lambda}r_Q R Q := 4Λ c ∗ r Q m ∗ m_* m ∗
28 2 m ∗ r Q ≤ δ 2 R Q . (345) \frac{28}{2^{m_*}}r_Q\leq\frac{\delta}{2}R_Q. \tag*{(345)} 2 m ∗ 28 r Q ≤ 2 δ R Q . (345) By the geometrically doubling property, 28 B ( Q ) 28B(Q) 28 B ( Q ) D m ∗ D^{m_*} D m ∗ 28 r Q / 2 m ∗ 28r_Q/2^{m_*} 28 r Q / 2 m ∗ F Q F_Q F Q F Q F_Q F Q δ R Q \delta R_Q δ R Q D m ∗ D^{m_*} D m ∗ σ ( F Q ) ≥ σ ( Q ) / 2 \sigma(F_Q)\geq\sigma(Q)/2 σ ( F Q ) ≥ σ ( Q ) /2 Lemma 2.6 , one of them, whose center we denote by x Q x_Q x Q E Q : = F Q ∩ B ( x Q , δ R Q ) E_Q:=F_Q\cap B(x_Q,\delta R_Q) E Q := F Q ∩ B ( x Q , δ R Q )
σ ( E Q ) ≥ σ ( Q ) 2 D m ∗ . (346) \sigma(E_Q)\geq\frac{\sigma(Q)}{2D^{m_*}}. \tag*{(346)} σ ( E Q ) ≥ 2 D m ∗ σ ( Q ) . (346) To define the radius of B ~ ( Q ) \widetilde{B}(Q) B ( Q ) ρ Q ∈ [ R Q , 2 R Q ] \rho_Q \in[R_Q,2R_Q] ρ Q ∈ [ R Q , 2 R Q ] B ~ ( Q ) : = B ( x Q , ρ Q ) \widetilde{B}(Q):=B(x_Q,\rho_Q) B ( Q ) := B ( x Q , ρ Q ) σ \sigma σ
To prove (2), it is enough to observe that
Λ ρ Q ≤ 2 Λ R Q = c ∗ 2 r Q < dist ( x Q , supp ( σ ) ∖ Q ) \Lambda\rho_Q \le2\Lambda R_Q = \frac{c_*}{2}r_Q < \operatorname{dist}(x_Q,\operatorname{supp}(\sigma)\setminus Q) Λ ρ Q ≤ 2Λ R Q = 2 c ∗ r Q < dist ( x Q , supp ( σ ) ∖ Q ) by Lemma 2.6 , since x Q ∈ F Q x_Q \in F_Q x Q ∈ F Q y ∈ Q y \in Q y ∈ Q x Q , y ∈ Q ⊂ 28 B ( Q ) x_Q,y \in Q \subset28B(Q) x Q , y ∈ Q ⊂ 28 B ( Q )
d ( y , x Q ) < 56 r Q ≤ 224 Λ c ∗ ρ Q = M ∗ ρ Q . d(y,x_Q)<56r_Q\le\frac{224\Lambda}{c_*}\rho_Q=M_*\rho_Q. d ( y , x Q ) < 56 r Q ≤ c ∗ 224Λ ρ Q = M ∗ ρ Q . Finally, the first inequality in (A.2) follows from (2). To prove the second one, note that the inequality ρ Q ≥ R Q \rho_Q\ge R_Q ρ Q ≥ R Q B ( x Q , δ R Q ) ⊂ δ B ~ ( Q ) B(x_Q,\delta R_Q)\subset\delta\widetilde{B}(Q) B ( x Q , δ R Q ) ⊂ δ B ( Q )
σ ( Q ) ≤ 2 D m ∗ σ ( δ B ~ ( Q ) ) = β ∗ σ ( δ B ~ ( Q ) ) . \sigma(Q)\le2D^{m_*}\sigma(\delta\widetilde{B}(Q))=\beta_*\sigma(\delta\widetilde{B}(Q)). σ ( Q ) ≤ 2 D m ∗ σ ( δ B ( Q )) = β ∗ σ ( δ B ( Q )) . Since, see Remark 2.7, we assume β > β ∗ \beta>\beta_* β > β ∗ B ~ ( R ) \widetilde{B}(R) B ( R ) R ∈ D σ d b R\in\mathcal{D}^{db}_\sigma R ∈ D σ d b
σ ( Λ B ~ ( R ) ) ≤ β σ ( δ B ~ ( R ) ) , \sigma(\Lambda\widetilde{B}(R))\le\beta\sigma(\delta\widetilde{B}(R)), σ ( Λ B ( R )) ≤ β σ ( δ B ( R )) , so they satisfy the doubling and small-boundary requirements of Theorem 1.3 . The radius requirement is automatic when supp σ \operatorname{supp}\sigma supp σ 0 < diam W < ∞ 0<\operatorname{diam}W<\infty 0 < diam W < ∞ W = supp σ W=\operatorname{supp}\sigma W = supp σ Lemma 2.3 and for the lattice of Lemma 2.5 with s ≤ 2 diam W s\le2\operatorname{diam}W s ≤ 2 diam W ρ Q ≤ c ∗ K 0 s / ( 2 Λ ) ≤ diam W \rho_Q\le c_*K_0s/(2\Lambda)\le\operatorname{diam}W ρ Q ≤ c ∗ K 0 s / ( 2Λ ) ≤ diam W § 4 .
By a standard application of Lemma 2.2, for any given non-negative Radon measure m m m X X X B ~ ( Q ) \widetilde{B}(Q) B ( Q ) Lemma A.1 can be selected so that the ball satisfies all the properties above, and an additional analogue of (A.3) with m m m σ \sigma σ σ \sigma σ m m m
Testing balls associated with doubling David-Mattila cells satisfy the following, additional properties.
Lemma A.2. For Q , R ∈ D σ d b Q,R\in\mathcal{D}^{db}_\sigma Q , R ∈ D σ d b Q ⊂ R Q\subset R Q ⊂ R J ( Q ) > J ( R ) J(Q)>J(R) J ( Q ) > J ( R )
ρ Q ρ R ≤ 2 K 0 A 0 − ( J ( Q ) − J ( R ) ) ≤ 2 K 0 A 0 < 1 2 . (347) \frac{\rho_Q}{\rho_R}\le2K_0A_0^{-(J(Q)-J(R))}\le\frac{2K_0}{A_0}<\frac{1}{2}. \tag*{(347)} ρ R ρ Q ≤ 2 K 0 A 0 − ( J ( Q ) − J ( R )) ≤ A 0 2 K 0 < 2 1 . (347) 2 M ∗ B ~ ( Q ) ⊂ 2 M ∗ B ~ ( R ) . (348) 2M_*\widetilde{B}(Q)\subset2M_*\widetilde{B}(R). \tag*{(348)} 2 M ∗ B ( Q ) ⊂ 2 M ∗ B ( R ) . (348) 84 B ( Q ) ⊂ 2 M ∗ B ~ ( Q ) and 100 B ( Q ) ⊂ 2 A Λ , ∗ B ~ ( Q ) . (349) 84B(Q)\subset2M_*\widetilde{B}(Q)\qquad\text{and}\qquad100B(Q)\subset2A_{\Lambda,*}\widetilde{B}(Q). \tag*{(349)} 84 B ( Q ) ⊂ 2 M ∗ B ( Q ) and 100 B ( Q ) ⊂ 2 A Λ , ∗ B ( Q ) . (349) Proof. The estimate (A.1) and the bounds s A 0 − J ( P ) ≤ r P ≤ K 0 s A 0 − J ( P ) sA_0^{-J(P)}\le r_P\le K_0sA_0^{-J(P)} s A 0 − J ( P ) ≤ r P ≤ K 0 s A 0 − J ( P ) P ∈ { Q , R } P\in\{Q,R\} P ∈ { Q , R } s = diam ( supp σ ) s=\operatorname{diam}(\operatorname{supp}\sigma) s = diam ( supp σ ) Lemma 2.3 , give ρ Q / ρ R ≤ 2 r Q / r R ≤ 2 K 0 A 0 − ( J ( Q ) − J ( R ) ) \rho_Q/\rho_R\le2r_Q/r_R\le2K_0A_0^{-(J(Q)-J(R))} ρ Q / ρ R ≤ 2 r Q / r R ≤ 2 K 0 A 0 − ( J ( Q ) − J ( R )) A 0 > 5000 K 0 A_0>5000K_0 A 0 > 5000 K 0 x Q ∈ Q x_Q\in Q x Q ∈ Q Q ⊂ R ⊂ 28 B ( R ) Q\subset R\subset28B(R) Q ⊂ R ⊂ 28 B ( R )
d ( x Q , x R ) < 56 r R ≤ M ∗ ρ R . (350) d(x_Q,x_R)<56r_R\le M_*\rho_R. \tag*{(350)} d ( x Q , x R ) < 56 r R ≤ M ∗ ρ R . (350) For y ∈ 2 M ∗ B ~ ( Q ) y\in2M_*\widetilde{B}(Q) y ∈ 2 M ∗ B ( Q ) d ( y , x R ) < 2 M ∗ ρ Q + M ∗ ρ R < 2 M ∗ ρ R d(y,x_R)<2M_*\rho_Q+M_*\rho_R<2M_*\rho_R d ( y , x R ) < 2 M ∗ ρ Q + M ∗ ρ R < 2 M ∗ ρ R x Q ∈ Q ⊂ 28 B ( Q ) x_Q\in Q\subset28B(Q) x Q ∈ Q ⊂ 28 B ( Q ) M ∗ M_* M ∗ A Λ , ∗ A_{\Lambda,*} A Λ , ∗ d ( y , x Q ) < 112 r Q ≤ 2 M ∗ ρ Q d(y,x_Q)<112r_Q\le2M_*\rho_Q d ( y , x Q ) < 112 r Q ≤ 2 M ∗ ρ Q y ∈ 84 B ( Q ) y\in84B(Q) y ∈ 84 B ( Q ) d ( y , x Q ) < 128 r Q ≤ 2 A Λ , ∗ ρ Q d(y,x_Q)<128r_Q\le2A_{\Lambda,*}\rho_Q d ( y , x Q ) < 128 r Q ≤ 2 A Λ , ∗ ρ Q y ∈ 100 B ( Q ) y\in100B(Q) y ∈ 100 B ( Q )
Properties of random cubes
First, we consider the dyadic cubes associated with ( X , d ) (X,d) ( X , d ) Theorem 2.9 .
Theorem A.3 ([39 ] , Theorem 5.6). Let ( Ω , P ) (\Omega,\mathbb{P}) ( Ω , P ) Theorem 2.11 with parameter D D D 144 ℓ ≤ 1 144\ell\le1 144 ℓ ≤ 1 Q Q Q
∂ ( Q , ε ) : = { x ∈ Q ~ : dist ( x , X ∖ Q ~ ) ≤ ε } , ε > 0. \partial(Q,\varepsilon):=\{x\in\widetilde{Q}:\operatorname{dist}(x,X\setminus\widetilde{Q})\le\varepsilon\},\qquad\varepsilon>0. ∂ ( Q , ε ) := { x ∈ Q : dist ( x , X ∖ Q ) ≤ ε } , ε > 0. Then there exist constants C 2 , η > 0 C_2,\eta>0 C 2 , η > 0 C 2 C_2 C 2 c 1 c_1 c 1 ℓ \ell ℓ η \eta η ℓ \ell ℓ τ 0 \tau_0 τ 0 Theorem 2.11 , such that, for every x ∈ X x\in X x ∈ X τ > 0 \tau>0 τ > 0 k ∈ Z k\in\mathbb{Z} k ∈ Z
P ( { w ∈ Ω : x ∈ ⋃ α ∈ A k ∂ Q α k ( w ) , τ ℓ k } ) ≤ C 2 τ η . \mathbb{P}\left(\left\{w\in\Omega:x\in\bigcup_{\alpha\in A_k}\partial Q_\alpha^k(w),\tau\ell^k\right\}\right)\le C_2\tau^\eta. P ( { w ∈ Ω : x ∈ α ∈ A k ⋃ ∂ Q α k ( w ) , τ ℓ k } ) ≤ C 2 τ η . Moreover, for every x ∈ X x\in X x ∈ X
P ( { w ∈ Ω : x ∈ ⋃ k , α ∂ Q α k ( w ) } ) = 0 , \mathbb{P}\left(\left\{w\in\Omega:x\in\bigcup_{k,\alpha}\partial Q_\alpha^k(w)\right\}\right)=0, P ⎩ ⎨ ⎧ w ∈ Ω : x ∈ k , α ⋃ ∂ Q α k ( w ) ⎭ ⎬ ⎫ = 0 , and, if μ \mu μ σ \sigma σ X X X
μ ( ⋃ k , α ∂ Q α k ( w ) ) = 0 , for P -a.e. w ∈ Ω . \mu\left(\bigcup_{k,\alpha}\partial Q_\alpha^k(w)\right)=0,\qquad\text{for }\mathbb{P}\text{-a.e. }w\in\Omega. μ k , α ⋃ ∂ Q α k ( w ) = 0 , for P -a.e. w ∈ Ω. For i = 1 , 2 i=1,2 i = 1 , 2 D i ( w i ) \mathbb{D}_i(w_i) D i ( w i ) Q ∈ D i ( w i ) Q\in\mathbb{D}_i(w_i) Q ∈ D i ( w i ) Q = Q α k ( w i ) Q=Q_\alpha^k(w_i) Q = Q α k ( w i ) k ∈ Z k\in\mathbb{Z} k ∈ Z ℓ ( Q ) = ℓ k \ell(Q)=\ell^k ℓ ( Q ) = ℓ k
We also set
D i , k ( w i ) : = { Q ∈ D i ( w i ) : ℓ ( Q ) = ℓ k } , k ∈ Z , \mathbb{D}_{i,k}(w_i):=\{Q\in\mathbb{D}_i(w_i):\ell(Q)=\ell^k\},\qquad k\in\mathbb{Z}, D i , k ( w i ) := { Q ∈ D i ( w i ) : ℓ ( Q ) = ℓ k } , k ∈ Z , and we refer to good and bad cubes as in Definition 6.1 . Recall that a cube Q ∈ D 1 ( w 1 ) Q\in\mathbb{D}_1(w_1) Q ∈ D 1 ( w 1 ) R ∈ D 2 ( w 2 ) R\in\mathbb{D}_2(w_2) R ∈ D 2 ( w 2 ) ℓ ( R ) ≥ ℓ − m ℓ ( Q ) \ell(R)\ge\ell^{-m}\ell(Q) ℓ ( R ) ≥ ℓ − m ℓ ( Q )
dist ( Q , R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α , dist ( Q , X ∖ R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α . \operatorname{dist}(Q,R)\le\ell(Q)^\alpha\ell(R)^{1-\alpha},\qquad\operatorname{dist}(Q,X\setminus R)\le\ell(Q)^\alpha\ell(R)^{1-\alpha}. dist ( Q , R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α , dist ( Q , X ∖ R ) ≤ ℓ ( Q ) α ℓ ( R ) 1 − α . The cube Q Q Q R R R
Lemma A.4. Let 0 < α < 1 0<\alpha<1 0 < α < 1 κ ∈ ( 0 , 1 ) \kappa\in(0,1) κ ∈ ( 0 , 1 ) L < ∞ L<\infty L < ∞ m 0 ∈ N m_0\in\mathbb{N} m 0 ∈ N ℓ \ell ℓ C 1 C_1 C 1 C 2 C_2 C 2 η \eta η α \alpha α κ \kappa κ M ∗ M_* M ∗ B i = B ( x i , r i ) B_i=B(x_i,r_i) B i = B ( x i , r i ) i = 1 , 2 i=1,2 i = 1 , 2 r 1 ≤ r 2 r_1\le r_2 r 1 ≤ r 2 2 M ∗ B 1 ⊂ 2 M ∗ B 2 2M_*B_1\subset2M_*B_2 2 M ∗ B 1 ⊂ 2 M ∗ B 2 k i k_i k i Q i ( w j ) Q_i(w_j) Q i ( w j ) D j , k i ( w j ) \mathbb{D}_{j,k_i}(w_j) D j , k i ( w j ) x i x_i x i m ≥ m 0 m\ge m_0 m ≥ m 0
( P ⊗ P ) ( { w ∈ Ω 2 : 6 M ∗ B i ⊂ Q i ( w j ) , diam Q i ( w j ) ≤ L r i , Q 1 ( w j ) ⊂ Q 2 ( w j ) , Q i ( w j ) ∈ D j g ( w , m ) , i , j = 1 , 2 } ) > 1 − κ . (\mathbb{P}\otimes\mathbb{P})\left(\left\{w\in\Omega^2:6M_*B_i\subset Q_i(w_j),\ \operatorname{diam}Q_i(w_j)\le Lr_i,\ Q_1(w_j)\subset Q_2(w_j),\ Q_i(w_j)\in\mathbb{D}_j^g(w,m),\ i,j=1,2\right\}\right)>1-\kappa. ( P ⊗ P ) ( { w ∈ Ω 2 : 6 M ∗ B i ⊂ Q i ( w j ) , diam Q i ( w j ) ≤ L r i , Q 1 ( w j ) ⊂ Q 2 ( w j ) , Q i ( w j ) ∈ D j g ( w , m ) , i , j = 1 , 2 } ) > 1 − κ . In particular, L L L m ≥ m 0 m\ge m_0 m ≥ m 0
Proof. We choose ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 )
2 ( C 2 + 1 ) ε η = κ 4 , 2(C_2+1)\varepsilon^\eta=\frac{\kappa}{4}, 2 ( C 2 + 1 ) ε η = 4 κ , and let k i k_i k i
ℓ k i ≥ 6 M ∗ r i ε . \ell^{k_i}\ge\frac{6M_*r_i}{\varepsilon}. ℓ k i ≥ ε 6 M ∗ r i . which yields ℓ k i < 6 M ∗ r i / ( ε ℓ ) \ell^{k_i}<6M_*r_i/(\varepsilon\ell) ℓ k i < 6 M ∗ r i / ( ε ℓ ) E i j : = { w j : 6 M ∗ B i ⊂ Q i ( w j ) } E_i^j:=\{w_j:6M_*B_i\subset Q_i(w_j)\} E i j := { w j : 6 M ∗ B i ⊂ Q i ( w j )} w j ∉ E i j w_j\notin E_i^j w j ∈ / E i j
dist ( x i , X ∖ Q i ( w j ) ) ≤ 6 M ∗ r i ≤ ε ℓ k i , \operatorname{dist}(x_i,X\setminus Q_i(w_j))\le6M_*r_i\le\varepsilon\ell^{k_i}, dist ( x i , X ∖ Q i ( w j )) ≤ 6 M ∗ r i ≤ ε ℓ k i , so x i ∈ ∂ ( Q i ( w j ) , ε ℓ k i ) x_i\in\partial(Q_i(w_j),\varepsilon\ell^{k_i}) x i ∈ ∂ ( Q i ( w j ) , ε ℓ k i ) Theorem A.3 , we deduce that
P ( E 1 j ∩ E 2 j ) ≥ 1 − 2 C 2 ε η > 1 − κ 4 , j = 1 , 2. (351) \mathbb{P}(E_1^j\cap E_2^j)\ge1-2C_2\varepsilon^\eta>1-\frac{\kappa}{4},\qquad j=1,2. \tag*{(351)} P ( E 1 j ∩ E 2 j ) ≥ 1 − 2 C 2 ε η > 1 − 4 κ , j = 1 , 2. (351) We also note that, if w j ∈ E 1 j ∩ E 2 j w_j\in E_1^j\cap E_2^j w j ∈ E 1 j ∩ E 2 j x 1 ∈ 2 M ∗ B 1 ⊂ 2 M ∗ B 2 ⊂ Q 2 ( w j ) x_1\in2M_*B_1\subset2M_*B_2\subset Q_2(w_j) x 1 ∈ 2 M ∗ B 1 ⊂ 2 M ∗ B 2 ⊂ Q 2 ( w j ) k 1 ≥ k 2 k_1\ge k_2 k 1 ≥ k 2 Q 1 ( w j ) ⊂ Q 2 ( w j ) Q_1(w_j)\subset Q_2(w_j) Q 1 ( w j ) ⊂ Q 2 ( w j )
diam Q i ( w j ) ≤ 2 C 1 ℓ k i < 12 M ∗ C 1 ε ℓ r i . \operatorname{diam}Q_i(w_j)\le2C_1\ell^{k_i}<\frac{12M_*C_1}{\varepsilon\ell}r_i. diam Q i ( w j ) ≤ 2 C 1 ℓ k i < ε ℓ 12 M ∗ C 1 r i . Fix w 1 w_1 w 1 Q = Q i ( w 1 ) Q=Q_i(w_1) Q = Q i ( w 1 ) k = k i k=k_i k = k i Q Q Q h ≤ k − m h\le k-m h ≤ k − m R ∈ D 2 , h ( w 2 ) R\in\mathcal{D}_{2,h}(w_2) R ∈ D 2 , h ( w 2 ) Q Q Q R R R X ∖ R X\setminus R X ∖ R ℓ α k ℓ ( 1 − α ) h \ell^{\alpha k}\ell^{(1-\alpha)h} ℓ α k ℓ ( 1 − α ) h x i ∈ Q x_i\in Q x i ∈ Q x i x_i x i
t h : = ℓ α k ℓ ( 1 − α ) h + 2 C 1 ℓ k . t_h:=\ell^{\alpha k}\ell^{(1-\alpha)h}+2C_1\ell^k. t h := ℓ α k ℓ ( 1 − α ) h + 2 C 1 ℓ k . Let S ∈ D 2 , h ( w 2 ) S\in\mathcal{D}_{2,h}(w_2) S ∈ D 2 , h ( w 2 ) x i x_i x i x i ∈ R x_i\in R x i ∈ R S = R S=R S = R dist ( x i , X ∖ S ) ≤ t h \operatorname{dist}(x_i,X\setminus S)\le t_h dist ( x i , X ∖ S ) ≤ t h R ⊂ X ∖ S R\subset X\setminus S R ⊂ X ∖ S dist ( x i , X ∖ S ) ≤ t h \operatorname{dist}(x_i,X\setminus S)\le t_h dist ( x i , X ∖ S ) ≤ t h x i ∈ ∂ ( S , t h ) x_i\in\partial(S,t_h) x i ∈ ∂ ( S , t h ) Theorem A.3 gives
P w 2 ( { Q i ( w 1 ) is bad } ) ≤ C 2 ∑ h ≤ k − m ( ℓ α ( k − h ) + 2 C 1 ℓ k − h ) η ≤ C 2 ( 1 + 2 C 1 ) η ∑ j = m ∞ ℓ α η j = C 2 ( 1 + 2 C 1 ) η 1 − ℓ α η ℓ α η m . (352) \begin{aligned}
\mathbb{P}_{w_2}\bigl(\{Q_i(w_1)\text{ is bad}\}\bigr)
\le C_2\sum_{h\le k-m}\left(\ell^{\alpha(k-h)}+2C_1\ell^{k-h}\right)^\eta\\
&\le C_2(1+2C_1)^\eta\sum_{j=m}^{\infty}\ell^{\alpha\eta j}
=\frac{C_2(1+2C_1)^\eta}{1-\ell^{\alpha\eta}}\ell^{\alpha\eta m}. \tag*{(352)}
\end{aligned} P w 2 ( { Q i ( w 1 ) is bad } ) ≤ C 2 h ≤ k − m ∑ ( ℓ α ( k − h ) + 2 C 1 ℓ k − h ) η ≤ C 2 ( 1 + 2 C 1 ) η j = m ∑ ∞ ℓ α η j = 1 − ℓ α η C 2 ( 1 + 2 C 1 ) η ℓ α η m . (352) Choose m 0 m_0 m 0 κ / 16 \kappa/16 κ /16 m ≥ m 0 m\ge m_0 m ≥ m 0
P w 2 ( { Q i ( w 1 ) is bad } ) < κ 16 and P w 1 ( { Q i ( w 2 ) is bad } ) < κ 16 , i = 1 , 2. (353) \mathbb{P}_{w_2}\bigl(\{Q_i(w_1)\text{ is bad}\}\bigr)<\frac{\kappa}{16}
\quad\text{and}\quad
\mathbb{P}_{w_1}\bigl(\{Q_i(w_2)\text{ is bad}\}\bigr)<\frac{\kappa}{16},
\qquad i=1,2. \tag*{(353)} P w 2 ( { Q i ( w 1 ) is bad } ) < 16 κ and P w 1 ( { Q i ( w 2 ) is bad } ) < 16 κ , i = 1 , 2. (353) By Fubini’s theorem, (A.10), and (A.12), we finally obtain
( P ⊗ P ) ( ⋂ j = 1 2 ( E 1 j ∩ E 2 j ) ∩ ⋂ i , j = 1 2 { Q i ( w j ) ∈ D j G ( w , m ) } ) > 1 − κ 2 − 4 κ 16 > 1 − κ . (\mathbb{P}\otimes\mathbb{P})\left(\bigcap_{j=1}^{2}(E_1^j\cap E_2^j)\cap\bigcap_{i,j=1}^{2}\{Q_i(w_j)\in\mathcal{D}_j^G(w,m)\}\right)>1-\frac{\kappa}{2}-\frac{4\kappa}{16}>1-\kappa. ( P ⊗ P ) ( j = 1 ⋂ 2 ( E 1 j ∩ E 2 j ) ∩ i , j = 1 ⋂ 2 { Q i ( w j ) ∈ D j G ( w , m )} ) > 1 − 2 κ − 16 4 κ > 1 − κ . This proves the lemma with L = 12 M ∗ C 1 / ( ε ℓ ) L=12M_*C_1/(\varepsilon\ell) L = 12 M ∗ C 1 / ( ε ℓ )
We conclude this appendix with the proof of Lemma 6.27 , which provides the Whitney -type decomposition of dyadic cubes used for the sum over S 4 , t e r m ′ \mathcal{S}'_{4,\mathrm{term}} S 4 , term ′ § 6 , see Lemma 6.28 .
Proof of Lemma 6.27
A.3. Proof of Lemma 6.27. Fix c W = 2 c_W=2 c W = 2 S ∈ D S\in\mathcal{D} S ∈ D S ⊂ Q S\subset Q S ⊂ Q c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) c_W\ell(S)\le\operatorname{dist}(S,X\setminus Q) c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) W ( Q ) W(Q) W ( Q ) Q Q Q x ∈ int Q x\in\operatorname{int}Q x ∈ int Q S k ( x ) S_k(x) S k ( x ) k k k x x x k k k Theorem 2.9 gives
dist ( S k ( x ) , X ∖ Q ) ≥ dist ( x , X ∖ Q ) − 2 C 1 ℓ k ≥ c W ℓ k . \operatorname{dist}(S_k(x),X\setminus Q)\ge\operatorname{dist}(x,X\setminus Q)-2C_1\ell^k\ge c_W\ell^k. dist ( S k ( x ) , X ∖ Q ) ≥ dist ( x , X ∖ Q ) − 2 C 1 ℓ k ≥ c W ℓ k . Thus S k ( x ) ⊂ Q S_k(x)\subset Q S k ( x ) ⊂ Q c W ℓ ( S k ( x ) ) ≤ dist ( S k ( x ) , X ∖ Q ) c_W\ell(S_k(x))\le\operatorname{dist}(S_k(x),X\setminus Q) c W ℓ ( S k ( x )) ≤ dist ( S k ( x ) , X ∖ Q ) S S S S k ( x ) S_k(x) S k ( x ) y ∈ X ∖ Q y\in X\setminus Q y ∈ X ∖ Q
c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) ≤ d ( x , y ) . c_W\ell(S)\le\operatorname{dist}(S,X\setminus Q)\le d(x,y). c W ℓ ( S ) ≤ dist ( S , X ∖ Q ) ≤ d ( x , y ) . There are only finitely many such ancestors, since their side lengths tend to infinity. Consequently, x x x W ( Q ) \mathcal{W}(Q) W ( Q ) W ( Q ) \mathcal{W}(Q) W ( Q ) int Q \operatorname{int} Q int Q
The lower bound in (2) follows from the definition of W ( Q ) \mathcal{W}(Q) W ( Q ) S ∈ W ( Q ) S \in\mathcal{W}(Q) S ∈ W ( Q ) S ^ \widehat{S} S S ^ ⊄ Q \widehat{S} \not\subset Q S ⊂ Q dist ( S ^ , X ∖ Q ) = 0 \operatorname{dist}(\widehat{S}, X \setminus Q) = 0 dist ( S , X ∖ Q ) = 0 dist ( S ^ , X ∖ Q ) < c W ℓ ( S ^ ) \operatorname{dist}(\widehat{S}, X \setminus Q) < c_W\ell(\widehat{S}) dist ( S , X ∖ Q ) < c W ℓ ( S )
dist ( S , X ∖ Q ) ≤ diam ( S ^ ) + dist ( S ^ , X ∖ Q ) ≤ ( 2 C 1 + c W ) ℓ ( S ^ ) = 2 C 1 + c W ℓ ℓ ( S ) . \operatorname{dist}(S, X \setminus Q) \le\operatorname{diam}(\widehat{S}) + \operatorname{dist}(\widehat{S}, X \setminus Q) \le(2C_1+c_W)\ell(\widehat{S}) = \frac{2C_1+c_W}{\ell}\ell(S). dist ( S , X ∖ Q ) ≤ diam ( S ) + dist ( S , X ∖ Q ) ≤ ( 2 C 1 + c W ) ℓ ( S ) = ℓ 2 C 1 + c W ℓ ( S ) . This proves (2) with C W = ( 2 C 1 + c W ) / ℓ C_W=(2C_1+c_W)/\ell C W = ( 2 C 1 + c W ) / ℓ
To prove (261) , we fix x ∈ X x \in X x ∈ X
W x : = { S ∈ W ( Q ) : x ∈ 2 S } , \mathcal{W}_x := \{S \in\mathcal{W}(Q): x \in2S\}, W x := { S ∈ W ( Q ) : x ∈ 2 S } , and we may assume that W x ≠ ∅ \mathcal{W}_x \ne\varnothing W x = ∅ S ∈ W x S \in\mathcal{W}_x S ∈ W x dist ( x , S ) < ℓ ( S ) \operatorname{dist}(x,S)<\ell(S) dist ( x , S ) < ℓ ( S ) Theorem 2.9 give
( c W − 1 ) ℓ ( S ) ≤ dist ( S , X ∖ Q ) − dist ( x , S ) ≤ dist ( x , X ∖ Q ) ≤ dist ( x , S ) + diam S + dist ( S , X ∖ Q ) ≤ ( C W + 1 + 2 C 1 ) ℓ ( S ) . \begin{aligned}
(c_W-1)\ell(S) \le\operatorname{dist}(S,X\setminus Q)-\operatorname{dist}(x,S) \le\operatorname{dist}(x,X\setminus Q) \\
&\le\operatorname{dist}(x,S)+\operatorname{diam} S+\operatorname{dist}(S,X\setminus Q) \le(C_W+1+2C_1)\ell(S).
\end{aligned} ( c W − 1 ) ℓ ( S ) ≤ dist ( S , X ∖ Q ) − dist ( x , S ) ≤ dist ( x , X ∖ Q ) ≤ dist ( x , S ) + diam S + dist ( S , X ∖ Q ) ≤ ( C W + 1 + 2 C 1 ) ℓ ( S ) . Hence, if we fix S 0 ∈ W x S_0 \in\mathcal{W}_x S 0 ∈ W x ρ W : = ( c W − 1 ) / ( C W + 1 + 2 C 1 ) \rho_W := (c_W-1)/(C_W+1+2C_1) ρ W := ( c W − 1 ) / ( C W + 1 + 2 C 1 )
ρ W ℓ ( S 0 ) ≤ ℓ ( S ) ≤ ρ W − 1 ℓ ( S 0 ) for every S ∈ W x . \rho_W\ell(S_0) \le\ell(S) \le\rho_W^{-1}\ell(S_0) \qquad\text{for every } S \in\mathcal{W}_x. ρ W ℓ ( S 0 ) ≤ ℓ ( S ) ≤ ρ W − 1 ℓ ( S 0 ) for every S ∈ W x . Thus, since B ( z S , c 1 ℓ ( S ) ) ⊂ S B(z_S,c_1\ell(S))\subset S B ( z S , c 1 ℓ ( S )) ⊂ S Theorem 2.9 , and the cubes in W x \mathcal{W}_x W x
d ( z S , z S ′ ) ≥ c 1 ρ W ℓ ( S 0 ) , S , S ′ ∈ W x , S ≠ S ′ . (354) d(z_S,z_{S'}) \ge c_1\rho_W\ell(S_0), \qquad S,S' \in\mathcal{W}_x,\quad S\ne S'. \tag*{(354)} d ( z S , z S ′ ) ≥ c 1 ρ W ℓ ( S 0 ) , S , S ′ ∈ W x , S = S ′ . (354) Moreover, since S ⊂ B ( z S , C 1 ℓ ( S ) ) S\subset B(z_S,C_1\ell(S)) S ⊂ B ( z S , C 1 ℓ ( S ))
d ( x , z S ) ≤ dist ( x , S ) + C 1 ℓ ( S ) < ( 1 + C 1 ) ρ W − 1 ℓ ( S 0 ) , S ∈ W x . d(x,z_S)\le\operatorname{dist}(x,S)+C_1\ell(S)<(1+C_1)\rho_W^{-1}\ell(S_0), \qquad S\in\mathcal{W}_x. d ( x , z S ) ≤ dist ( x , S ) + C 1 ℓ ( S ) < ( 1 + C 1 ) ρ W − 1 ℓ ( S 0 ) , S ∈ W x . Now, we denote
N : = ⌈ log 2 3 ( 2 + C 1 ) c 1 ρ W ⌉ + 1. N:=\left\lceil\log_2\frac{3(2+C_1)}{c_1\rho_W}\right\rceil+1. N := ⌈ log 2 c 1 ρ W 3 ( 2 + C 1 ) ⌉ + 1. By the geometric doubling property, the ball B ( x , ( 2 + C 1 ) ρ W − 1 ℓ ( S 0 ) ) B(x,(2+C_1)\rho_W^{-1}\ell(S_0)) B ( x , ( 2 + C 1 ) ρ W − 1 ℓ ( S 0 )) z S z_S z S S ∈ W x S\in\mathcal{W}_x S ∈ W x D N D^N D N ( 2 + C 1 ) ρ W − 1 ℓ ( S 0 ) / 2 N < c 1 ρ W ℓ ( S 0 ) / 3 (2+C_1)\rho_W^{-1}\ell(S_0)/2^N<c_1\rho_W\ell(S_0)/3 ( 2 + C 1 ) ρ W − 1 ℓ ( S 0 ) / 2 N < c 1 ρ W ℓ ( S 0 ) /3 z S z_S z S # W x ≤ D N \#\mathcal{W}_x\le D^N # W x ≤ D N x ∈ X x\in X x ∈ X (261) , with c : = D N c:=D^N c := D N Lemma 6.27 .
Appendix B. Suppressed kernels associated with a dominating function
Dominating functions
B.1. Dominating functions. Throughout this appendix, as in [37 ] , we assume ( X , d , μ ) (X,d,\mu) ( X , d , μ ) μ ≠ 0 \mu\ne0 μ = 0 λ : X × ( 0 , ∞ ) → ( 0 , ∞ ) \lambda:X\times(0,\infty)\to(0,\infty) λ : X × ( 0 , ∞ ) → ( 0 , ∞ ) dominating function such that, for some C λ > 0 C_\lambda>0 C λ > 0
(i) The function λ ( x , ⋅ ) \lambda(x,\cdot) λ ( x , ⋅ ) μ ( B ( x , r ) ) ≤ λ ( x , r ) \mu(B(x,r))\le\lambda(x,r) μ ( B ( x , r )) ≤ λ ( x , r ) x ∈ X x\in X x ∈ X r > 0 r>0 r > 0
(ii) We have that
λ ( x , 2 r ) ≤ C λ λ ( x , r ) , for all x ∈ X , r > 0. (355) \lambda(x,2r)\le C_\lambda\lambda(x,r), \qquad\text{for all }x\in X,r>0. \tag*{(355)} λ ( x , 2 r ) ≤ C λ λ ( x , r ) , for all x ∈ X , r > 0. (355) We first construct a dominating function comparable to λ \lambda λ
Lemma B.1. Let ( X , d , μ ) (X,d,\mu) ( X , d , μ ) λ \lambda λ
λ ~ ( x , r ) : = 1 log 2 ∫ r 2 r λ ( x , ρ ) d ρ ρ , for x ∈ X , r > 0. \widetilde{\lambda}(x,r) := \frac{1}{\log2}\int_r^{2r}\lambda(x,\rho)\frac{\mathrm{d}\rho}{\rho}, \qquad\text{for }x\in X,r>0. λ ( x , r ) := log 2 1 ∫ r 2 r λ ( x , ρ ) ρ d ρ , for x ∈ X , r > 0. Then, for every x ∈ X x\in X x ∈ X r > 0 r>0 r > 0
λ ( x , r ) ≤ λ ~ ( x , r ) ≤ C λ λ ( x , r ) and λ ~ ( x , 2 r ) ≤ C λ λ ~ ( x , r ) . (356) \lambda(x,r)\leq\widetilde{\lambda}(x,r)\leq C_\lambda\lambda(x,r) \qquad\text{and}\qquad\widetilde{\lambda}(x,2r)\leq C_\lambda\widetilde{\lambda}(x,r). \tag*{(356)} λ ( x , r ) ≤ λ ( x , r ) ≤ C λ λ ( x , r ) and λ ( x , 2 r ) ≤ C λ λ ( x , r ) . (356) Moreover, there exists a constant C > 0 C>0 C > 0 C λ C_\lambda C λ x ∈ X x\in X x ∈ X m > 0 m>0 m > 0 s , t ∈ [ m / 2 , m ] s,t\in[m/2,m] s , t ∈ [ m /2 , m ]
∣ λ ~ ( x , t ) − λ ~ ( x , s ) ∣ λ ~ ( x , m ) ≤ C ∣ t − s ∣ m . (357) \frac{\left|\widetilde{\lambda}(x,t)-\widetilde{\lambda}(x,s)\right|}{\widetilde{\lambda}(x,m)}\leq C\frac{|t-s|}{m}. \tag*{(357)} λ ( x , m ) λ ( x , t ) − λ ( x , s ) ≤ C m ∣ t − s ∣ . (357) Proof. Let ρ ∈ [ r , 2 r ] \rho\in[r,2r] ρ ∈ [ r , 2 r ]
λ ( x , r ) ≤ λ ( x , ρ ) ≤ λ ( x , 2 r ) ≤ C λ λ ( x , r ) , \lambda(x,r)\leq\lambda(x,\rho)\leq\lambda(x,2r)\leq C_\lambda\lambda(x,r), λ ( x , r ) ≤ λ ( x , ρ ) ≤ λ ( x , 2 r ) ≤ C λ λ ( x , r ) , which readily implies the first estimate in (B.2). Moreover, a change of variables and (B.1) give the second one.
We now prove the regularity estimate (B.3). By symmetry, we assume that s < t s<t s < t λ ~ \widetilde{\lambda} λ
λ ~ ( x , t ) − λ ~ ( x , s ) = 1 log 2 ( ∫ t 2 t λ ( x , ρ ) d ρ ρ − ∫ s 2 s λ ( x , ρ ) d ρ ρ ) . \widetilde{\lambda}(x,t)-\widetilde{\lambda}(x,s)=\frac{1}{\log2}\left(\int_t^{2t}\lambda(x,\rho)\frac{\mathrm{d}\rho}{\rho}-\int_s^{2s}\lambda(x,\rho)\frac{\mathrm{d}\rho}{\rho}\right). λ ( x , t ) − λ ( x , s ) = log 2 1 ( ∫ t 2 t λ ( x , ρ ) ρ d ρ − ∫ s 2 s λ ( x , ρ ) ρ d ρ ) . Hence, denoting by △ \triangle △
∣ λ ~ ( x , t ) − λ ~ ( x , s ) ∣ ≤ 1 log 2 ∫ [ s , 2 s ] △ [ t , 2 t ] λ ( x , ρ ) d ρ ρ . \left|\widetilde{\lambda}(x,t)-\widetilde{\lambda}(x,s)\right|\leq\frac{1}{\log2}\int_{[s,2s]\triangle[t,2t]}\lambda(x,\rho)\frac{\mathrm{d}\rho}{\rho}. λ ( x , t ) − λ ( x , s ) ≤ log 2 1 ∫ [ s , 2 s ] △ [ t , 2 t ] λ ( x , ρ ) ρ d ρ . Since s , t ∈ [ m / 2 , m ] s,t\in[m/2,m] s , t ∈ [ m /2 , m ] [ s , 2 s ] △ [ t , 2 t ] ⊂ [ m / 2 , 2 m ] [s,2s]\triangle[t,2t]\subset[m/2,2m] [ s , 2 s ] △ [ t , 2 t ] ⊂ [ m /2 , 2 m ] ρ ∈ [ m / 2 , 2 m ] \rho\in[m/2,2m] ρ ∈ [ m /2 , 2 m ]
λ ( x , ρ ) ≤ λ ( x , 2 m ) ≤ C λ λ ( x , m ) ≤ (B.2) C λ λ ~ ( x , m ) . (358) \lambda(x,\rho)\leq\lambda(x,2m)\leq C_\lambda\lambda(x,m)\overset{\text{\tiny(B.2)}}{\leq}C_\lambda\widetilde{\lambda}(x,m). \tag*{(358)} λ ( x , ρ ) ≤ λ ( x , 2 m ) ≤ C λ λ ( x , m ) ≤ (B.2) C λ λ ( x , m ) . (358) Therefore, we obtain
∣ λ ~ ( x , t ) − λ ~ ( x , s ) ∣ ≤ C λ log 2 λ ~ ( x , m ) ∫ [ s , 2 s ] △ [ t , 2 t ] d ρ ρ ≤ 2 C λ log 2 log ( t s ) λ ~ ( x , m ) , (359) \left|\widetilde{\lambda}(x,t)-\widetilde{\lambda}(x,s)\right|\leq\frac{C_\lambda}{\log2}\widetilde{\lambda}(x,m)\int_{[s,2s]\triangle[t,2t]}\frac{\mathrm{d}\rho}{\rho}\leq2\frac{C_\lambda}{\log2}\log\left(\frac{t}{s}\right)\widetilde{\lambda}(x,m), \tag*{(359)} λ ( x , t ) − λ ( x , s ) ≤ log 2 C λ λ ( x , m ) ∫ [ s , 2 s ] △ [ t , 2 t ] ρ d ρ ≤ 2 log 2 C λ log ( s t ) λ ( x , m ) , (359) where the last inequality follows from the assumptions s < t s<t s < t s , t ∈ [ m / 2 , m ] s,t\in[m/2,m] s , t ∈ [ m /2 , m ]
Furthermore, since s ≥ m / 2 s\geq m/2 s ≥ m /2
log t s = log ( 1 + t − s s ) ≤ t − s s ≤ 2 t − s m . (360) \log\frac{t}{s}=\log\left(1+\frac{t-s}{s}\right)\leq\frac{t-s}{s}\leq2\frac{t-s}{m}. \tag*{(360)} log s t = log ( 1 + s t − s ) ≤ s t − s ≤ 2 m t − s . (360) Gathering (B.4), (B.5), and (B.6) finally implies (B.3) with C : = 4 C λ / log 2 C:=4C_\lambda/\log2 C := 4 C λ / log 2
To prove the bounds for suppressed kernels, we also need to introduce a dominating function Λ \Lambda Λ Λ ( ⋅ , r ) \Lambda(\cdot,r) Λ ( ⋅ , r ) r > 0 r>0 r > 0 [45 Proposition 1.3] .
Lemma B.2. Let ( X , d , μ ) (X,d,\mu) ( X , d , μ ) λ : X × ( 0 , ∞ ) → ( 0 , ∞ ) \lambda:X\times(0,\infty)\to(0,\infty) λ : X × ( 0 , ∞ ) → ( 0 , ∞ )
For
λ ~ ( x , r ) : = 1 log 2 ∫ r 2 r λ ( x , ρ ) d ρ ρ , x ∈ X , r > 0 , \widetilde{\lambda}(x,r):=\frac{1}{\log2}\int_r^{2r}\lambda(x,\rho)\frac{\mathrm{d}\rho}{\rho},\qquad x\in X,\ r>0, λ ( x , r ) := log 2 1 ∫ r 2 r λ ( x , ρ ) ρ d ρ , x ∈ X , r > 0 , we define
Λ ( x , r ) : = inf z ∈ X λ ~ ( z , r + d ( x , z ) ) , x ∈ X , r > 0. \Lambda(x,r):=\inf_{z\in X}\widetilde{\lambda}(z,r+d(x,z)),\qquad x\in X,\ r>0. Λ ( x , r ) := z ∈ X inf λ ( z , r + d ( x , z )) , x ∈ X , r > 0. We have that
Λ ( x , r ) ≤ C λ λ ( x , r ) , for every x ∈ X , r > 0. (361) \Lambda(x,r)\le C_\lambda\lambda(x,r),\qquad\text{for every }x\in X,\ r>0. \tag*{(361)} Λ ( x , r ) ≤ C λ λ ( x , r ) , for every x ∈ X , r > 0. (361) Furthermore, Λ \Lambda Λ
(a) For every x ∈ X x\in X x ∈ X Λ ( x , ⋅ ) \Lambda(x,\cdot) Λ ( x , ⋅ ) Λ ( x , r ) > 0 \Lambda(x,r)>0 Λ ( x , r ) > 0 μ ( B ( x , r ) ) ≤ Λ ( x , r ) \mu(B(x,r))\le\Lambda(x,r) μ ( B ( x , r )) ≤ Λ ( x , r ) x ∈ X x\in X x ∈ X r > 0 r>0 r > 0
(b) For every x ∈ X x\in X x ∈ X r > 0 r>0 r > 0
Λ ( x , 2 r ) ≤ C λ 2 Λ ( x , r ) . (362) \Lambda(x,2r)\le C_\lambda^2\Lambda(x,r). \tag*{(362)} Λ ( x , 2 r ) ≤ C λ 2 Λ ( x , r ) . (362) (c) For every x ∈ X x\in X x ∈ X m > 0 m>0 m > 0 s , t ∈ [ m / 2 , m ] s,t\in[m/2,m] s , t ∈ [ m /2 , m ]
∣ Λ ( x , t ) − Λ ( x , s ) ∣ ≤ 2 C C λ ∣ t − s ∣ m Λ ( x , m ) , (363) \left|\Lambda(x,t)-\Lambda(x,s)\right|\le2CC_\lambda\frac{|t-s|}{m}\Lambda(x,m), \tag*{(363)} ∣ Λ ( x , t ) − Λ ( x , s ) ∣ ≤ 2 C C λ m ∣ t − s ∣ Λ ( x , m ) , (363) where C > 0 C>0 C > 0 Lemma B.1 .
(d) For every r > 0 r>0 r > 0 x , y ∈ X x,y\in X x , y ∈ X d ( x , y ) ≤ r d(x,y)\le r d ( x , y ) ≤ r
Λ ( x , r ) ≤ C λ Λ ( y , r ) . \Lambda(x,r)\le C_\lambda\Lambda(y,r). Λ ( x , r ) ≤ C λ Λ ( y , r ) . (e) If r > 0 r>0 r > 0 x , x ′ ∈ X x,x'\in X x , x ′ ∈ X d ( x , x ′ ) ≤ r d(x,x')\le r d ( x , x ′ ) ≤ r
∣ Λ ( x , r ) − Λ ( x ′ , r ) ∣ ≤ C C λ d ( x , x ′ ) r Λ ( x , r ) . (364) \left|\Lambda(x,r)-\Lambda(x',r)\right|\le CC_\lambda\frac{d(x,x')}{r}\Lambda(x,r). \tag*{(364)} ∣ Λ ( x , r ) − Λ ( x ′ , r ) ∣ ≤ C C λ r d ( x , x ′ ) Λ ( x , r ) . (364) Proof. Let x ∈ X x\in X x ∈ X r > 0 r>0 r > 0 z ∈ X z\in X z ∈ X B ( x , r ) ⊂ B ( z , r + d ( x , z ) ) B(x,r)\subset B(z,r+d(x,z)) B ( x , r ) ⊂ B ( z , r + d ( x , z ))
μ ( B ( x , r ) ) ≤ λ ( z , r + d ( x , z ) ) ≤ λ ~ ( z , r + d ( x , z ) ) (365) \mu(B(x,r))\le\lambda(z,r+d(x,z))\le\widetilde{\lambda}(z,r+d(x,z)) \tag*{(365)} μ ( B ( x , r )) ≤ λ ( z , r + d ( x , z )) ≤ λ ( z , r + d ( x , z )) (365) and, taking the infimum over z ∈ X z\in X z ∈ X μ ( B ( x , r ) ) ≤ Λ ( x , r ) \mu(B(x,r))\le\Lambda(x,r) μ ( B ( x , r )) ≤ Λ ( x , r )
To prove (B.7), we choose z = x z=x z = x Λ \Lambda Λ
Λ ( x , r ) ≤ λ ~ ( x , r ) ≤ C λ λ ( x , r ) . (366) \Lambda(x,r)\le\widetilde{\lambda}(x,r)\le C_\lambda\lambda(x,r). \tag*{(366)} Λ ( x , r ) ≤ λ ( x , r ) ≤ C λ λ ( x , r ) . (366) We now prove properties (a)-(e). For every x ∈ X x\in X x ∈ X Λ ( x , ⋅ ) \Lambda(x,\cdot) Λ ( x , ⋅ ) λ ~ ( z , d ( x , z ) + ⋅ ) \widetilde{\lambda}(z,d(x,z)+\cdot) λ ( z , d ( x , z ) + ⋅ ) z ∈ X z\in X z ∈ X
To prove (B.8), it suffices to note that, for every x , z ∈ X x,z\in X x , z ∈ X r > 0 r>0 r > 0
λ ~ ( z , 2 r + d ( x , z ) ) ≤ λ ~ ( z , 2 ( r + d ( x , z ) ) ) ≤ C λ 2 λ ~ ( z , r + d ( x , z ) ) , \widetilde{\lambda}(z,2r+d(x,z))\le\widetilde{\lambda}(z,2(r+d(x,z)))\le C_\lambda^2\widetilde{\lambda}(z,r+d(x,z)), λ ( z , 2 r + d ( x , z )) ≤ λ ( z , 2 ( r + d ( x , z ))) ≤ C λ 2 λ ( z , r + d ( x , z )) , where in the last inequality we used (B.1) and (B.2). Hence, we take the infimum over z ∈ X z\in X z ∈ X μ ≠ 0 \mu\ne0 μ = 0 x x x Λ ( x , r ) > 0 \Lambda(x,r)>0 Λ ( x , r ) > 0 r > 0 r>0 r > 0
Let us prove (B.9). By symmetry, we assume that s < t s<t s < t z ∈ X z\in X z ∈ X
s ~ : = s + d ( x , z ) , t ~ : = t + d ( x , z ) , m ~ : = t ~ . \widetilde{s}:=s+d(x,z),\qquad\widetilde{t}:=t+d(x,z),\qquad\widetilde{m}:=\widetilde{t}. s := s + d ( x , z ) , t := t + d ( x , z ) , m := t . Then s ~ , t ~ ∈ [ m ~ / 2 , m ~ ] \tilde{s},\tilde{t}\in[\tilde{m}/2,\tilde{m}] s ~ , t ~ ∈ [ m ~ /2 , m ~ ] m ~ ≥ m / 2 \tilde{m}\ge m/2 m ~ ≥ m /2 t ~ ≤ 2 s ~ \tilde{t}\le2\tilde{s} t ~ ≤ 2 s ~ λ ~ \tilde{\lambda} λ ~
0 ≤ λ ~ ( z , t ~ ) − λ ~ ( z , s ~ ) ≤ C t − s m ~ λ ~ ( z , m ~ ) ≤ 2 C C λ t − s m λ ~ ( z , s ~ ) . (367) 0\le\tilde{\lambda}(z,\tilde{t})-\tilde{\lambda}(z,\tilde{s})\le C\frac{t-s}{\tilde{m}}\tilde{\lambda}(z,\tilde{m})\le2CC_{\lambda}\frac{t-s}{m}\tilde{\lambda}(z,\tilde{s}). \tag*{(367)} 0 ≤ λ ~ ( z , t ~ ) − λ ~ ( z , s ~ ) ≤ C m ~ t − s λ ~ ( z , m ~ ) ≤ 2 C C λ m t − s λ ~ ( z , s ~ ) . (367) Thus λ ~ ( z , t ~ ) ≤ ( 1 + 2 C C λ ( t − s ) / m ) λ ~ ( z , s ~ ) \tilde{\lambda}(z,\tilde{t})\le(1+2CC_{\lambda}(t-s)/m)\tilde{\lambda}(z,\tilde{s}) λ ~ ( z , t ~ ) ≤ ( 1 + 2 C C λ ( t − s ) / m ) λ ~ ( z , s ~ ) z z z
0 ≤ Λ ( x , t ) − Λ ( x , s ) ≤ 2 C C λ t − s m Λ ( x , s ) ≤ 2 C C λ t − s m Λ ( x , m ) , 0\le\Lambda(x,t)-\Lambda(x,s)\le2CC_{\lambda}\frac{t-s}{m}\Lambda(x,s)\le2CC_{\lambda}\frac{t-s}{m}\Lambda(x,m), 0 ≤ Λ ( x , t ) − Λ ( x , s ) ≤ 2 C C λ m t − s Λ ( x , s ) ≤ 2 C C λ m t − s Λ ( x , m ) , as required.
To prove (d), if d ( x , y ) ≤ r d(x,y)\le r d ( x , y ) ≤ r r + d ( x , z ) ≤ 2 ( r + d ( y , z ) ) r+d(x,z)\le2(r+d(y,z)) r + d ( x , z ) ≤ 2 ( r + d ( y , z )) λ ~ \tilde{\lambda} λ ~ λ ~ ( z , r + d ( x , z ) ) ≤ C λ λ ~ ( z , r + d ( y , z ) ) \tilde{\lambda}(z,r+d(x,z))\le C_{\lambda}\tilde{\lambda}(z,r+d(y,z)) λ ~ ( z , r + d ( x , z )) ≤ C λ λ ~ ( z , r + d ( y , z )) z z z
We are left with proving property (e). Let x , x ′ ∈ X x,x'\in X x , x ′ ∈ X d ( x , x ′ ) ≤ r d(x,x')\le r d ( x , x ′ ) ≤ r z ∈ X z\in X z ∈ X
s : = r + d ( x , z ) and t : = r + d ( x ′ , z ) , s:=r+d(x,z) \qquad\text{and}\qquad t:=r+d(x',z), s := r + d ( x , z ) and t := r + d ( x ′ , z ) , so that, by the triangle inequality,
∣ s − t ∣ ≤ d ( x , x ′ ) ≤ r ≤ min { s , t } . |s-t|\le d(x,x')\le r\le\min\{s,t\}. ∣ s − t ∣ ≤ d ( x , x ′ ) ≤ r ≤ min { s , t } . Moreover, if we denote m : = max { s , t } m:=\max\{s,t\} m := max { s , t } s , t ∈ [ m / 2 , m ] s,t\in[m/2,m] s , t ∈ [ m /2 , m ] λ ~ \tilde{\lambda} λ ~
∣ λ ~ ( z , s ) − λ ~ ( z , t ) ∣ ≤ C ∣ s − t ∣ m λ ~ ( z , m ) ≤ C d ( x , x ′ ) r λ ~ ( z , m ) ≤ C C λ d ( x , x ′ ) r λ ~ ( z , t ) , |\tilde{\lambda}(z,s)-\tilde{\lambda}(z,t)|\le C\frac{|s-t|}{m}\tilde{\lambda}(z,m)\le C\frac{d(x,x')}{r}\tilde{\lambda}(z,m)\le CC_{\lambda}\frac{d(x,x')}{r}\tilde{\lambda}(z,t), ∣ λ ~ ( z , s ) − λ ~ ( z , t ) ∣ ≤ C m ∣ s − t ∣ λ ~ ( z , m ) ≤ C r d ( x , x ′ ) λ ~ ( z , m ) ≤ C C λ r d ( x , x ′ ) λ ~ ( z , t ) , so we have
λ ~ ( z , s ) ≤ ( 1 + C C λ d ( x , x ′ ) r ) λ ~ ( z , t ) . \tilde{\lambda}(z,s)\le\left(1+CC_{\lambda}\frac{d(x,x')}{r}\right)\tilde{\lambda}(z,t). λ ~ ( z , s ) ≤ ( 1 + C C λ r d ( x , x ′ ) ) λ ~ ( z , t ) . Hence, we finally get
Λ ( x , r ) ≤ ( 1 + C C λ d ( x , x ′ ) r ) Λ ( x ′ , r ) and Λ ( x ′ , r ) ≤ ( 1 + C C λ d ( x , x ′ ) r ) Λ ( x , r ) , \Lambda(x,r)\le\left(1+CC_{\lambda}\frac{d(x,x')}{r}\right)\Lambda(x',r)\qquad\text{and}\qquad\Lambda(x',r)\le\left(1+CC_{\lambda}\frac{d(x,x')}{r}\right)\Lambda(x,r), Λ ( x , r ) ≤ ( 1 + C C λ r d ( x , x ′ ) ) Λ ( x ′ , r ) and Λ ( x ′ , r ) ≤ ( 1 + C C λ r d ( x , x ′ ) ) Λ ( x , r ) , which proves (B.10). □
CZ-type bounds for suppressed kernels
In this subsection, we define suppressions of Calderón-Zygmund kernels in a metric measure space ( X , d , μ ) (X,d,\mu) ( X , d , μ ) λ \lambda λ K : ( X × X ) ∖ { ( x , x ) : x ∈ X } → C K:(X\times X)\setminus\{(x,x):x\in X\}\to\mathbb{C} K : ( X × X ) ∖ {( x , x ) : x ∈ X } → C γ ∈ ( 0 , 1 ] \gamma\in(0,1] γ ∈ ( 0 , 1 ] C > 0 C>0 C > 0
∣ K ( x , y ) ∣ ≤ C λ ( x , d ( x , y ) ) , x , y ∈ X , x ≠ y , (368) |K(x,y)|\le\frac{C}{\lambda(x,d(x,y))},\qquad x,y\in X,\quad x\ne y, \tag*{(368)} ∣ K ( x , y ) ∣ ≤ λ ( x , d ( x , y )) C , x , y ∈ X , x = y , (368) and, for all x , x ′ , y ∈ X x,x',y\in X x , x ′ , y ∈ X m : = max { d ( y , x ) , d ( y , x ′ ) } m:=\max\{d(y,x),d(y,x')\} m := max { d ( y , x ) , d ( y , x ′ )} 2 d ( x , x ′ ) ≤ m 2d(x,x')\le m 2 d ( x , x ′ ) ≤ m
∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ C λ ( y , m ) ( d ( x , x ′ ) m ) γ . (369) |K(x,y)-K(x',y)|+|K(y,x)-K(y,x')|\le\frac{C}{\lambda(y,m)}\left(\frac{d(x,x')}{m}\right)^{\gamma}. \tag*{(369)} ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ λ ( y , m ) C ( m d ( x , x ′ ) ) γ . (369) By Lemma B.2 -(a) and (b), Λ \Lambda Λ 1 / λ ≤ C λ / Λ 1/\lambda\le C_{\lambda}/\Lambda 1/ λ ≤ C λ /Λ Λ \Lambda Λ λ \lambda λ C C C Λ ≤ C λ λ \Lambda\le C_{\lambda}\lambda Λ ≤ C λ λ λ \lambda λ μ \mu μ Λ \Lambda Λ Λ \Lambda Λ Λ \Lambda Λ λ \lambda λ C λ C_{\lambda} C λ Lemma B.2 , so that the following properties hold:
(i) The function λ ( x , ⋅ ) \lambda(x,\cdot) λ ( x , ⋅ ) μ ( B ( x , r ) ) ≤ λ ( x , r ) \mu(B(x,r))\le\lambda(x,r) μ ( B ( x , r )) ≤ λ ( x , r ) x ∈ X x\in X x ∈ X r > 0 r>0 r > 0
(ii) We have that
λ ( x , 2 r ) ≤ C λ λ ( x , r ) , for every x ∈ X , r > 0. (370) \lambda(x,2r)\le C_{\lambda}\lambda(x,r),\qquad\text{for every }x\in X,\ r>0. \tag*{(370)} λ ( x , 2 r ) ≤ C λ λ ( x , r ) , for every x ∈ X , r > 0. (370) (iii) For every x ∈ X x \in X x ∈ X m > 0 m > 0 m > 0 s , t ∈ [ m / 2 , m ] s,t \in[m/2,m] s , t ∈ [ m /2 , m ]
∣ λ ( x , t ) − λ ( x , s ) ∣ ≤ C λ ∣ t − s ∣ m λ ( x , m ) . (371) |\lambda(x,t)-\lambda(x,s)| \le C_\lambda\frac{|t-s|}{m}\lambda(x,m). \tag*{(371)} ∣ λ ( x , t ) − λ ( x , s ) ∣ ≤ C λ m ∣ t − s ∣ λ ( x , m ) . (371) (iv) For every r > 0 r > 0 r > 0 x , y ∈ X x,y \in X x , y ∈ X d ( x , y ) ≤ r d(x,y) \le r d ( x , y ) ≤ r
λ ( x , r ) ≤ C λ λ ( y , r ) . (372) \lambda(x,r) \le C_\lambda\lambda(y,r). \tag*{(372)} λ ( x , r ) ≤ C λ λ ( y , r ) . (372) (v) If r > 0 r > 0 r > 0 x , x ′ ∈ X x,x' \in X x , x ′ ∈ X d ( x , x ′ ) ≤ r d(x,x') \le r d ( x , x ′ ) ≤ r
∣ λ ( x , r ) − λ ( x ′ , r ) ∣ ≤ C λ d ( x , x ′ ) r λ ( x , r ) . (373) |\lambda(x,r)-\lambda(x',r)| \le C_\lambda\frac{d(x,x')}{r}\lambda(x,r). \tag*{(373)} ∣ λ ( x , r ) − λ ( x ′ , r ) ∣ ≤ C λ r d ( x , x ′ ) λ ( x , r ) . (373) Let d λ ≥ 0 d_\lambda\ge0 d λ ≥ 0 λ ( x , 2 r ) ≤ 2 d λ λ ( x , r ) \lambda(x,2r) \le2^{d_\lambda}\lambda(x,r) λ ( x , 2 r ) ≤ 2 d λ λ ( x , r ) x ∈ X x \in X x ∈ X r > 0 r > 0 r > 0 d λ = log 2 C λ d_\lambda=\log_2 C_\lambda d λ = log 2 C λ
λ ( z , t ) ≤ 2 d λ ( t / s ) d λ λ ( z , s ) , z ∈ X , 0 < s ≤ t . \lambda(z,t) \le2^{d_\lambda}(t/s)^{d_\lambda}\lambda(z,s), \qquad z \in X,\quad0 < s \le t. λ ( z , t ) ≤ 2 d λ ( t / s ) d λ λ ( z , s ) , z ∈ X , 0 < s ≤ t . We fix M ≥ max { 1 , d λ / 2 } M \ge\max\{1,d_\lambda/2\} M ≥ max { 1 , d λ /2 } § 3.5 , where λ ( x , r ) = r n \lambda(x,r)=r^n λ ( x , r ) = r n d λ = n d_\lambda=n d λ = n (68) corresponds to M = n M=n M = n n ≥ 1 n \ge1 n ≥ 1
Definition B.3 (Product-suppressed kernel). Let Ξ : X → [ 0 , ∞ ) \Xi:X \to[0,\infty) Ξ : X → [ 0 , ∞ )
∣ Ξ ( x ) − Ξ ( x ′ ) ∣ ≤ d ( x , x ′ ) , x , x ′ ∈ X . (374) |\Xi(x)-\Xi(x')| \le d(x,x'), \qquad x,x' \in X. \tag*{(374)} ∣Ξ ( x ) − Ξ ( x ′ ) ∣ ≤ d ( x , x ′ ) , x , x ′ ∈ X . (374) For x ≠ y x \ne y x = y r = d ( x , y ) r=d(x,y) r = d ( x , y )
A Ξ ( x , y ) : = r 2 M r 2 M + Ξ ( x ) M Ξ ( y ) M . (375) A_\Xi(x,y):=\frac{r^{2M}}{r^{2M}+\Xi(x)^M\Xi(y)^M}. \tag*{(375)} A Ξ ( x , y ) := r 2 M + Ξ ( x ) M Ξ ( y ) M r 2 M . (375) The product-suppressed kernel associated with K K K
K Ξ ( x , y ) : = K ( x , y ) A Ξ ( x , y ) , x , y ∈ X , x ≠ y . (376) K_\Xi(x,y):=K(x,y)A_\Xi(x,y), \qquad x,y \in X,\quad x \ne y. \tag*{(376)} K Ξ ( x , y ) := K ( x , y ) A Ξ ( x , y ) , x , y ∈ X , x = y . (376) The factor in (B.19) is symmetric and takes values in ( 0 , 1 ] (0,1] ( 0 , 1 ] Ξ ( x ) = 0 \Xi(x)=0 Ξ ( x ) = 0 Ξ ( y ) = 0 \Xi(y)=0 Ξ ( y ) = 0 A Ξ ( x , y ) = 1 A_\Xi(x,y)=1 A Ξ ( x , y ) = 1 K Ξ ( x , y ) = K ( x , y ) K_\Xi(x,y)=K(x,y) K Ξ ( x , y ) = K ( x , y ) Ξ ( x ) = 0 \Xi(x)=0 Ξ ( x ) = 0 Ξ ( y ) = 0 \Xi(y)=0 Ξ ( y ) = 0
Lemma B.4 (Upper-doubling product suppression). Let K K K Ξ \Xi Ξ K Ξ K_\Xi K Ξ γ \gamma γ C C C C λ C_\lambda C λ M M M γ \gamma γ
(i) For every x ≠ y x \ne y x = y
∣ K Ξ ( x , y ) ∣ ≤ C λ ( x , d ( x , y ) ) . (377) |K_\Xi(x,y)| \le\frac{C}{\lambda(x,d(x,y))}. \tag*{(377)} ∣ K Ξ ( x , y ) ∣ ≤ λ ( x , d ( x , y )) C . (377) (ii) With the convention 1 / λ ( z , 0 ) : = + ∞ 1/\lambda(z,0):=+\infty 1/ λ ( z , 0 ) := + ∞
∣ K Ξ ( x , y ) ∣ ≲ min { 1 λ ( x , Ξ ( x ) ) , 1 λ ( y , Ξ ( y ) ) } . (378) |K_\Xi(x,y)| \lesssim\min\left\{\frac{1}{\lambda(x,\Xi(x))},\frac{1}{\lambda(y,\Xi(y))}\right\}. \tag*{(378)} ∣ K Ξ ( x , y ) ∣ ≲ min { λ ( x , Ξ ( x )) 1 , λ ( y , Ξ ( y )) 1 } . (378) (iii) If x , x ′ , y ∈ X x,x',y \in X x , x ′ , y ∈ X 2 d ( x , x ′ ) ≤ m : = max { d ( y , x ) , d ( y , x ′ ) } 2d(x,x') \le m:=\max\{d(y,x),d(y,x')\} 2 d ( x , x ′ ) ≤ m := max { d ( y , x ) , d ( y , x ′ )}
∣ K Ξ ( x , y ) − K Ξ ( x ′ , y ) ∣ + ∣ K Ξ ( y , x ) − K Ξ ( y , x ′ ) ∣ ≲ 1 λ ( y , m ) ( d ( x , x ′ ) m ) γ . (379) |K_\Xi(x,y)-K_\Xi(x',y)|+|K_\Xi(y,x)-K_\Xi(y,x')|
\lesssim\frac{1}{\lambda(y,m)}\left(\frac{d(x,x')}{m}\right)^\gamma. \tag*{(379)} ∣ K Ξ ( x , y ) − K Ξ ( x ′ , y ) ∣ + ∣ K Ξ ( y , x ) − K Ξ ( y , x ′ ) ∣ ≲ λ ( y , m ) 1 ( m d ( x , x ′ ) ) γ . (379) (iv) The identity
K Ξ ( x , y ) = K ( x , y ) (380) K_\Xi(x,y)=K(x,y) \tag*{(380)} K Ξ ( x , y ) = K ( x , y ) (380) holds whenever Ξ ( x ) = 0 \Xi(x)=0 Ξ ( x ) = 0 Ξ ( y ) = 0 \Xi(y)=0 Ξ ( y ) = 0
Proof. Let x ≠ y x \ne y x = y r : = d ( x , y ) r := d(x,y) r := d ( x , y ) 0 < A Ξ ≤ 1 0 < A_{\Xi} \le1 0 < A Ξ ≤ 1 x x x y y y
We set a : = Ξ ( x ) a := \Xi(x) a := Ξ ( x ) b : = Ξ ( y ) b := \Xi(y) b := Ξ ( y ) a > 0 a > 0 a > 0 r ≥ a / 4 r \ge a/4 r ≥ a /4 λ ( x , a ) ≤ C λ 2 λ ( x , r ) \lambda(x,a) \le C_{\lambda}^{2}\lambda(x,r) λ ( x , a ) ≤ C λ 2 λ ( x , r ) ∣ K Ξ ( x , y ) ∣ ≤ C C λ 2 / λ ( x , a ) |K_{\Xi}(x,y)| \le CC_{\lambda}^{2}/\lambda(x,a) ∣ K Ξ ( x , y ) ∣ ≤ C C λ 2 / λ ( x , a ) r < a / 4 r < a/4 r < a /4 b ≥ a − r ≥ a / 2 b \ge a-r \ge a/2 b ≥ a − r ≥ a /2
A Ξ ( x , y ) ≤ 2 M ( r a ) 2 M . A_{\Xi}(x,y) \le2^{M}\left(\frac{r}{a}\right)^{2M}. A Ξ ( x , y ) ≤ 2 M ( a r ) 2 M . Hence, by (B.12),
∣ K Ξ ( x , y ) ∣ ≤ C 2 M λ ( x , r ) ( r a ) 2 M ≤ C 2 M + d λ λ ( x , a ) ( r a ) 2 M − d λ ≤ C 2 M + d λ λ ( x , a ) , |K_{\Xi}(x,y)| \le\frac{C2^{M}}{\lambda(x,r)}\left(\frac{r}{a}\right)^{2M} \le\frac{C2^{M+d_{\lambda}}}{\lambda(x,a)}\left(\frac{r}{a}\right)^{2M-d_{\lambda}} \le\frac{C2^{M+d_{\lambda}}}{\lambda(x,a)}, ∣ K Ξ ( x , y ) ∣ ≤ λ ( x , r ) C 2 M ( a r ) 2 M ≤ λ ( x , a ) C 2 M + d λ ( a r ) 2 M − d λ ≤ λ ( x , a ) C 2 M + d λ , where the last inequality follows from 2 M ≥ d λ 2M \ge d_{\lambda} 2 M ≥ d λ r < a r < a r < a
To prove (B.23), we fix x , x ′ , y ∈ X x,x',y \in X x , x ′ , y ∈ X 2 d ( x , x ′ ) ≤ m 2d(x,x') \le m 2 d ( x , x ′ ) ≤ m δ : = d ( x , x ′ ) \delta:= d(x,x') δ := d ( x , x ′ ) r : = d ( x , y ) r := d(x,y) r := d ( x , y ) r ′ : = d ( x ′ , y ) r' := d(x',y) r ′ := d ( x ′ , y ) a : = Ξ ( x ) a := \Xi(x) a := Ξ ( x ) a ′ : = Ξ ( x ′ ) a' := \Xi(x') a ′ := Ξ ( x ′ ) b : = Ξ ( y ) b := \Xi(y) b := Ξ ( y ) r , r ′ ∈ [ m / 2 , m ] r,r' \in[m/2,m] r , r ′ ∈ [ m /2 , m ] ∣ a − a ′ ∣ ≤ δ |a-a'| \le\delta ∣ a − a ′ ∣ ≤ δ ∣ a − b ∣ ≤ r |a-b| \le r ∣ a − b ∣ ≤ r ∣ a ′ − b ∣ ≤ r ′ |a'-b| \le r' ∣ a ′ − b ∣ ≤ r ′ A Ξ A_{\Xi} A Ξ
∣ K Ξ ( x , y ) − K Ξ ( x ′ , y ) ∣ + ∣ K Ξ ( y , x ) − K Ξ ( y , x ′ ) ∣ ≤ ( ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ) A Ξ ( x , y ) + ( ∣ K ( x ′ , y ) ∣ + ∣ K ( y , x ′ ) ∣ ) ∣ A Ξ ( x , y ) − A Ξ ( x ′ , y ) ∣ = : I 1 + I 2 . \begin{aligned}
&|K_{\Xi}(x,y)-K_{\Xi}(x',y)|+|K_{\Xi}(y,x)-K_{\Xi}(y,x')|\\
&\le\bigl(|K(x,y)-K(x',y)|+|K(y,x)-K(y,x')|\bigr)A_{\Xi}(x,y)\\
&\quad+\bigl(|K(x',y)|+|K(y,x')|\bigr)|A_{\Xi}(x,y)-A_{\Xi}(x',y)| =: \mathcal{I}_1+\mathcal{I}_2.
\end{aligned} ∣ K Ξ ( x , y ) − K Ξ ( x ′ , y ) ∣ + ∣ K Ξ ( y , x ) − K Ξ ( y , x ′ ) ∣ ≤ ( ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ) A Ξ ( x , y ) + ( ∣ K ( x ′ , y ) ∣ + ∣ K ( y , x ′ ) ∣ ) ∣ A Ξ ( x , y ) − A Ξ ( x ′ , y ) ∣ =: I 1 + I 2 . Note that A Ξ ≤ 1 A_{\Xi} \le1 A Ξ ≤ 1 I 1 ≤ C λ ( y , m ) − 1 ( δ / m ) γ \mathcal{I}_1 \le C\lambda(y,m)^{-1}(\delta/m)^{\gamma} I 1 ≤ C λ ( y , m ) − 1 ( δ / m ) γ
To estimate I 2 \mathcal{I}_2 I 2 r ′ ∈ [ m / 2 , m ] r' \in[m/2,m] r ′ ∈ [ m /2 , m ] ∣ K ( x ′ , y ) ∣ + ∣ K ( y , x ′ ) ∣ ≲ λ ( y , m ) − 1 |K(x',y)|+|K(y,x')| \lesssim\lambda(y,m)^{-1} ∣ K ( x ′ , y ) ∣ + ∣ K ( y , x ′ ) ∣ ≲ λ ( y , m ) − 1 A Ξ ( x , y ) = F ( r , a , b ) A_{\Xi}(x,y)=F(r,a,b) A Ξ ( x , y ) = F ( r , a , b ) A Ξ ( x ′ , y ) = F ( r ′ , a ′ , b ) A_{\Xi}(x',y)=F(r',a',b) A Ξ ( x ′ , y ) = F ( r ′ , a ′ , b ) F ( ρ , s , t ) : = ρ 2 M / ( ρ 2 M + s M t M ) F(\rho,s,t):=\rho^{2M}/(\rho^{2M}+s^{M}t^{M}) F ( ρ , s , t ) := ρ 2 M / ( ρ 2 M + s M t M )
∣ ∂ ρ F ( ρ , s , t ) ∣ + ∣ ∂ s F ( ρ , s , t ) ∣ ≲ M 1 ρ if ρ > 0 and ∣ s − t ∣ ≤ ρ . |\partial_{\rho}F(\rho,s,t)|+|\partial_sF(\rho,s,t)|\lesssim_M \frac{1}{\rho}\qquad\text{if }\rho>0\text{ and }|s-t|\le\rho. ∣ ∂ ρ F ( ρ , s , t ) ∣ + ∣ ∂ s F ( ρ , s , t ) ∣ ≲ M ρ 1 if ρ > 0 and ∣ s − t ∣ ≤ ρ . For ∂ s F \partial_sF ∂ s F ρ = 1 \rho=1 ρ = 1 t ≥ s / 2 t\ge s/2 t ≥ s /2 s > 4 s>4 s > 4 { ( ρ , s ) : ρ ≥ m / 2 , ∣ s − b ∣ ≤ ρ } \{(\rho,s):\rho\ge m/2,\ |s-b|\le\rho\} {( ρ , s ) : ρ ≥ m /2 , ∣ s − b ∣ ≤ ρ } ( r , a ) (r,a) ( r , a ) ( r ′ , a ′ ) (r',a') ( r ′ , a ′ )
∣ A Ξ ( x , y ) − A Ξ ( x ′ , y ) ∣ ≲ M ∣ r − r ′ ∣ + ∣ a − a ′ ∣ m ≤ 2 δ m . |A_{\Xi}(x,y)-A_{\Xi}(x',y)|\lesssim_M \frac{|r-r'|+|a-a'|}{m}\le2\frac{\delta}{m}. ∣ A Ξ ( x , y ) − A Ξ ( x ′ , y ) ∣ ≲ M m ∣ r − r ′ ∣ + ∣ a − a ′ ∣ ≤ 2 m δ . Hence, since δ ≤ m \delta\le m δ ≤ m γ ≤ 1 \gamma\le1 γ ≤ 1 I 2 ≲ λ ( y , m ) − 1 ( δ / m ) γ \mathcal{I}_2\lesssim\lambda(y,m)^{-1}(\delta/m)^{\gamma} I 2 ≲ λ ( y , m ) − 1 ( δ / m ) γ I 1 \mathcal{I}_1 I 1 □ \square □
For a finite complex Radon measure ν \nu ν t > 0 t>0 t > 0
T t ν ( x ) : = ∫ d ( x , y ) > t K ( x , y ) d ν ( y ) , T Ξ , t ν ( x ) : = ∫ d ( x , y ) > t K Ξ ( x , y ) d ν ( y ) . (381) T_t\nu(x):=\int_{d(x,y)>t}K(x,y)\,d\nu(y),\qquad T_{\Xi,t}\nu(x):=\int_{d(x,y)>t}K_{\Xi}(x,y)\,d\nu(y). \tag*{(381)} T t ν ( x ) := ∫ d ( x , y ) > t K ( x , y ) d ν ( y ) , T Ξ , t ν ( x ) := ∫ d ( x , y ) > t K Ξ ( x , y ) d ν ( y ) . (381) T t ∗ ν ( x ) : = ∫ d ( x , y ) > t K ( y , x ) d ν ( y ) , T Ξ , t ∗ ν ( x ) : = ∫ d ( x , y ) > t K Ξ ( y , x ) d ν ( y ) . (382) T_t^*\nu(x):=\int_{d(x,y)>t}K(y,x)\,d\nu(y),\qquad T_{\Xi,t}^*\nu(x):=\int_{d(x,y)>t}K_{\Xi}(y,x)\,d\nu(y). \tag*{(382)} T t ∗ ν ( x ) := ∫ d ( x , y ) > t K ( y , x ) d ν ( y ) , T Ξ , t ∗ ν ( x ) := ∫ d ( x , y ) > t K Ξ ( y , x ) d ν ( y ) . (382) and define the truncated upper-doubling maximal function by
M λ , t ν ( x ) : = sup s ≥ t ∣ ν ∣ ( B ( x , s ) ) λ ( x , s ) . (383) \mathcal{M}_{\lambda,t}\nu(x):=\sup_{s\ge t}\frac{|\nu|(B(x,s))}{\lambda(x,s)}. \tag*{(383)} M λ , t ν ( x ) := s ≥ t sup λ ( x , s ) ∣ ν ∣ ( B ( x , s )) . (383) Lemma B.5. Let ν \nu ν t > 0 t>0 t > 0 x ∈ X x\in X x ∈ X Ξ ( x ) ≤ t \Xi(x)\le t Ξ ( x ) ≤ t
∣ T Ξ , t ν ( x ) − T t ν ( x ) ∣ + ∣ T Ξ , t ∗ ν ( x ) − T t ∗ ν ( x ) ∣ ≲ M λ , t ν ( x ) . (384) \lvert T_{\Xi,t}\nu(x)-T_t\nu(x)\rvert+\lvert T_{\Xi,t}^{*}\nu(x)-T_t^{*}\nu(x)\rvert\lesssim\mathcal{M}_{\lambda,t}\nu(x). \tag*{(384)} ∣ T Ξ , t ν ( x ) − T t ν ( x )∣ + ∣ T Ξ , t ∗ ν ( x ) − T t ∗ ν ( x )∣ ≲ M λ , t ν ( x ) . (384) Proof. We write a : = Ξ ( x ) a:=\Xi(x) a := Ξ ( x ) y y y r : = d ( x , y ) > t r:=d(x,y)>t r := d ( x , y ) > t Ξ ( y ) ≤ a + r ≤ 2 r \Xi(y)\le a+r\le2r Ξ ( y ) ≤ a + r ≤ 2 r
0 ≤ 1 − A Ξ ( x , y ) ≤ a M Ξ ( y ) M r 2 M ≤ 2 M ( a r ) M . (385) 0\le1-A_{\Xi}(x,y)\le\frac{a^{M}\Xi(y)^{M}}{r^{2M}}\le2^{M}\left(\frac{a}{r}\right)^{M}. \tag*{(385)} 0 ≤ 1 − A Ξ ( x , y ) ≤ r 2 M a M Ξ ( y ) M ≤ 2 M ( r a ) M . (385) On the annulus 2 k t < d ( x , y ) ≤ 2 k + 1 t 2^{k}t<d(x,y)\le2^{k+1}t 2 k t < d ( x , y ) ≤ 2 k + 1 t
2 − k M ∣ ν ∣ ( B ( x , 2 k + 2 t ) ) λ ( x , 2 k t ) ≲ C λ 2 2 − k M M λ , t ν ( x ) . 2^{-kM}\frac{\lvert\nu\rvert(B(x,2^{k+2}t))}{\lambda(x,2^{k}t)}\lesssim C_{\lambda}^{2}2^{-kM}\mathcal{M}_{\lambda,t}\nu(x). 2 − k M λ ( x , 2 k t ) ∣ ν ∣ ( B ( x , 2 k + 2 t )) ≲ C λ 2 2 − k M M λ , t ν ( x ) . Summation over k ≥ 0 k\ge0 k ≥ 0 λ ( y , d ( x , y ) ) ≈ C λ λ ( x , d ( x , y ) ) \lambda(y,d(x,y))\approx_{C_{\lambda}}\lambda(x,d(x,y)) λ ( y , d ( x , y )) ≈ C λ λ ( x , d ( x , y )) □ \square □
We define
H Ξ , ε ν ( x ) : = T max { ε , Ξ ( x ) } ν ( x ) and H Ξ , ε ∗ ν ( x ) : = T max { ε , Ξ ( x ) } ∗ ν ( x ) . (386) H_{\Xi,\varepsilon}\nu(x):=T_{\max\{\varepsilon,\Xi(x)\}}\nu(x)
\qquad\text{and}\qquad
H_{\Xi,\varepsilon}^{*}\nu(x):=T_{\max\{\varepsilon,\Xi(x)\}}^{*}\nu(x). \tag*{(386)} H Ξ , ε ν ( x ) := T m a x { ε , Ξ ( x )} ν ( x ) and H Ξ , ε ∗ ν ( x ) := T m a x { ε , Ξ ( x )} ∗ ν ( x ) . (386) Lemmas B.4 and B.5 imply, for every ε > 0 \varepsilon>0 ε > 0
∣ T Ξ , ε ν ( x ) − H Ξ , ε ν ( x ) ∣ + ∣ T Ξ , ε ∗ ν ( x ) − H Ξ , ε ∗ ν ( x ) ∣ ≲ M λ , max { ε , Ξ ( x ) } ν ( x ) . (387) \lvert T_{\Xi,\varepsilon}\nu(x)-H_{\Xi,\varepsilon}\nu(x)\rvert+\lvert T_{\Xi,\varepsilon}^{*}\nu(x)-H_{\Xi,\varepsilon}^{*}\nu(x)\rvert\lesssim\mathcal{M}_{\lambda,\max\{\varepsilon,\Xi(x)\}}\nu(x). \tag*{(387)} ∣ T Ξ , ε ν ( x ) − H Ξ , ε ν ( x )∣ + ∣ T Ξ , ε ∗ ν ( x ) − H Ξ , ε ∗ ν ( x )∣ ≲ M λ , m a x { ε , Ξ ( x )} ν ( x ) . (387) Indeed, if ε ≥ Ξ ( x ) \varepsilon\ge\Xi(x) ε ≥ Ξ ( x ) ε < Ξ ( x ) \varepsilon<\Xi(x) ε < Ξ ( x ) T Ξ , ε ν ( x ) T_{\Xi,\varepsilon}\nu(x) T Ξ , ε ν ( x ) ε < d ( x , y ) ≤ Ξ ( x ) \varepsilon<d(x,y)\le\Xi(x) ε < d ( x , y ) ≤ Ξ ( x )
∣ ν ∣ ( B ( x , 2 Ξ ( x ) ) ) λ ( x , Ξ ( x ) ) ≤ C λ M λ , Ξ ( x ) ν ( x ) , \frac{\lvert\nu\rvert(B(x,2\Xi(x)))}{\lambda(x,\Xi(x))}\le C_{\lambda}\mathcal{M}_{\lambda,\Xi(x)}\nu(x), λ ( x , Ξ ( x )) ∣ ν ∣ ( B ( x , 2Ξ ( x ))) ≤ C λ M λ , Ξ ( x ) ν ( x ) , and the remaining part is treated by Lemma B.5 with t = Ξ ( x ) t=\Xi(x) t = Ξ ( x )
For a non-negative Borel function Ξ \Xi Ξ
M Ξ λ ν ( x ) : = sup s ≥ Ξ ( x ) s > 0 ∣ ν ∣ ( B ( x , s ) ) λ ( x , s ) (388) \mathcal{M}_{\Xi}^{\lambda}\nu(x):=\sup_{\substack{s\ge\Xi(x)\\s>0}}\frac{\lvert\nu\rvert(B(x,s))}{\lambda(x,s)} \tag*{(388)} M Ξ λ ν ( x ) := s ≥ Ξ ( x ) s > 0 sup λ ( x , s ) ∣ ν ∣ ( B ( x , s )) (388) and prove its boundedness properties in the following lemma.
Lemma B.6. Let Ξ : X → [ 0 , ∞ ) \Xi:X\to[0,\infty) Ξ : X → [ 0 , ∞ ) m 1 , m 2 m_1,m_2 m 1 , m 2 c 1 , c 2 > 0 c_1,c_2>0 c 1 , c 2 > 0
m j ( B ( x , r ) ) ≤ c j λ ( x , r ) , for x ∈ X , r > 0 , r ≥ Ξ ( x ) , and j = 1 , 2. (389) m_j(B(x,r))\le c_j\lambda(x,r),\qquad\text{for }x\in X,\ r>0,\ r\ge\Xi(x),\qquad\text{and }j=1,2. \tag*{(389)} m j ( B ( x , r )) ≤ c j λ ( x , r ) , for x ∈ X , r > 0 , r ≥ Ξ ( x ) , and j = 1 , 2. (389) Then M Ξ λ \mathcal{M}_{\Xi}^{\lambda} M Ξ λ L 1 ( m 2 ) L^1(m_2) L 1 ( m 2 ) L 1 , ∞ ( m 1 ) L^{1,\infty}(m_1) L 1 , ∞ ( m 1 ) L p ( m 2 ) L^p(m_2) L p ( m 2 ) L p ( m 1 ) L^p(m_1) L p ( m 1 ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ p p p C λ C_{\lambda} C λ c 1 c_1 c 1 c 2 c_2 c 2
Proof. If M Ξ λ ( f m 2 ) ( x ) > t \mathcal{M}_{\Xi}^{\lambda}(fm_2)(x)>t M Ξ λ ( f m 2 ) ( x ) > t r x > 0 r_x>0 r x > 0 r x ≥ Ξ ( x ) r_x\ge\Xi(x) r x ≥ Ξ ( x )
∫ B ( x , r x ) ∣ f ∣ d m 2 > t λ ( x , r x ) . \int_{B(x,r_x)}\lvert f\rvert\,dm_2>t\lambda(x,r_x). ∫ B ( x , r x ) ∣ f ∣ d m 2 > t λ ( x , r x ) . Since we obtain estimates uniform on the radii, without loss of generality we can consider them bounded. Then, we apply the 5 r 5r 5 r { B j = \{B_j= { B j =
B ( x j , r j ) } j B(x_j,r_j)\}_{j} B ( x j , r j ) } j { x ∈ X : M Ξ λ ( f m 2 ) ( x ) > t } \{x\in X:\mathcal{M}^{\lambda}_{\Xi}(fm_2)(x)>t\} { x ∈ X : M Ξ λ ( f m 2 ) ( x ) > t } ⋃ j 5 B j \bigcup_j 5B_j ⋃ j 5 B j
m 1 ( { M Ξ λ ( f m 2 ) > t } ) ≤ ∑ j m 1 ( 5 B j ) ≲ c 1 ∑ j λ ( x j , r j ) ≤ c 1 t ∑ j ∫ B j ∣ f ∣ d m 2 ≤ c 1 t ∥ f ∥ L 1 ( m 2 ) , \begin{aligned}
m_1\left(\left\{\mathcal{M}^{\lambda}_{\Xi}(fm_2)>t\right\}\right)\leq\sum_jm_1(5B_j)\lesssim c_1\sum_j\lambda(x_j,r_j) \\
&\leq\frac{c_1}{t}\sum_j\int_{B_j}|f|\,dm_2\leq\frac{c_1}{t}\|f\|_{L^1(m_2)},
\end{aligned} m 1 ( { M Ξ λ ( f m 2 ) > t } ) ≤ j ∑ m 1 ( 5 B j ) ≲ c 1 j ∑ λ ( x j , r j ) ≤ t c 1 j ∑ ∫ B j ∣ f ∣ d m 2 ≤ t c 1 ∥ f ∥ L 1 ( m 2 ) , which proves that M Ξ λ \mathcal{M}^{\lambda}_{\Xi} M Ξ λ L 1 ( m 2 ) L^1(m_2) L 1 ( m 2 ) L 1 , ∞ ( m 1 ) L^{1,\infty}(m_1) L 1 , ∞ ( m 1 ) m 2 m_2 m 2 ∥ M Ξ λ ( f m 2 ) ∥ L ∞ ( m 1 ) ≤ c 2 ∥ f ∥ L ∞ ( m 2 ) \|\mathcal{M}^{\lambda}_{\Xi}(fm_2)\|_{L^\infty(m_1)}\leq c_2\|f\|_{L^\infty(m_2)} ∥ M Ξ λ ( f m 2 ) ∥ L ∞ ( m 1 ) ≤ c 2 ∥ f ∥ L ∞ ( m 2 )
We conclude with the analogue of Lemma 3.14 . Let ν \nu ν x ∈ X x\in X x ∈ X R 0 ≥ 0 R_0\geq0 R 0 ≥ 0 a 0 > 0 a_0>0 a 0 > 0 c 0 > 0 c_0>0 c 0 > 0
∣ ν ∣ ( B ( x , r ) ) ≤ c 0 λ ( x , r ) and ∣ T t ν ( x ) ∣ ≤ a 0 , (390) |\nu|(B(x,r))\leq c_0\lambda(x,r)\qquad\text{and}\qquad|T_t\nu(x)|\leq a_0, \tag*{(390)} ∣ ν ∣ ( B ( x , r )) ≤ c 0 λ ( x , r ) and ∣ T t ν ( x ) ∣ ≤ a 0 , (390) for t , r > 0 t,r>0 t , r > 0 t , r ≥ R 0 t,r\geq R_0 t , r ≥ R 0 Ξ ( x ) ≥ R 0 \Xi(x)\geq R_0 Ξ ( x ) ≥ R 0 ε > 0 \varepsilon>0 ε > 0 t : = max { ε , Ξ ( x ) } ≥ R 0 t:=\max\{\varepsilon,\Xi(x)\}\geq R_0 t := max { ε , Ξ ( x )} ≥ R 0
∣ T Ξ , ε ν ( x ) ∣ ≤ ∣ T t ν ( x ) ∣ + C M λ , t ν ( x ) ≤ a 0 + C c 0 , (391) |T_{\Xi,\varepsilon}\nu(x)|\leq|T_t\nu(x)|+C\mathcal{M}_{\lambda,t}\nu(x)\leq a_0+Cc_0, \tag*{(391)} ∣ T Ξ , ε ν ( x ) ∣ ≤ ∣ T t ν ( x ) ∣ + C M λ , t ν ( x ) ≤ a 0 + C c 0 , (391) and the same bound holds for T Ξ , ε ∗ ν ( x ) T^*_{\Xi,\varepsilon}\nu(x) T Ξ , ε ∗ ν ( x ) ∣ T t ∗ ν ( x ) ∣ ≤ a 0 |T^*_t\nu(x)|\leq a_0 ∣ T t ∗ ν ( x ) ∣ ≤ a 0 t > 0 t>0 t > 0 t ≥ R 0 t\geq R_0 t ≥ R 0 Lemma 3.13 is the case λ ( x , r ) = r n \lambda(x,r)=r^n λ ( x , r ) = r n M = n M=n M = n Lemmas B.4 and B.5, Lemma B.6 is the analogue of Lemma 3.12 , and (B.35) is the analogue of Lemma 3.14 . On the other hand, we remark that the mixed Cotlar inequality of Lemma 3.16 , § 4 –§ 7 , and Theorem 1.3 are stated and proved only under polynomial-growth assumptions. ANDREA MERLO, DEPARTAMENTO DE MATEMÁTICAS, UNIVERSIDAD DEL PAÍS VASCO (UPV/EHU), BARRIO SARRIENA S/N 48940 LEIOA, SPAIN AND, IKERBASQUE, BASQUE FOUNDATION FOR SCIENCE, BILBAO, SPAIN. Email address: andrea.merlo@ehu.eus
MIHALIS MOURGOGLOU, DEPARTAMENTO DE MATEMÁTICAS, UNIVERSIDAD DEL PAÍS VASCO (UPV/EHU), BARRIO SARRIENA S/N 48940 LEIOA, SPAIN AND, IKERBASQUE, BASQUE FOUNDATION FOR SCIENCE, BILBAO, SPAIN. Email address: michail.mourgoglou@ehu.eus
CARMELO PULIATTI, DEPARTAMENTO DE MATEMÀTIQUES, UNIVERSITAT AUTÒNOMA DE BARCELONA, 08193 BELLATERRA (BARCELONA), CATALONIA. Email address: carmelo.puliatti@uab.cat
