Introduction and main results
Maximal inequalities for convolution operators provide a means of passing from convergence on a dense class to almost everywhere convergence. For integrable approximation kernels, Zó [30 ] proved weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) L 1 L_1 L 1 [28 Chapter II, Section 4] .
We study these maximal inequalities for operator-valued functions. Let M \mathcal{M} M τ \tau τ
N = L ∞ ( R d ) ⊗ ‾ M , φ = ∫ R d d x ⊗ τ . \mathcal{N}=L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M},\qquad\varphi=\int_{\mathbb{R}^{d}} dx\otimes\tau. N = L ∞ ( R d ) ⊗ M , φ = ∫ R d d x ⊗ τ . In this setting, the noncommutative Hardy–Littlewood maximal inequality was established by Mei [17 ] . Mei’s approach was built on previous foundational works on noncommutative martingale and ergodic inequalities and vector-valued noncommutative L p L_p L p [5 , 24 , 25 , 13 , 14 ] ; see also [15 Yea77] for earlier contributions. The first noncommutative Calderón–Zygmund decomposition was obtained by Parcet [21 ] , who exploited Cuculescu’s projections from noncommutative martingale theory in place of classical stopping times. There are four terms in Parcet’s decomposition:
f = g + b + g o f f + b o f f . f=g+b+g_{\mathrm{off}}+b_{\mathrm{off}}. f = g + b + g off + b off . The additional terms g o f f g_{\mathrm{off}} g off b o f f b_{\mathrm{off}} b off g g g b b b g o f f g_{\mathrm{off}} g off [21 ] explored the T T ∗ TT^* T T ∗ [19 ] . This method does not directly apply under the Hörmander condition, so he left the sufficiency of this condition as an open problem in [21 Remark 5.5] , see also [22 Remark 1.3] .
Subsequently, partial progress was made by Cadilhac, Conde-Alonso and Parcet in [2 ] , where they constructed a new Calderón–Zygmund decomposition by modifying b o f f b_{\mathrm{off}} b off g o f f g_{\mathrm{off}} g off L 2 L_2 L 2 L 2 L_2 L 2 [11 ] proved the noncommutative maximal weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 1 ) (1,1) ( 1 , 1 )
With many applications in mind, the original purpose of the present paper is to build a noncommutative version of Zo’s maximal inequalities for integrable approximation kernels. Then we found that the entirely new strategy designed to overcome the limitations of the classical vector-valued Calderón–Zygmund approach also works for singular convolution kernels satisfying the Hörmander condition. This provides a positive answer to the previously mentioned problem for convolution kernels.
Our first result is a noncommutative analogue of Zo’s maximal inequalities. The unexplained notation will be specified in § 2.2 .
Theorem 1.1. Let ( T n ) n ≥ 1 (T_n)_{n\geq1} ( T n ) n ≥ 1 N \mathcal{N} N K n ∈ L 1 , l o c ( R d ∖ { 0 } ) K_n\in L_{1,\mathrm{loc}}(\mathbb{R}^{d}\setminus\{0\}) K n ∈ L 1 , loc ( R d ∖ { 0 }) D \mathcal{D} D
sup y ≠ 0 ∫ ∣ x ∣ > 2 ∣ y ∣ sup n ≥ 1 ∣ K n ( x − y ) − K n ( x ) ∣ d x ≤ C H . (1) \sup_{y\neq0}\int_{\lvert x\rvert>2\lvert y\rvert}\sup_{n\geq1}\lvert K_n(x-y)-K_n(x)\rvert\,dx\leq C_H. \tag*{(1)} y = 0 sup ∫ ∣ x ∣ > 2 ∣ y ∣ n ≥ 1 sup ∣ K n ( x − y ) − K n ( x )∣ d x ≤ C H . (1) Assume also that, for some 1 < r ≤ ∞ 1<r\leq\infty 1 < r ≤ ∞
∥ ( T n f ) n ≥ 1 ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r . \lVert(T_nf)_{n\geq1}\rVert_{L_r(\mathcal{N};\ell_\infty)}\leq M_r\lVert f\rVert_r. ∥( T n f ) n ≥ 1 ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r . Then ( T n ) n ≥ 1 (T_n)_{n\ge1} ( T n ) n ≥ 1 ( 1 , 1 ) (1,1) ( 1 , 1 ) f ∈ L 1 ( N ) f \in L_1(\mathcal{N}) f ∈ L 1 ( N ) t > 0 t > 0 t > 0 e ∈ N e \in\mathcal{N} e ∈ N
sup n ≥ 1 ∥ e T n ( f ) e ∥ ∞ ≤ t , t φ ( e ⊥ ) ≤ C d , r ( M r + C H ) ∥ f ∥ 1 . \sup_{n\ge1}\|eT_n(f)e\|_\infty\le t,\qquad t\varphi(e^\perp)\le C_{d,r}(M_r+C_H)\|f\|_1. n ≥ 1 sup ∥ e T n ( f ) e ∥ ∞ ≤ t , tφ ( e ⊥ ) ≤ C d , r ( M r + C H ) ∥ f ∥ 1 . Our second result concerns all sharp truncations of one singular convolution kernel.
Theorem 1.2. Let T T T L 2 ( N ) L_2(\mathcal{N}) L 2 ( N ) K : R d ∖ { 0 } → C K:\mathbb{R}^d\setminus\{0\}\to\mathbb{C} K : R d ∖ { 0 } → C L 1 ( N ) L_1(\mathcal{N}) L 1 ( N )
∣ K ( x ) ∣ ≤ A ∣ x ∣ − d , sup y ≠ 0 ∫ ∣ x ∣ > 2 ∣ y ∣ ∣ K ( x − y ) − K ( x ) ∣ d x ≤ C H . |K(x)|\le A|x|^{-d},\qquad\sup_{y\ne0}\int_{|x|>2|y|}|K(x-y)-K(x)|\,dx\le C_H. ∣ K ( x ) ∣ ≤ A ∣ x ∣ − d , y = 0 sup ∫ ∣ x ∣ > 2∣ y ∣ ∣ K ( x − y ) − K ( x ) ∣ d x ≤ C H . For ε > 0 \varepsilon>0 ε > 0
T ε f ( x ) = ∫ ∣ x − y ∣ > ε K ( x − y ) f ( y ) d y . T_\varepsilon f(x)=\int_{|x-y|>\varepsilon}K(x-y)f(y)\,dy. T ε f ( x ) = ∫ ∣ x − y ∣ > ε K ( x − y ) f ( y ) d y . Assume that, for some 1 < p 0 < ∞ 1<p_0<\infty 1 < p 0 < ∞
∥ ( T ε f ) ε > 0 ∥ L p 0 ( N ; ℓ ∞ ) ≤ M p 0 ∥ f ∥ p 0 . (2) \|(T_\varepsilon f)_{\varepsilon>0}\|_{L_{p_0}(\mathcal{N};\ell_\infty)}\le M_{p_0}\|f\|_{p_0}. \tag*{(2)} ∥ ( T ε f ) ε > 0 ∥ L p 0 ( N ; ℓ ∞ ) ≤ M p 0 ∥ f ∥ p 0 . (2) Then ( T ε ) ε > 0 (T_\varepsilon)_{\varepsilon>0} ( T ε ) ε > 0 ( 1 , 1 ) (1,1) ( 1 , 1 ) f ∈ L 1 ( N ) f\in L_1(\mathcal{N}) f ∈ L 1 ( N ) t > 0 t>0 t > 0 e ∈ N e\in\mathcal{N} e ∈ N
sup ε > 0 ∥ e T ε ( f ) e ∥ ∞ ≤ t , t φ ( e ⊥ ) ≤ C d , p 0 ( A + C H + M p 0 ) ∥ f ∥ 1 . \sup_{\varepsilon>0}\|eT_\varepsilon(f)e\|_\infty\le t,\qquad t\varphi(e^\perp)\le C_{d,p_0}(A+C_H+M_{p_0})\|f\|_1. ε > 0 sup ∥ e T ε ( f ) e ∥ ∞ ≤ t , tφ ( e ⊥ ) ≤ C d , p 0 ( A + C H + M p 0 ) ∥ f ∥ 1 . In many cases, the assumption {(xref)}(1.1) is redundant, because it can be deduced from the Cotlar inequality; see [11 Theorem 1.3] . Given one singular integral operator T T T (1.1) in Theorem 1.2 , applying Theorem 1.1 to T n = T T_n=T T n = T n n n ( 1 , 1 ) (1,1) ( 1 , 1 ) T T T [21 Remark 5.5] and reiterated in Parcet’s recent ICM survey [22 Remark 1.3] . Finally, the deduction from Theorem 1.1 to Theorem 1.2 is not automatic since the kernels of ( T ε ) ε > 0 (T_\varepsilon)_{\varepsilon>0} ( T ε ) ε > 0 (H); see § 6.4 for this non-trivial passage.
As with Zó’s result (see, e.g., [28 Chapter II, §§5.10 and 5.16] ), Theorem 1.1 gives lots of applications in § 7 , such as approximations of the identity, kernels with monotone L log L L\log L L log L § 8 .
Strategy and novelties. As mentioned previously, the proof of Theorem 1.1 requires substantial new ideas. We will exploit the effective decomposition introduced in [2 ] . The main obstruction to using the usual L 1 L_1 L 1 b o f f , Q b_{\mathrm{off},Q} b off , Q Q Q Q § 2.3 . Estimating these terms separately using the L 1 L_1 L 1
∑ Q ∥ b o f f , Q ∥ 1 ≲ ∥ f ∥ 1 , \sum_Q\|b_{\mathrm{off},Q}\|_1\lesssim\|f\|_1, Q ∑ ∥ b off , Q ∥ 1 ≲ ∥ f ∥ 1 , which is unavailable for the noncommutative decomposition; see [2 Remark 2.4] . Their proof instead uses an annular L 2 L_2 L 2 [11 ] for maximal truncations.
Our new approach will not require the preceding L 1 L_1 L 1 ( 1 , 1 ) (1,1) ( 1 , 1 )
The weak type estimate for dyadic series. In § 4 , by assuming a maximal strong type ( r , r ) (r,r) ( r , r ) 1 < r ≤ ∞ 1 < r \le\infty 1 < r ≤ ∞ ( 1 , 1 ) (1,1) ( 1 , 1 )
S n f = ∑ Q ∈ Q b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q , \mathcal{S}_n f = \sum_{Q \in\mathcal{Q}} b_{n,Q} \mathsf{E}_Q f, \qquad|b_{n,Q}| \le m_Q, S n f = Q ∈ Q ∑ b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q , provided additionally
Γ : = sup P ∈ Q ess sup y ∈ P ∑ Q ⊆ P y ∈ Q 1 ∣ Q ∣ ∫ ( β P ) c m Q ( x ) d x < ∞ . (3) \Gamma:= \sup_{P \in\mathcal{Q}} \operatorname*{ess\,sup}_{y \in P} \sum_{\substack{Q \subseteq P \\ y \in Q}} \frac{1}{|Q|} \int_{(\beta P)^c} m_Q(x)\,\mathrm{d}x < \infty. \tag*{(3)} Γ := P ∈ Q sup y ∈ P ess sup Q ⊆ P y ∈ Q ∑ ∣ Q ∣ 1 ∫ ( β P ) c m Q ( x ) d x < ∞. (3) where β ≥ 3 \beta\ge3 β ≥ 3 Proposition 3.2 . We briefly explain this decomposition and how it yields the weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) t > 0 t > 0 t > 0 g g g λ > 0 \lambda> 0 λ > 0 t t t w w w
w S n ( f − g ) w = B n + B ℓ , n + B r , n , w\mathcal{S}_n(f-g)w = B_n + B_{\ell,n} + B_{r,n}, w S n ( f − g ) w = B n + B ℓ , n + B r , n , where the three bad terms depend on w w w w w w λ / 4 \lambda/4 λ /4 n n n φ ( 1 − w ) \varphi(1-w) φ ( 1 − w ) t t t λ \lambda λ
We now turn to the construction of w w w [2 ] . Although the good function g g g Lemma 2.5 , we use the coefficient bound to choose one projection w w w n n n φ ( 1 − w ) \varphi(1-w) φ ( 1 − w ) B ℓ , n B_{\ell,n} B ℓ , n B r , n B_{r,n} B r , n L 1 L_1 L 1
To explain this construction, write f Q = E Q f f_Q = \mathsf{E}_Q f f Q = E Q f q Q q_Q q Q t t t
Π Q ( x ) = ∑ P ∈ P ( Q ) 1 ( β P ) c ( x ) p P , \Pi_Q(x) = \sum_{P \in\mathcal{P}(Q)} \mathbf{1}_{(\beta P)^c}(x)p_P, Π Q ( x ) = P ∈ P ( Q ) ∑ 1 ( β P ) c ( x ) p P , where P ( Q ) \mathcal{P}(Q) P ( Q ) p P p_P p P Proposition 3.2 . The coefficient bound (3) gives
φ ( ∑ Q m Q Π Q ) ≤ Γ t ∥ f ∥ 1 , ∑ Q φ ( m Q Π Q f Q Π Q ) ≤ Γ ∥ f ∥ 1 . \varphi\left(\sum_Q m_Q\Pi_Q\right) \le\frac{\Gamma}{t}\|f\|_1, \qquad\sum_Q \varphi(m_Q\Pi_Q f_Q\Pi_Q) \le\Gamma\|f\|_1. φ Q ∑ m Q Π Q ≤ t Γ ∥ f ∥ 1 , Q ∑ φ ( m Q Π Q f Q Π Q ) ≤ Γ∥ f ∥ 1 . After a spectral cut, we obtain a projection w 0 w_0 w 0 φ ( 1 − w 0 ) ≤ C d , β ∥ f ∥ 1 / t \varphi(1-w_0) \le C_{d,\beta}\lVert f\rVert_1/t φ ( 1 − w 0 ) ≤ C d , β ∥ f ∥ 1 / t
∑ Q b n , Q w 0 Π Q f Q q Q w 0 = R n C , \sum_Q b_{n,Q}w_0\Pi_Qf_Qq_Qw_0=R_nC, Q ∑ b n , Q w 0 Π Q f Q q Q w 0 = R n C , where
R n = ( b n , Q m Q w 0 Π Q ) Q , C = ( m Q Π Q f Q q Q w 0 ) Q c o l u m n , R_n=\left(\frac{b_{n,Q}}{\sqrt{m_Q}}w_0\Pi_Q\right)_Q,\qquad C=\left(\sqrt{m_Q}\Pi_Qf_Qq_Qw_0\right)^{\mathrm{column}}_Q, R n = ( m Q b n , Q w 0 Π Q ) Q , C = ( m Q Π Q f Q q Q w 0 ) Q column , with the quotient defined to be zero when m Q = 0 m_Q=0 m Q = 0 f Q q Q f Q ≤ t f Q f_Qq_Qf_Q\leq tf_Q f Q q Q f Q ≤ t f Q
( R n C ) ∗ ( R n C ) ≤ Γ C ∗ C , φ ( C ∗ C ) ≤ t Γ ∥ f ∥ 1 . (R_nC)^*(R_nC)\leq\Gamma C^*C,\qquad\varphi(C^*C)\leq t\Gamma\lVert f\rVert_1. ( R n C ) ∗ ( R n C ) ≤ Γ C ∗ C , φ ( C ∗ C ) ≤ t Γ ∥ f ∥ 1 . The same bound holds for the adjoint of the right off-diagonal sum. Therefore, starting from w 0 w_0 w 0 C ∗ C C^*C C ∗ C λ / 4 \lambda/4 λ /4 16 t Γ 2 λ − 2 ∥ f ∥ 1 16t\Gamma^2\lambda^{-2}\lVert f\rVert_1 16 t Γ 2 λ − 2 ∥ f ∥ 1 B n B_n B n w ≤ w 0 w\leq w_0 w ≤ w 0
The simultaneous kernel decomposition and dyadic representations of operators. To apply this abstract estimate to the operators in Theorem 1.1 , we first regularize them such that the regularized kernels are bounded and Lipschitz; see Lemma 5.2 .
For a fixed regularization parameter ε > 0 \varepsilon>0 ε > 0 T n , ε T_{n,\varepsilon} T n , ε K n , ε K_{n,\varepsilon} K n , ε § 5 gives one family D \mathcal{D} D a n , Q a_{n,Q} a n , Q K n , ε = ∑ Q ∈ D a n , Q 1 Q K_{n,\varepsilon}=\sum_{Q\in\mathcal{D}}a_{n,Q}1_Q K n , ε = ∑ Q ∈ D a n , Q 1 Q P ∈ D P\in\mathcal{D} P ∈ D
∑ Q ∈ D Q ⊆ P ∣ Q ∣ sup n ∣ a n , Q ∣ ≲ ∫ P sup n ∣ K n , ε − E P K n , ε ∣ ; \sum_{\substack{Q\in\mathcal{D}\\Q\subseteq P}}|Q|\sup_n|a_{n,Q}|\lesssim\int_P\sup_n|K_{n,\varepsilon}-\mathbb{E}_PK_{n,\varepsilon}|; Q ∈ D Q ⊆ P ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≲ ∫ P n sup ∣ K n , ε − E P K n , ε ∣ ; see Proposition 4.1 .
For each cube Q ∈ D Q\in\mathcal{D} Q ∈ D x ∈ R d x\in\mathbb{R}^d x ∈ R d x − Q x-Q x − Q 1 Q ∗ f ( x ) 1_Q*f(x) 1 Q ∗ f ( x ) f f f
T n , ε f = ∑ Q ∈ D b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q . T_{n,\varepsilon}f=\sum_{Q\in\mathcal{D}}b_{n,Q}\mathbb{E}_Qf,\qquad|b_{n,Q}|\leq m_Q. T n , ε f = Q ∈ D ∑ b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q . The coefficient estimates for the kernel expansion, together with the joint Hörmander condition and the geometry of translated cubes, yield Γ ≤ C d , r ( M r + C H ) \Gamma\leq C_{d,r}(M_r+C_H) Γ ≤ C d , r ( M r + C H ) Proposition 4.4 . This is the bound needed in Theorem 3.1 . Its uniformity in the regularization parameter allows us to recover the original operators and complete the proof of Theorem 1.1 in § 6 .
Limitation of the method. Convolution allows us to combine the kernel decomposition with the geometry of translated cubes to obtain Γ < ∞ \Gamma<\infty Γ < ∞ x x x
Organization of the paper. § 2 collects the definitions and preliminary results used below. § 4 proves a weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) § 5 constructs the simultaneous kernel decomposition and expresses the convolution operators in terms of dyadic averages. § 6 discusses the joint Hörmander condition and regularization, and proves Theorems 1.1 and 1.2 . § 7 studies maximal inequalities and approximation of the identity. It first treats integrable kernels and dyadic approximation, then monotone L log L L\log L L log L L 1 L_1 L 1
Preliminaries
In this section, we introduce the notation and recall basic facts about noncommutative measure spaces and convolution operators. We also recall the noncommutative Calderón–Zygmund decomposition and transference results used below.
Noncommutative measure spaces.
Noncommutative L p L_p L p
Let M \mathcal{M} M τ \tau τ M + \mathcal{M}_+ M + S ( M ) + S(\mathcal{M})_+ S ( M ) + x ∈ M + x \in\mathcal{M}_+ x ∈ M + S ( M ) S(\mathcal{M}) S ( M ) S ( M ) + S(\mathcal{M})_+ S ( M ) + 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ L p ( M ) L_p(\mathcal{M}) L p ( M ) S ( M ) S(\mathcal{M}) S ( M )
∥ x ∥ p = τ ( ∣ x ∣ p ) 1 / p , ∣ x ∣ = ( x ∗ x ) 1 / 2 . \lVert x\rVert_p=\tau(|x|^p)^{1/p}, \qquad|x|=(x^*x)^{1/2}. ∥ x ∥ p = τ ( ∣ x ∣ p ) 1/ p , ∣ x ∣ = ( x ∗ x ) 1/2 . As usual, L ∞ ( M ) = M L_\infty(\mathcal{M})=\mathcal{M} L ∞ ( M ) = M
A closed densely defined operator x x x M \mathcal{M} M τ \tau τ
τ ( 1 ( t , ∞ ) ( ∣ x ∣ ) ) < ∞ for some t > 0. \tau\bigl(\mathbf{1}_{(t,\infty)}(|x|)\bigr)<\infty\qquad\text{for some } t>0. τ ( 1 ( t , ∞ ) ( ∣ x ∣ ) ) < ∞ for some t > 0. We denote the algebra of these operators by L 0 ( M ) L_0(\mathcal{M}) L 0 ( M )
L p ( M ) = { x ∈ L 0 ( M ) : τ ( ∣ x ∣ p ) < ∞ } , 1 ≤ p < ∞ . L_p(\mathcal{M})=\{x\in L_0(\mathcal{M}):\tau(|x|^p)<\infty\}, \qquad1\le p<\infty. L p ( M ) = { x ∈ L 0 ( M ) : τ ( ∣ x ∣ p ) < ∞ } , 1 ≤ p < ∞. And the weak L p L_p L p L p , ∞ ( M ) L_{p,\infty}(\mathcal{M}) L p , ∞ ( M ) x ∈ L 0 ( M ) x\in L_0(\mathcal{M}) x ∈ L 0 ( M )
∥ x ∥ p , ∞ : = sup t > 0 t d x ( t ) 1 / p < ∞ , \lVert x\rVert_{p,\infty}:=\sup_{t>0}t d_x(t)^{1/p}<\infty, ∥ x ∥ p , ∞ := t > 0 sup t d x ( t ) 1/ p < ∞ , where d x ( t ) = τ ( 1 ( t , ∞ ) ( ∣ x ∣ ) ) d_x(t)=\tau\bigl(\mathbf{1}_{(t,\infty)}(|x|)\bigr) d x ( t ) = τ ( 1 ( t , ∞ ) ( ∣ x ∣ ) ) x x x
∥ x ∥ p , ∞ ≤ ∥ x ∥ p , x ∈ L p ( M ) . \lVert x\rVert_{p,\infty}\le\lVert x\rVert_p,\qquad x\in L_p(\mathcal{M}). ∥ x ∥ p , ∞ ≤ ∥ x ∥ p , x ∈ L p ( M ) . We refer to [26 ] for further details on noncommutative L p L_p L p
Vector-valued noncommutative L p L_p L p
For 1 ≤ p ≤ ∞ 1\le p\le\infty 1 ≤ p ≤ ∞ L p ( M ; ℓ ∞ ( Λ ) ) L_p(\mathcal{M};\ell_\infty(\Lambda)) L p ( M ; ℓ ∞ ( Λ )) x = ( x λ ) λ ∈ Λ x=(x_\lambda)_{\lambda\in\Lambda} x = ( x λ ) λ ∈ Λ
x λ = a y λ b , λ ∈ Λ , x_\lambda=ay_\lambda b,\qquad\lambda\in\Lambda, x λ = a y λ b , λ ∈ Λ , where a , b ∈ L 2 p ( M ) a,b\in L_{2p}(\mathcal{M}) a , b ∈ L 2 p ( M ) λ \lambda λ ( y λ ) λ ∈ Λ (y_\lambda)_{\lambda\in\Lambda} ( y λ ) λ ∈ Λ M \mathcal{M} M Λ \Lambda Λ x x x L p ( M ; ℓ ∞ ( Λ ) ) L_p(\mathcal{M};\ell_\infty(\Lambda)) L p ( M ; ℓ ∞ ( Λ ))
∥ x ∥ L p ( M ; ℓ ∞ ( Λ ) ) = inf { ∥ a ∥ 2 p sup λ ∈ Λ ∥ y λ ∥ ∞ ∥ b ∥ 2 p } , \lVert x\rVert_{L_p(\mathcal{M};\ell_\infty(\Lambda))}
=\inf\left\{\lVert a\rVert_{2p}\sup_{\lambda\in\Lambda}\lVert y_\lambda\rVert_\infty\lVert b\rVert_{2p}\right\}, ∥ x ∥ L p ( M ; ℓ ∞ ( Λ )) = inf { ∥ a ∥ 2 p λ ∈ Λ sup ∥ y λ ∥ ∞ ∥ b ∥ 2 p } , where the infimum runs over all such factorizations. For p = ∞ p=\infty p = ∞
∥ x ∥ L ∞ ( M ; ℓ ∞ ( Λ ) ) = sup λ ∈ Λ ∥ x λ ∥ ∞ . \lVert x\rVert_{L_\infty(\mathcal{M};\ell_\infty(\Lambda))}
=\sup_{\lambda\in\Lambda}\lVert x_\lambda\rVert_\infty. ∥ x ∥ L ∞ ( M ; ℓ ∞ ( Λ )) = λ ∈ Λ sup ∥ x λ ∥ ∞ . We refer to [24 , 13 , 14 ] for further details.
For 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ Λ p , ∞ ( M ; ℓ ∞ ( Λ ) ) \Lambda_{p,\infty}(\mathcal{M};\ell_\infty(\Lambda)) Λ p , ∞ ( M ; ℓ ∞ ( Λ )) x = ( x λ ) λ ∈ Λ ⊂ L 0 ( M ) x=(x_\lambda)_{\lambda\in\Lambda}\subset L_0(\mathcal{M}) x = ( x λ ) λ ∈ Λ ⊂ L 0 ( M )
∥ x ∥ Λ p , ∞ ( M ; ℓ ∞ ( Λ ) ) : = sup t > 0 t inf { τ ( 1 − e ) 1 / p : e ∈ P ( M ) , sup λ ∈ Λ ∥ e x λ e ∥ ∞ ≤ t } < ∞ . \|x\|_{\Lambda_{p,\infty}(\mathcal{M};\ell_\infty(\Lambda))}:=\sup_{t>0}t\inf\left\{\tau(1-e)^{1/p}:e\in\mathcal{P}(\mathcal{M}),\ \sup_{\lambda\in\Lambda}\|ex_\lambda e\|_\infty\le t\right\}<\infty. ∥ x ∥ Λ p , ∞ ( M ; ℓ ∞ ( Λ )) := t > 0 sup t inf { τ ( 1 − e ) 1/ p : e ∈ P ( M ) , λ ∈ Λ sup ∥ e x λ e ∥ ∞ ≤ t } < ∞. where P ( M ) \mathcal{P}(\mathcal{M}) P ( M ) M \mathcal{M} M x x x x λ x_\lambda x λ sup λ ∈ Λ ∥ e x λ e ∥ ∞ ≤ t \sup_{\lambda\in\Lambda}\|ex_\lambda e\|_\infty\le t sup λ ∈ Λ ∥ e x λ e ∥ ∞ ≤ t
− t e ≤ e x λ e ≤ t e for every λ ∈ Λ . -te\le ex_\lambda e\le te\qquad\text{for every }\lambda\in\Lambda. − t e ≤ e x λ e ≤ t e for every λ ∈ Λ. We refer to [11 Section 2.2] for further details.
Definition 2.2. Let 1 ≤ p ≤ ∞ 1\le p\le\infty 1 ≤ p ≤ ∞ S = ( S λ ) λ ∈ Λ S=(S_\lambda)_{\lambda\in\Lambda} S = ( S λ ) λ ∈ Λ L p ( M ) L_p(\mathcal{M}) L p ( M ) L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) x ∈ L p ( M ) x\in L_p(\mathcal{M}) x ∈ L p ( M ) S x = ( S λ x ) λ ∈ Λ Sx=(S_\lambda x)_{\lambda\in\Lambda} S x = ( S λ x ) λ ∈ Λ S S S ( p , p ) (p,p) ( p , p ) C p C_p C p
∥ S x ∥ L p ( M ; ℓ ∞ ( Λ ) ) ≤ C p ∥ x ∥ p , x ∈ L p ( M ) . \|Sx\|_{L_p(\mathcal{M};\ell_\infty(\Lambda))}\le C_p\|x\|_p,\qquad x\in L_p(\mathcal{M}). ∥ S x ∥ L p ( M ; ℓ ∞ ( Λ )) ≤ C p ∥ x ∥ p , x ∈ L p ( M ) . When p < ∞ p<\infty p < ∞ S S S ( p , p ) (p,p) ( p , p ) C p C_p C p
∥ S x ∥ Λ p , ∞ ( M ; ℓ ∞ ( Λ ) ) ≤ C p ∥ x ∥ p , x ∈ L p ( M ) . \|Sx\|_{\Lambda_{p,\infty}(\mathcal{M};\ell_\infty(\Lambda))}\le C_p\|x\|_p,\qquad x\in L_p(\mathcal{M}). ∥ S x ∥ Λ p , ∞ ( M ; ℓ ∞ ( Λ )) ≤ C p ∥ x ∥ p , x ∈ L p ( M ) . The following consequence of the factorization defining L p ( M ; ℓ ∞ ) L_p(\mathcal{M};\ell_\infty) L p ( M ; ℓ ∞ )
Lemma 2.3. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ x = ( x n ) n ≥ 1 ∈ L p ( M ; ℓ ∞ ) x=(x_n)_{n\ge1}\in L_p(\mathcal{M};\ell_\infty) x = ( x n ) n ≥ 1 ∈ L p ( M ; ℓ ∞ ) t > 0 t>0 t > 0 e ∈ P ( M ) e\in\mathcal{P}(\mathcal{M}) e ∈ P ( M )
sup n ∥ e x n e ∥ ∞ ≤ t , τ ( 1 − e ) ≤ 2 p + 1 t p ∥ x ∥ L p ( M ; ℓ ∞ ) p . \sup_n\|ex_ne\|_\infty\le t,\qquad\tau(1-e)\le\frac{2^{p+1}}{t^p}\|x\|_{L_p(\mathcal{M};\ell_\infty)}^p. n sup ∥ e x n e ∥ ∞ ≤ t , τ ( 1 − e ) ≤ t p 2 p + 1 ∥ x ∥ L p ( M ; ℓ ∞ ) p . We recall the noncommutative Marcinkiewicz interpolation theorem of Junge and Xu.
Lemma 2.4 ([14 Theorem 3.1] ). Let 1 ≤ p 0 < p 1 ≤ ∞ 1\le p_0<p_1\le\infty 1 ≤ p 0 < p 1 ≤ ∞ S = ( S n ) n ≥ 1 S=(S_n)_{n\ge1} S = ( S n ) n ≥ 1 L p 0 ( M ) + + L p 1 ( M ) + L_{p_0}(\mathcal{M})_++L_{p_1}(\mathcal{M})_+ L p 0 ( M ) + + L p 1 ( M ) + L 0 ( M ) + L_0(\mathcal{M})_+ L 0 ( M ) +
S n ( x + y ) ≤ S n ( x ) + S n ( y ) ( n ≥ 1 ) . S_n(x+y)\le S_n(x)+S_n(y)\qquad(n\ge1). S n ( x + y ) ≤ S n ( x ) + S n ( y ) ( n ≥ 1 ) . Suppose that S S S ( p 0 , p 0 ) (p_0,p_0) ( p 0 , p 0 ) ( p 1 , p 1 ) (p_1,p_1) ( p 1 , p 1 ) B 0 B_0 B 0 B 1 B_1 B 1 p 0 < p < p 1 p_0<p<p_1 p 0 < p < p 1
∥ S x ∥ L p ( M ; ℓ ∞ ) ≤ C B 0 1 − θ B 1 θ ( 1 p 0 − 1 p ) − 2 ∥ x ∥ p , x ∈ L p ( M ) + . \|Sx\|_{L_p(\mathcal{M};\ell_\infty)}\le C B_0^{1-\theta}B_1^\theta\left(\frac{1}{p_0}-\frac{1}{p}\right)^{-2}\|x\|_p,\qquad x\in L_p(\mathcal{M})_+. ∥ S x ∥ L p ( M ; ℓ ∞ ) ≤ C B 0 1 − θ B 1 θ ( p 0 1 − p 1 ) − 2 ∥ x ∥ p , x ∈ L p ( M ) + . where
1 p = 1 − θ p 0 + θ p 1 , \frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}, p 1 = p 0 1 − θ + p 1 θ , and C C C
Bilateral almost uniform convergence
A sequence ( x n ) ⊂ L 0 ( M ) (x_n) \subset L_0(\mathcal{M}) ( x n ) ⊂ L 0 ( M ) x ∈ L 0 ( M ) x \in L_0(\mathcal{M}) x ∈ L 0 ( M ) ε > 0 \varepsilon> 0 ε > 0 e ∈ P ( M ) e \in\mathcal{P}(\mathcal{M}) e ∈ P ( M )
τ ( 1 − e ) < ε , ∥ e ( x n − x ) e ∥ ∞ ⟶ 0. \tau(1-e)<\varepsilon,\qquad\lVert e(x_n-x)e\rVert_\infty\longrightarrow0. τ ( 1 − e ) < ε , ∥ e ( x n − x ) e ∥ ∞ ⟶ 0. We write x n → x x_n \to x x n → x [14 Definition 6.1(i)] .
For operator-valued functions on R d \mathbb{R}^d R d
N = L ∞ ( R d ) ⊗ ‾ M , φ = ∫ R d d x ⊗ τ . \mathcal{N}=L_\infty(\mathbb{R}^d)\overline{\otimes}\mathcal{M},\qquad\varphi=\int_{\mathbb{R}^d}dx\otimes\tau. N = L ∞ ( R d ) ⊗ M , φ = ∫ R d d x ⊗ τ . Suppose that the restrictions of f n f_n f n f f f f n → f f_n \to f f n → f E ⊂ R d E\subset\mathbb{R}^d E ⊂ R d ε > 0 \varepsilon>0 ε > 0 e ≤ p E : = 1 E ⊗ 1 e\leq p_E:=1_E\otimes1 e ≤ p E := 1 E ⊗ 1
φ ( p E − e ) < ε , ∥ e ( f n − f ) e ∥ ∞ ⟶ 0. \varphi(p_E-e)<\varepsilon,\qquad\lVert e(f_n-f)e\rVert_\infty\longrightarrow0. φ ( p E − e ) < ε , ∥ e ( f n − f ) e ∥ ∞ ⟶ 0. In particular, this definition applies to functions in L 1 , l o c ( R d ; L 1 ( M ) ) L_{1,\mathrm{loc}}(\mathbb{R}^d;L_1(\mathcal{M})) L 1 , loc ( R d ; L 1 ( M ))
Convolution operators
For 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ L p ( N ) L_p(\mathcal{N}) L p ( N ) L p ( R d ; L p ( M ) ) L_p(\mathbb{R}^d;L_p(\mathcal{M})) L p ( R d ; L p ( M )) K ∈ L 1 ( R d ) K\in L_1(\mathbb{R}^d) K ∈ L 1 ( R d ) f ∈ L p ( N ) f\in L_p(\mathcal{N}) f ∈ L p ( N ) K K K f f f
K ∗ f = ∫ R d K ( y ) f ( ⋅ − y ) d y , K*f=\int_{\mathbb{R}^d}K(y)f(\mathord{\cdot}-y)\,dy, K ∗ f = ∫ R d K ( y ) f ( ⋅ − y ) d y , where the integral is a Bochner integral in L p ( N ) L_p(\mathcal{N}) L p ( N ) p = ∞ p=\infty p = ∞ K K K ∗ ^* ∗
∥ K ∗ f ∥ p ≤ ∥ K ∥ 1 ∥ f ∥ p , 1 ≤ p ≤ ∞ . (4) \lVert K*f\rVert_p\leq\lVert K\rVert_1\lVert f\rVert_p,\qquad1\leq p\leq\infty. \tag*{(4)} ∥ K ∗ f ∥ p ≤ ∥ K ∥ 1 ∥ f ∥ p , 1 ≤ p ≤ ∞. (4) More generally, let σ \sigma σ R d \mathbb{R}^d R d R d ∖ { 0 } \mathbb{R}^d\setminus\{0\} R d ∖ { 0 } K ∈ L 1 , l o c ( R d ∖ { 0 } ) K\in L_{1,\mathrm{loc}}(\mathbb{R}^d\setminus\{0\}) K ∈ L 1 , loc ( R d ∖ { 0 }) T T T
D = C c ∞ ( R d ) ⊗ S ( M ) \mathcal{D}=C^\infty_c(\mathbb{R}^d)\otimes\mathcal{S}(\mathcal{M}) D = C c ∞ ( R d ) ⊗ S ( M ) of L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) f = ∑ k = 1 m u k ⊗ a k ∈ D f=\sum_{k=1}^{m}u_k\otimes a_k\in\mathcal{D} f = ∑ k = 1 m u k ⊗ a k ∈ D
T f = ∑ k = 1 m ( σ ∗ u k ) ⊗ a k , ( σ ∗ u k ) ( x ) = ⟨ σ , u k ( x − ⋅ ) ⟩ . Tf=\sum_{k=1}^{m}(\sigma*u_k)\otimes a_k,\qquad(\sigma*u_k)(x)=\langle\sigma,u_k(x-\mathord{\cdot})\rangle. T f = k = 1 ∑ m ( σ ∗ u k ) ⊗ a k , ( σ ∗ u k ) ( x ) = ⟨ σ , u k ( x − ⋅ )⟩ . Consequently,
T f ( x ) = ∫ R d K ( x − y ) f ( y ) d y for almost every x ∉ supp R d f . (5) Tf(x)=\int_{\mathbb{R}^d}K(x-y)f(y)\,dy\quad\text{for almost every }x\notin\operatorname{supp}_{\mathbb{R}^d}f. \tag*{(5)} T f ( x ) = ∫ R d K ( x − y ) f ( y ) d y for almost every x ∈ / supp R d f . (5) Noncommutative Calderón–Zygmund decomposition
The first noncommutative Calderón–Zygmund decomposition was introduced by Parcet [21 ] . But the decomposition to be used in the present paper is closer to that in [2 Sections 1 and 2.1] .
For j ∈ Z j\in\mathbb{Z} j ∈ Z Q j \mathcal{Q}_j Q j 2 − j 2^{-j} 2 − j
Q j = { 2 − j ( m + [ 0 , 1 ) d ) : m ∈ Z d } , Q = ⋃ j ∈ Z Q j . \mathcal{Q}_j=\{2^{-j}(m+[0,1)^d):m\in\mathbb{Z}^d\},\qquad\mathcal{Q}=\bigcup_{j\in\mathbb{Z}}\mathcal{Q}_j. Q j = { 2 − j ( m + [ 0 , 1 ) d ) : m ∈ Z d } , Q = j ∈ Z ⋃ Q j . For an axis-parallel cube Q Q Q ℓ ( Q ) \ell(Q) ℓ ( Q ) c Q c_Q c Q c > 0 c>0 c > 0 c Q cQ c Q c ℓ ( Q ) c\ell(Q) c ℓ ( Q ) f f f Q Q Q
E Q f = 1 ∣ Q ∣ ∫ Q f ( y ) d y . \mathbb{E}_Qf=\frac{1}{|Q|}\int_Q f(y)\,\mathrm{d}y. E Q f = ∣ Q ∣ 1 ∫ Q f ( y ) d y . Let N j \mathcal{N}_j N j N \mathcal{N} N Q ∈ Q j Q\in\mathcal{Q}_j Q ∈ Q j
E j f = ∑ Q ∈ Q j 1 Q ⊗ E Q f . (6) \mathbb{E}_jf=\sum_{Q\in\mathcal{Q}_j}1_Q\otimes\mathbb{E}_Qf. \tag*{(6)} E j f = Q ∈ Q j ∑ 1 Q ⊗ E Q f . (6) These expectations are contractive on L 1 ( N ) L_1(\mathcal{N}) L 1 ( N )
E j f ≤ 2 d E j − 1 f , f ∈ L 1 ( N ) + . (7) \mathbb{E}_jf\leq2^d\mathbb{E}_{j-1}f,\qquad f\in L_1(\mathcal{N})_+. \tag*{(7)} E j f ≤ 2 d E j − 1 f , f ∈ L 1 ( N ) + . (7) Fix f ∈ L 1 ( N ) + f\in L_1(\mathcal{N})_+ f ∈ L 1 ( N ) + t > 0 t>0 t > 0 j 0 ∈ Z j_0\in\mathbb{Z} j 0 ∈ Z f j = E j f f_j=\mathbb{E}_jf f j = E j f [5 ] , as presented in [2 Section 1] , define
q j 0 − 1 = 1 , q j = 1 [ 0 , t ] ( q j − 1 f j q j − 1 ) , p j = q j − 1 − q j ( j ≥ j 0 ) , (8) q_{j_0-1}=1,\qquad q_j=1_{[0,t]}(q_{j-1}f_jq_{j-1}),\qquad p_j=q_{j-1}-q_j\quad(j\geq j_0), \tag*{(8)} q j 0 − 1 = 1 , q j = 1 [ 0 , t ] ( q j − 1 f j q j − 1 ) , p j = q j − 1 − q j ( j ≥ j 0 ) , (8) where the spectral projection is taken in q j − 1 N j q j − 1 q_{j-1}\mathcal{N}_jq_{j-1} q j − 1 N j q j − 1 q j ∈ N j q_j\in\mathcal{N}_j q j ∈ N j q j ≤ q j − 1 q_j\leq q_{j-1} q j ≤ q j − 1
q j f j q j ≤ t q j , p j f j q j = q j f j p j = 0 , p j f j p j ≥ t p j . (9) q_jf_jq_j\leq tq_j,\qquad p_jf_jq_j=q_jf_jp_j=0,\qquad p_jf_jp_j\geq tp_j. \tag*{(9)} q j f j q j ≤ t q j , p j f j q j = q j f j p j = 0 , p j f j p j ≥ t p j . (9) The projections p j p_j p j q j 1 + ∑ j = j 0 j 1 p j = 1 q_{j_1}+\sum_{j=j_0}^{j_1}p_j=1 q j 1 + ∑ j = j 0 j 1 p j = 1 j 1 ≥ j 0 j_1\geq j_0 j 1 ≥ j 0
t ∑ j ≥ j 0 φ ( p j ) ≤ ∥ f ∥ 1 . (10) t\sum_{j\geq j_0}\varphi(p_j)\leq\lVert f\rVert_1. \tag*{(10)} t j ≥ j 0 ∑ φ ( p j ) ≤ ∥ f ∥ 1 . (10) For each j 1 ≥ j 0 j_1\geq j_0 j 1 ≥ j 0
f j 1 = g + b d + b o f f , (11) f_{j_1}=g+b_{\mathrm{d}}+b_{\mathrm{off}}, \tag*{(11)} f j 1 = g + b d + b off , (11) where
g = q j 1 f j 1 q j 1 + ∑ j = j 0 j 1 p j f j p j , b d = ∑ j = j 0 j 1 p j ( f j 1 − f j ) p j , g=q_{j_1}f_{j_1}q_{j_1}+\sum_{j=j_0}^{j_1}p_jf_jp_j,\qquad b_{\mathrm{d}}=\sum_{j=j_0}^{j_1}p_j(f_{j_1}-f_j)p_j, g = q j 1 f j 1 q j 1 + j = j 0 ∑ j 1 p j f j p j , b d = j = j 0 ∑ j 1 p j ( f j 1 − f j ) p j , and
b o f f = ∑ j = j 0 j 1 [ p j ( f j 1 − f j ) q j + q j ( f j 1 − f j ) p j ] . b_{\mathrm{off}}=\sum_{j=j_0}^{j_1}\left[p_j(f_{j_1}-f_j)q_j+q_j(f_{j_1}-f_j)p_j\right]. b off = j = j 0 ∑ j 1 [ p j ( f j 1 − f j ) q j + q j ( f j 1 − f j ) p j ] . The following properties can be found in [2 Equation (1.1) and Lemma 1.1] .
Lemma 2.5. The terms in (11) satisfy
g ≥ 0 , ∥ g ∥ 1 = ∥ f ∥ 1 , ∑ j = j 0 j 1 ∥ p j ( f j 1 − f j ) p j ∥ 1 ≤ 2 ∥ f ∥ 1 . (12) g\geq0,\qquad\lVert g\rVert_1=\lVert f\rVert_1,\qquad\sum_{j=j_0}^{j_1}\lVert p_j(f_{j_1}-f_j)p_j\rVert_1\leq2\lVert f\rVert_1. \tag*{(12)} g ≥ 0 , ∥ g ∥ 1 = ∥ f ∥ 1 , j = j 0 ∑ j 1 ∥ p j ( f j 1 − f j ) p j ∥ 1 ≤ 2 ∥ f ∥ 1 . (12) Moreover, for j 0 ≤ j ≤ j 1 j_0\leq j\leq j_1 j 0 ≤ j ≤ j 1
E j ( p j ( f j 1 − f j ) p j ) = 0 , \mathbb{E}_j\bigl(p_j(f_{j_1}-f_j)p_j\bigr)=0, E j ( p j ( f j 1 − f j ) p j ) = 0 , E j ( p j ( f j 1 − f j ) q j + q j ( f j 1 − f j ) p j ) = 0. \mathbb{E}_j\bigl(p_j(f_{j_1}-f_j)q_j+q_j(f_{j_1}-f_j)p_j\bigr)=0. E j ( p j ( f j 1 − f j ) q j + q j ( f j 1 − f j ) p j ) = 0. If, in addition, f j 0 − 1 ≤ t 1 f_{j_0-1}\leq t1 f j 0 − 1 ≤ t 1
p j f j p j ≤ 2 d t p j , ∥ g ∥ ∞ ≤ 2 d t . (13) p_jf_jp_j\leq2^dtp_j,\qquad\lVert g\rVert_\infty\leq2^dt. \tag*{(13)} p j f j p j ≤ 2 d t p j , ∥ g ∥ ∞ ≤ 2 d t . (13) Since p j ∈ N j p_j \in\mathcal{N}_j p j ∈ N j p Q ∈ P ( M ) p_Q \in\mathcal{P}(\mathcal{M}) p Q ∈ P ( M )
p j = ∑ Q ∈ Q j 1 Q ⊗ p Q . p_j = \sum_{Q \in\mathcal{Q}_j} 1_Q \otimes p_Q. p j = Q ∈ Q j ∑ 1 Q ⊗ p Q . For β ≥ 1 \beta\ge1 β ≥ 1
ζ = 1 − ⋁ j ≥ j 0 ⋁ Q ∈ Q j ( 1 β Q ⊗ p Q ) . (14) \zeta= 1 - \bigvee_{j \ge j_0} \bigvee_{Q \in\mathcal{Q}_j} (1_{\beta Q} \otimes p_Q). \tag*{(14)} ζ = 1 − j ≥ j 0 ⋁ Q ∈ Q j ⋁ ( 1 β Q ⊗ p Q ) . (14) The following properties are given in [1 Lemma 3.4] for odd positive integer dilation factors. The same proof applies to every β ≥ 1 \beta\ge1 β ≥ 1
Lemma 2.6. The projection ζ \zeta ζ
φ ( 1 − ζ ) ≤ β d t ∥ f ∥ 1 . \varphi(1-\zeta) \le\frac{\beta^d}{t}\|f\|_1. φ ( 1 − ζ ) ≤ t β d ∥ f ∥ 1 . Moreover, for every Q ∈ Q j Q \in\mathcal{Q}_j Q ∈ Q j j ≥ j 0 j \ge j_0 j ≥ j 0
ζ ( x ) p Q = p Q ζ ( x ) = 0 for almost every x ∈ β Q . \zeta(x)p_Q = p_Q\zeta(x) = 0 \quad\text{for almost every } x \in\beta Q. ζ ( x ) p Q = p Q ζ ( x ) = 0 for almost every x ∈ β Q . Auxiliary lemmas and transference
We collect three lemmas used in the maximal estimates below, and then recall a transference principle for positive convolution operators.
Positive operator matrices and compression estimates
We first prove a factorization for positive operator matrices. The following factorization is classical for bounded operators; see [18 Theorem 5.11] . We include a proof that also applies to measurable operators.
Lemma 2.7. Let 1 ≤ p ≤ ∞ 1 \le p \le\infty 1 ≤ p ≤ ∞ u , v ∈ L p ( M ) + u, v \in L_p(\mathcal{M})_+ u , v ∈ L p ( M ) + x ∈ L p ( M ) x \in L_p(\mathcal{M}) x ∈ L p ( M )
( u x x ∗ v ) ≥ 0. \begin{pmatrix} u & x \\ x^* & v \end{pmatrix} \ge0. ( u x ∗ x v ) ≥ 0. Then there is a contraction z ∈ M z \in\mathcal{M} z ∈ M x = u 1 / 2 z v 1 / 2 x = u^{1/2}zv^{1/2} x = u 1/2 z v 1/2 u , v ≤ t e u, v \le te u , v ≤ t e t ≥ 0 t \ge0 t ≥ 0 e ∈ P ( M ) e \in\mathcal{P}(\mathcal{M}) e ∈ P ( M ) ∥ x ∥ ∞ ≤ t \|x\|_\infty\le t ∥ x ∥ ∞ ≤ t
Proof. Write
R = ( u x x ∗ v ) 1 / 2 = ( R 1 R 2 ) , R = \begin{pmatrix} u & x \\ x^* & v \end{pmatrix}^{1/2} = (R_1\ R_2), R = ( u x ∗ x v ) 1/2 = ( R 1 R 2 ) , where R 1 R_1 R 1 R 2 R_2 R 2 R R R
R 1 ∗ R 1 = u , R 2 ∗ R 2 = v , R 1 ∗ R 2 = x . R_1^*R_1 = u,\qquad R_2^*R_2 = v,\qquad R_1^*R_2 = x. R 1 ∗ R 1 = u , R 2 ∗ R 2 = v , R 1 ∗ R 2 = x . By polar decomposition,
R 1 = w 1 u 1 / 2 , R 2 = w 2 v 1 / 2 , R_1 = w_1u^{1/2},\qquad R_2 = w_2v^{1/2}, R 1 = w 1 u 1/2 , R 2 = w 2 v 1/2 , where w 1 , w 2 ∈ M 2 , 1 ( M ) w_1,w_2 \in M_{2,1}(\mathcal{M}) w 1 , w 2 ∈ M 2 , 1 ( M ) M 2 , 1 ( M ) M_{2,1}(\mathcal{M}) M 2 , 1 ( M ) 2 × 1 2 \times1 2 × 1 M \mathcal{M} M H H H H ⊕ H H \oplus H H ⊕ H M \mathcal{M} M H H H
Set z = w 1 ∗ w 2 z = w_1^*w_2 z = w 1 ∗ w 2
z ∗ z = w 2 ∗ w 1 w 1 ∗ w 2 ≤ w 2 ∗ w 2 ≤ 1 , z^*z = w_2^*w_1w_1^*w_2 \le w_2^*w_2 \le1, z ∗ z = w 2 ∗ w 1 w 1 ∗ w 2 ≤ w 2 ∗ w 2 ≤ 1 , we have ∥ z ∥ ∞ ≤ 1 \|z\|_\infty\le1 ∥ z ∥ ∞ ≤ 1
x = R 1 ∗ R 2 = u 1 / 2 z v 1 / 2 . x = R_1^*R_2 = u^{1/2}zv^{1/2}. x = R 1 ∗ R 2 = u 1/2 z v 1/2 . If u , v ≤ t e u,v \le te u , v ≤ t e
∥ x ∥ ∞ ≤ ∥ u 1 / 2 ∥ ∞ ∥ z ∥ ∞ ∥ v 1 / 2 ∥ ∞ ≤ t . \|x\|_\infty\le\|u^{1/2}\|_\infty\|z\|_\infty\|v^{1/2}\|_\infty\le t. ∥ x ∥ ∞ ≤ ∥ u 1/2 ∥ ∞ ∥ z ∥ ∞ ∥ v 1/2 ∥ ∞ ≤ t . Lemma 2.8. Let t ≥ 0 t \ge0 t ≥ 0 x j ∈ L 1 ( M ) ∩ M x_j \in L_1(\mathcal{M}) \cap\mathcal{M} x j ∈ L 1 ( M ) ∩ M x j → x x_j \to x x j → x L 1 ( M ) L_1(\mathcal{M}) L 1 ( M ) ∥ x j ∥ ∞ ≤ t \lVert x_j\rVert_\infty\le t ∥ x j ∥ ∞ ≤ t x ∈ M x \in\mathcal{M} x ∈ M ∥ x ∥ ∞ ≤ t \lVert x\rVert_\infty\le t ∥ x ∥ ∞ ≤ t x j ≥ 0 x_j \ge0 x j ≥ 0 j j j x ≥ 0 x \ge0 x ≥ 0
Proof. Let x = w ∣ x ∣ x = w|x| x = w ∣ x ∣ x x x ε > 0 \varepsilon> 0 ε > 0 q = 1 ( t + ε , ∞ ) ( ∣ x ∣ ) q = \mathbf{1}_{(t+\varepsilon,\infty)}(|x|) q = 1 ( t + ε , ∞ ) ( ∣ x ∣ ) x ∈ L 1 ( M ) x \in L_1(\mathcal{M}) x ∈ L 1 ( M )
τ ( q ) ≤ ∥ x ∥ 1 t + ε < ∞ . \tau(q) \le\frac{\lVert x\rVert_1}{t+\varepsilon} < \infty. τ ( q ) ≤ t + ε ∥ x ∥ 1 < ∞. Moreover, ∥ q w ∗ ∥ ∞ ≤ 1 \lVert qw^*\rVert_\infty\le1 ∥ q w ∗ ∥ ∞ ≤ 1 ∥ q w ∗ ∥ 1 ≤ τ ( q ) \lVert qw^*\rVert_1 \le\tau(q) ∥ q w ∗ ∥ 1 ≤ τ ( q )
( t + ε ) τ ( q ) ≤ τ ( q ∣ x ∣ ) ≤ ∣ τ ( q w ∗ x j ) ∣ + ∥ x − x j ∥ 1 ≤ t τ ( q ) + ∥ x − x j ∥ 1 . (t+\varepsilon)\tau(q) \le\tau(q|x|) \le|\tau(qw^*x_j)| + \lVert x-x_j\rVert_1 \le t\tau(q) + \lVert x-x_j\rVert_1. ( t + ε ) τ ( q ) ≤ τ ( q ∣ x ∣ ) ≤ ∣ τ ( q w ∗ x j ) ∣ + ∥ x − x j ∥ 1 ≤ t τ ( q ) + ∥ x − x j ∥ 1 . Letting j → ∞ j \to\infty j → ∞ τ ( q ) = 0 \tau(q) = 0 τ ( q ) = 0 q = 0 q = 0 q = 0 ε > 0 \varepsilon> 0 ε > 0 ∣ x ∣ ≤ t 1 |x| \le t1 ∣ x ∣ ≤ t 1 L 1 ( M ) + L_1(\mathcal{M})_+ L 1 ( M ) +
The next lemma follows by adapting the compactness argument in the proof of [12 Lemma 3.2] .
Lemma 2.9. Let ( x n ) n ≥ 1 ⊂ L 1 ( M ) + (x_n)_{n\ge1} \subset L_1(\mathcal{M})_+ ( x n ) n ≥ 1 ⊂ L 1 ( M ) + e N ∈ M e_N \in\mathcal{M} e N ∈ M t N → t > 0 t_N \to t > 0 t N → t > 0 D N → D < ∞ D_N \to D < \infty D N → D < ∞
e N x n e N ≤ t N e N ( 1 ≤ n ≤ N ) , τ ( 1 − e N ) ≤ D N . e_Nx_ne_N \le t_Ne_N \quad(1 \le n \le N), \qquad\tau(1-e_N) \le D_N. e N x n e N ≤ t N e N ( 1 ≤ n ≤ N ) , τ ( 1 − e N ) ≤ D N . Then there is a projection e ∈ M e \in\mathcal{M} e ∈ M
e x n e ≤ 2 t e ( n ≥ 1 ) , τ ( 1 − e ) ≤ 2 D . ex_ne \le2te \quad(n \ge1), \qquad\tau(1-e) \le2D. e x n e ≤ 2 t e ( n ≥ 1 ) , τ ( 1 − e ) ≤ 2 D . Proof. By the Banach–Alaoglu theorem, a subnet ( e N α ) (e_{N_\alpha}) ( e N α ) a ∈ M a \in\mathcal{M} a ∈ M 0 ≤ a ≤ 1 0 \le a \le1 0 ≤ a ≤ 1 τ \tau τ
τ ( 1 − a ) ≤ lim inf α τ ( 1 − e N α ) ≤ D . \tau(1-a) \le\liminf_{\alpha} \tau(1-e_{N_\alpha}) \le D. τ ( 1 − a ) ≤ α lim inf τ ( 1 − e N α ) ≤ D . Set e = 1 [ 1 / 2 , 1 ] ( a ) e = \mathbf{1}_{[1/2,1]}(a) e = 1 [ 1/2 , 1 ] ( a ) τ ( 1 − e ) ≤ τ ( 1 − 2 a ) ≤ 2 D \tau(1-e) \le\tau(1-2a) \le2D τ ( 1 − e ) ≤ τ ( 1 − 2 a ) ≤ 2 D
Fix n ≥ 1 n \ge1 n ≥ 1 x n , k = f k ( x n ) x_{n,k} = f_k(x_n) x n , k = f k ( x n ) f k ( s ) = min { s , k } f_k(s) = \min\{s,k\} f k ( s ) = min { s , k } s ≥ 0 s \ge0 s ≥ 0 0 ≤ x n , k ≤ x n 0 \le x_{n,k} \le x_n 0 ≤ x n , k ≤ x n
x n , k ∈ L 1 ( M ) ∩ M , ∥ x n , k − x n ∥ 1 ⟶ 0. x_{n,k} \in L_1(\mathcal{M}) \cap\mathcal{M}, \qquad\lVert x_{n,k} - x_n\rVert_1 \longrightarrow0. x n , k ∈ L 1 ( M ) ∩ M , ∥ x n , k − x n ∥ 1 ⟶ 0. Moreover, for fixed k k k α \alpha α
e N α x n , k e N α ≤ t N α e N α . e_{N_\alpha}x_{n,k}e_{N_\alpha} \le t_{N_\alpha}e_{N_\alpha}. e N α x n , k e N α ≤ t N α e N α . Note that for b ∈ M + b \in\mathcal{M}_+ b ∈ M + p ∈ P ( M ) p \in\mathcal{P}(\mathcal{M}) p ∈ P ( M ) c ≥ 0 c \ge0 c ≥ 0 (15)
p b p ≤ c p ⟺ b 1 / 2 p b 1 / 2 ≤ c 1 , (15) pbp \le cp \Longleftrightarrow b^{1/2}pb^{1/2} \le c1, \tag*{(15)} p b p ≤ c p ⟺ b 1/2 p b 1/2 ≤ c 1 , (15) since both inequalities are equivalent to ∥ b 1 / 2 p ∥ ∞ 2 ≤ c \lVert b^{1/2}p\rVert_\infty^2 \le c ∥ b 1/2 p ∥ ∞ 2 ≤ c
x n , k 1 / 2 e N α x n , k 1 / 2 ≤ t N α 1. x_{n,k}^{1/2}e_{N_\alpha}x_{n,k}^{1/2} \le t_{N_\alpha}1. x n , k 1/2 e N α x n , k 1/2 ≤ t N α 1. Since x n , k 1 / 2 x_{n,k}^{1/2} x n , k 1/2 e ≤ 2 a e \le2a e ≤ 2 a
x n , k 1 / 2 e x n , k 1 / 2 ≤ 2 x n , k 1 / 2 a x n , k 1 / 2 ≤ 2 t 1. x_{n,k}^{1/2}ex_{n,k}^{1/2} \le2x_{n,k}^{1/2}ax_{n,k}^{1/2} \le2t1. x n , k 1/2 e x n , k 1/2 ≤ 2 x n , k 1/2 a x n , k 1/2 ≤ 2 t 1. Another application of (15) gives e x n , k e ≤ 2 t e ex_{n,k}e \le2te e x n , k e ≤ 2 t e
Finally, note that
∥ e x n , k e − e x n e ∥ 1 ≤ ∥ x n , k − x n ∥ 1 ⟶ 0 , \lVert ex_{n,k}e-ex_ne\rVert_1 \le\lVert x_{n,k}-x_n\rVert_1 \longrightarrow0, ∥ e x n , k e − e x n e ∥ 1 ≤ ∥ x n , k − x n ∥ 1 ⟶ 0 , we have ∥ e x n e ∥ ∞ ≤ 2 t \lVert ex_ne\rVert_\infty\le2t ∥ e x n e ∥ ∞ ≤ 2 t Lemma 2.8 . Since e x n e ex_ne e x n e e e e e x n e ≤ 2 t e ex_ne \le2te e x n e ≤ 2 t e e e e n n n
Transference for positive convolution operators
We now recall a transference principle that transfers maximal inequalities for positive convolution operators on R d \mathbb{R}^{d} R d [10 Theorems 3.1 and 3.3] . Although those theorems are stated for probability measures, their proofs apply equally to finite positive measures K n ( y ) d y K_n(y)\,\mathrm{d}y K n ( y ) d y ∫ R d K n = 1 \int_{\mathbb{R}^{d}} K_n = 1 ∫ R d K n = 1
Let α = ( α y ) y ∈ R d \alpha=(\alpha_y)_{y\in\mathbb{R}^{d}} α = ( α y ) y ∈ R d ∗ * ∗ M \mathcal{M} M α 0 = id M \alpha_0=\operatorname{id}_{\mathcal{M}} α 0 = id M α y + z = α y ∘ α z \alpha_{y+z}=\alpha_y\circ\alpha_z α y + z = α y ∘ α z y , z ∈ R d y,z\in\mathbb{R}^{d} y , z ∈ R d τ ( α y ( x ) ) = τ ( x ) \tau(\alpha_y(x))=\tau(x) τ ( α y ( x )) = τ ( x ) x ∈ M + x\in\mathcal{M}_{+} x ∈ M + y ∈ R d y\in\mathbb{R}^{d} y ∈ R d ( K n ) n ≥ 1 ⊂ L 1 ( R d ) + (K_n)_{n\geq1}\subset L_1(\mathbb{R}^{d})_{+} ( K n ) n ≥ 1 ⊂ L 1 ( R d ) +
T n F ( s ) = ∫ R d K n ( y ) F ( s + y ) d y , T n α x = ∫ R d K n ( y ) α y ( x ) d y . T_nF(s)=\int_{\mathbb{R}^{d}}K_n(y)F(s+y)\,\mathrm{d}y,\qquad T_n^\alpha x=\int_{\mathbb{R}^{d}}K_n(y)\alpha_y(x)\,\mathrm{d}y. T n F ( s ) = ∫ R d K n ( y ) F ( s + y ) d y , T n α x = ∫ R d K n ( y ) α y ( x ) d y . The following statement concerns these families on the positive cones of L p ( N ) L_p(\mathcal{N}) L p ( N ) L p ( M ) L_p(\mathcal{M}) L p ( M )
Lemma 2.10. For 1 < p < ∞ 1<p<\infty 1 < p < ∞ ( p , p ) (p,p) ( p , p ) ( T n ) (T_n) ( T n ) C p C_p C p ( T n α ) (T_n^\alpha) ( T n α )
Similarly, a weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( T n ) (T_n) ( T n ) C 1 C_1 C 1 ( 1 , 1 ) (1,1) ( 1 , 1 ) ( T n α ) (T_n^\alpha) ( T n α ) C C 1 CC_1 C C 1 C C C
A weak type (1, 1) estimate for dyadic series
In this section, we prove a weak type ( 1 , 1 ) (1,1) ( 1 , 1 )
The weak type (1, 1) estimate
Let D 0 \mathcal{D}_0 D 0 f ∈ L 1 ( N ) ∩ L ∞ ( N ) f\in L_1(\mathcal{N})\cap L_\infty(\mathcal{N}) f ∈ L 1 ( N ) ∩ L ∞ ( N ) β ≥ 3 \beta\geq3 β ≥ 3 ( S n ) n ≥ 1 (\mathcal{S}_n)_{n\geq1} ( S n ) n ≥ 1
S n f = ∑ Q ∈ Q b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q , f ∈ D 0 , \mathcal{S}_n f=\sum_{Q\in\mathcal{Q}}b_{n,Q}E_Qf,\qquad\lvert b_{n,Q}\rvert\leq m_Q,\qquad f\in\mathcal{D}_0, S n f = Q ∈ Q ∑ b n , Q E Q f , ∣ b n , Q ∣ ≤ m Q , f ∈ D 0 , with absolute convergence in local L 1 L_1 L 1 b n , Q b_{n,Q} b n , Q m Q m_Q m Q n n n
Γ = sup P ∈ Q ess sup y ∈ P ∑ Q ⊂ P y ∈ Q 1 ∣ Q ∣ ∫ ( β P ) c m Q ( x ) d x . (16) \Gamma=\sup_{P\in\mathcal{Q}}\operatorname*{ess\,sup}_{y\in P}\sum_{\substack{Q\subset P\\y\in Q}}\frac{1}{\lvert Q\rvert}\int_{(\beta P)^c}m_Q(x)\,\mathrm{d}x. \tag*{(16)} Γ = P ∈ Q sup y ∈ P ess sup Q ⊂ P y ∈ Q ∑ ∣ Q ∣ 1 ∫ ( β P ) c m Q ( x ) d x . (16) Theorem 3.1. Suppose that Γ < ∞ \Gamma<\infty Γ < ∞ 1 < r ≤ ∞ 1<r\leq\infty 1 < r ≤ ∞
∥ ( S n f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . \left\|(\mathcal{S}_n f)_n\right\|_{L_r(\mathcal{N};\ell_\infty)}\leq M_r\lVert f\rVert_r,\qquad f\in L_r(\mathcal{N}). ∥ ( S n f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . Then, for every f ∈ D 0 f\in\mathcal{D}_0 f ∈ D 0 λ > 0 \lambda>0 λ > 0 e ∈ N e\in\mathcal{N} e ∈ N
sup n ∥ e S n ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ C d , β , r ( M r + Γ ) ∥ f ∥ 1 . \sup_n\lVert e\mathcal{S}_n(f)e\rVert_\infty\leq\lambda,\qquad\lambda\varphi(1-e)\leq C_{d,\beta,r}(M_r+\Gamma)\lVert f\rVert_1. n sup ∥ e S n ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ C d , β , r ( M r + Γ ) ∥ f ∥ 1 . The theorem follows from the following decomposition proposition, whose proof is given in the next subsection. As in the earlier noncommutative Calderón–Zygmund decomposition recalled in Lemma 2.5 , we use Cuculescu’s projections to construct a good function g g g b d b_{\mathrm{d}} b d b o f f b_{\mathrm{off}} b off S n \mathcal{S}_n S n (16) allows us to control these sums on one projection w w w n n n L 1 L_1 L 1
Proposition 3.2. Suppose that Γ < ∞ \Gamma<\infty Γ < ∞ For every f ∈ D 0 f\in\mathcal{D}_0 f ∈ D 0 with f ≥ 0 f\ge0 f ≥ 0 and every t , λ > 0 t,\lambda>0 t , λ > 0 there exist g ∈ D 0 g\in\mathcal{D}_0 g ∈ D 0 a projection w ∈ N w\in\mathcal{N} w ∈ N and operators B n , B ℓ , n , B r , n ∈ w N w B_n,B_{\ell,n},B_{r,n}\in w\mathcal{N}w B n , B ℓ , n , B r , n ∈ w N w such that
w S n ( f ) w = w S n ( g ) w + B n + B ℓ , n + B r , n , n ≥ 1. (17) w\mathcal{S}_n(f)w=w\mathcal{S}_n(g)w+B_n+B_{\ell,n}+B_{r,n},\qquad n\ge1. \tag*{(17)} w S n ( f ) w = w S n ( g ) w + B n + B ℓ , n + B r , n , n ≥ 1. (17) Moreover,
0 ≤ g ≤ 2 d t 1 , ∥ g ∥ 1 = ∥ f ∥ 1 , (18) 0\le g\le2^d t1,\qquad\lVert g\rVert_1=\lVert f\rVert_1, \tag*{(18)} 0 ≤ g ≤ 2 d t 1 , ∥ g ∥ 1 = ∥ f ∥ 1 , (18) sup n ≥ 1 max { ∥ B n ∥ ∞ , ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ } ≤ λ / 4 , (19) \sup_{n\ge1}\max\left\{\lVert B_n\rVert_\infty,\lVert B_{\ell,n}\rVert_\infty,\lVert B_{r,n}\rVert_\infty\right\}\le\lambda/4, \tag*{(19)} n ≥ 1 sup max { ∥ B n ∥ ∞ , ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ } ≤ λ /4 , (19) and
φ ( 1 − w ) ≤ C d , β ( 1 t + Γ λ + t Γ 2 λ 2 ) ∥ f ∥ 1 . (20) \varphi(1-w)\le C_{d,\beta}\left(\frac{1}{t}+\frac{\Gamma}{\lambda}+\frac{t\Gamma^2}{\lambda^2}\right)\lVert f\rVert_1. \tag*{(20)} φ ( 1 − w ) ≤ C d , β ( t 1 + λ Γ + λ 2 t Γ 2 ) ∥ f ∥ 1 . (20) Proof of Theorem 3.1. We first assume that f ≥ 0 f\ge0 f ≥ 0 M r + Γ = 0 M_r+\Gamma=0 M r + Γ = 0 t , λ > 0 t,\lambda>0 t , λ > 0
Suppose first that 1 < r < ∞ 1<r<\infty 1 < r < ∞ (18) , we have
∥ g ∥ r r ≤ ∥ g ∥ ∞ r − 1 ∥ g ∥ 1 ≤ ( 2 d t ) r − 1 ∥ f ∥ 1 . \lVert g\rVert_r^r\le\lVert g\rVert_\infty^{r-1}\lVert g\rVert_1\le(2^d t)^{r-1}\lVert f\rVert_1. ∥ g ∥ r r ≤ ∥ g ∥ ∞ r − 1 ∥ g ∥ 1 ≤ ( 2 d t ) r − 1 ∥ f ∥ 1 . Hence
∥ ( S n g ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r r ( 2 d t ) r − 1 ∥ f ∥ 1 . \lVert(\mathcal{S}_n g)_n\rVert_{L_r(\mathcal{N};\ell_\infty)}\le M_r^r(2^d t)^{r-1}\lVert f\rVert_1. ∥( S n g ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r r ( 2 d t ) r − 1 ∥ f ∥ 1 . Lemma 2.3 gives a projection e g e_g e g
sup n ∥ e g S n ( g ) e g ∥ ∞ ≤ λ / 4 , φ ( 1 − e g ) ≤ C d , r M r r t r − 1 λ − r ∥ f ∥ 1 . \sup_n\lVert e_g\mathcal{S}_n(g)e_g\rVert_\infty\le\lambda/4,\qquad\varphi(1-e_g)\le C_{d,r}M_r^r t^{r-1}\lambda^{-r}\lVert f\rVert_1. n sup ∥ e g S n ( g ) e g ∥ ∞ ≤ λ /4 , φ ( 1 − e g ) ≤ C d , r M r r t r − 1 λ − r ∥ f ∥ 1 . Set e = w ∧ e g e=w\wedge e_g e = w ∧ e g (17) and (19) , we get
sup n ∥ e S n ( f ) e ∥ ∞ ≤ λ . \sup_n\lVert e\mathcal{S}_n(f)e\rVert_\infty\le\lambda. n sup ∥ e S n ( f ) e ∥ ∞ ≤ λ . Moreover, we obtain
λ φ ( 1 − e ) ≤ λ φ ( 1 − w ) + λ φ ( 1 − e g ) ≤ C d , β , r [ λ t + Γ + t Γ 2 λ + M r r ( t λ ) r − 1 ] ∥ f ∥ 1 . \begin{aligned}
\lambda\varphi(1-e)\le\lambda\varphi(1-w)+\lambda\varphi(1-e_g) \\
&\le C_{d,\beta,r}\left[\frac{\lambda}{t}+\Gamma+\frac{t\Gamma^2}{\lambda}+M_r^r\left(\frac{t}{\lambda}\right)^{r-1}\right]\lVert f\rVert_1.
\end{aligned} λ φ ( 1 − e ) ≤ λ φ ( 1 − w ) + λ φ ( 1 − e g ) ≤ C d , β , r [ t λ + Γ + λ t Γ 2 + M r r ( λ t ) r − 1 ] ∥ f ∥ 1 . Taking t = λ / ( M r + Γ ) t=\lambda/(M_r+\Gamma) t = λ / ( M r + Γ )
λ φ ( 1 − e ) ≤ C d , β , r ( M r + Γ ) ∥ f ∥ 1 . \lambda\varphi(1-e)\le C_{d,\beta,r}(M_r+\Gamma)\lVert f\rVert_1. λ φ ( 1 − e ) ≤ C d , β , r ( M r + Γ ) ∥ f ∥ 1 . For r = ∞ r=\infty r = ∞
t = λ 2 d + 2 ( M ∞ + Γ ) . t=\frac{\lambda}{2^{d+2}(M_\infty+\Gamma)}. t = 2 d + 2 ( M ∞ + Γ ) λ . Then, we have
sup n ∥ S n g ∥ ∞ ≤ M ∞ ∥ g ∥ ∞ ≤ 2 d t M ∞ ≤ λ / 4. \sup_n\lVert\mathcal{S}_n g\rVert_\infty\le M_\infty\lVert g\rVert_\infty\le2^d tM_\infty\le\lambda/4. n sup ∥ S n g ∥ ∞ ≤ M ∞ ∥ g ∥ ∞ ≤ 2 d t M ∞ ≤ λ /4. Thus by (17) and (19) ,
sup n ∥ w S n ( f ) w ∥ ∞ ≤ λ , \sup_n\lVert w\mathcal{S}_n(f)w\rVert_\infty\le\lambda, n sup ∥ w S n ( f ) w ∥ ∞ ≤ λ , and (20) yields
λ φ ( 1 − w ) ≤ C d , β ( M ∞ + Γ ) ∥ f ∥ 1 . \lambda\varphi(1-w)\le C_{d,\beta}(M_\infty+\Gamma)\lVert f\rVert_1. λ φ ( 1 − w ) ≤ C d , β ( M ∞ + Γ ) ∥ f ∥ 1 . For arbitrary f ∈ D 0 f \in\mathcal{D}_0 f ∈ D 0
f = ( Re f ) + − ( Re f ) − + i ( Im f ) + − i ( Im f ) − . f = (\operatorname{Re} f)_+ - (\operatorname{Re} f)_- + i(\operatorname{Im} f)_+ - i(\operatorname{Im} f)_-. f = ( Re f ) + − ( Re f ) − + i ( Im f ) + − i ( Im f ) − . The four positive parts belong to D 0 \mathcal{D}_0 D 0 L 1 L_1 L 1 2 ∥ f ∥ 1 2\lVert f\rVert_1 2 ∥ f ∥ 1 λ / 4 \lambda/4 λ /4
Proof of the decomposition
Before proving Proposition 3.2, we record two operator inequalities that will be used to estimate the mixed terms. For x ∈ M + x \in\mathcal{M}_+ x ∈ M + q ∈ P ( M ) q \in\mathcal{P}(\mathcal{M}) q ∈ P ( M ) t ≥ 0 t \ge0 t ≥ 0
q x q ≤ t q ⟹ x q x ≤ t x . (21) qxq \le tq \quad\Longrightarrow\quad xqx \le tx. \tag*{(21)} q x q ≤ tq ⟹ x q x ≤ t x . (21) Indeed, ∥ x 1 / 2 q x 1 / 2 ∥ ∞ = ∥ q x q ∥ ∞ ≤ t \lVert x^{1/2}qx^{1/2}\rVert_\infty= \lVert qxq\rVert_\infty\le t ∥ x 1/2 q x 1/2 ∥ ∞ = ∥ q x q ∥ ∞ ≤ t
x q x = x 1 / 2 ( x 1 / 2 q x 1 / 2 ) x 1 / 2 ≤ t x . xqx = x^{1/2}(x^{1/2}qx^{1/2})x^{1/2} \le tx. x q x = x 1/2 ( x 1/2 q x 1/2 ) x 1/2 ≤ t x . Lemma 3.3. Let ( a k ) k ≥ 1 ⊂ M (a_k)_{k\ge1} \subset\mathcal{M} ( a k ) k ≥ 1 ⊂ M ( b k ) k ≥ 1 ⊂ L 2 ( M ) (b_k)_{k\ge1} \subset L_2(\mathcal{M}) ( b k ) k ≥ 1 ⊂ L 2 ( M )
∑ k = 1 N a k a k ∗ ≤ C 1 , ∑ k ≥ 1 ∥ b k ∥ 2 2 < ∞ , \sum_{k=1}^{N} a_k a_k^* \le C1,\qquad\sum_{k\ge1}\lVert b_k\rVert_2^2 < \infty, k = 1 ∑ N a k a k ∗ ≤ C 1 , k ≥ 1 ∑ ∥ b k ∥ 2 2 < ∞ , where C ≥ 0 C \ge0 C ≥ 0 z n , k ∈ M z_{n,k} \in\mathcal{M} z n , k ∈ M ∥ z n , k ∥ ∞ ≤ 1 \lVert z_{n,k}\rVert_\infty\le1 ∥ z n , k ∥ ∞ ≤ 1 n , k ≥ 1 n,k \ge1 n , k ≥ 1
x n = ∑ k ≥ 1 a k z n , k b k x_n = \sum_{k\ge1} a_k z_{n,k}b_k x n = k ≥ 1 ∑ a k z n , k b k converges in L 2 ( M ) L_2(\mathcal{M}) L 2 ( M )
x n ∗ x n ≤ C ∑ k ≥ 1 b k ∗ b k ( n ≥ 1 ) . (22) x_n^*x_n \le C\sum_{k\ge1}b_k^*b_k\qquad(n\ge1). \tag*{(22)} x n ∗ x n ≤ C k ≥ 1 ∑ b k ∗ b k ( n ≥ 1 ) . (22) Proof. Let ( e i j ) (e_{ij}) ( e ij ) B ( ℓ 2 ) B(\ell_2) B ( ℓ 2 ) F ⊂ N F \subset\mathbb{N} F ⊂ N
( ∑ k ∈ F a k z n , k b k ) ⊗ e 11 = ( ∑ k ∈ F a k z n , k ⊗ e 1 k ) ( ∑ k ∈ F b k ⊗ e k 1 ) . \left(\sum_{k\in F}a_kz_{n,k}b_k\right)\otimes e_{11}
=
\left(\sum_{k\in F}a_kz_{n,k}\otimes e_{1k}\right)
\left(\sum_{k\in F}b_k\otimes e_{k1}\right). ( k ∈ F ∑ a k z n , k b k ) ⊗ e 11 = ( k ∈ F ∑ a k z n , k ⊗ e 1 k ) ( k ∈ F ∑ b k ⊗ e k 1 ) . Since ∥ z n , k ∥ ∞ ≤ 1 \lVert z_{n,k}\rVert_\infty\le1 ∥ z n , k ∥ ∞ ≤ 1
∥ ∑ k ∈ F a k z n , k ⊗ e 1 k ∥ ∞ 2 = ∥ ∑ k ∈ F a k z n , k z n , k ∗ a k ∗ ∥ ∞ ≤ ∥ ∑ k ∈ F a k a k ∗ ∥ ∞ ≤ C . \left\lVert\sum_{k\in F}a_kz_{n,k}\otimes e_{1k}\right\rVert_\infty^2
=
\left\lVert\sum_{k\in F}a_kz_{n,k}z_{n,k}^*a_k^*\right\rVert_\infty
\le
\left\lVert\sum_{k\in F}a_ka_k^*\right\rVert_\infty
\le C. k ∈ F ∑ a k z n , k ⊗ e 1 k ∞ 2 = k ∈ F ∑ a k z n , k z n , k ∗ a k ∗ ∞ ≤ k ∈ F ∑ a k a k ∗ ∞ ≤ C . Consequently,
∣ ( ∑ k ∈ F a k z n , k b k ) ⊗ e 11 ∣ 2 ≤ C ∣ ∑ k ∈ F b k ⊗ e k 1 ∣ 2 = C ( ∑ k ∈ F b k ∗ b k ) ⊗ e 11 , \left|\left(\sum_{k\in F}a_kz_{n,k}b_k\right)\otimes e_{11}\right|^2
\le C\left|\sum_{k\in F}b_k\otimes e_{k1}\right|^2
=
C\left(\sum_{k\in F}b_k^*b_k\right)\otimes e_{11}, ( k ∈ F ∑ a k z n , k b k ) ⊗ e 11 2 ≤ C k ∈ F ∑ b k ⊗ e k 1 2 = C ( k ∈ F ∑ b k ∗ b k ) ⊗ e 11 , and hence
∣ ∑ k ∈ F a k z n , k b k ∣ 2 ≤ C ∑ k ∈ F b k ∗ b k . \left|\sum_{k\in F}a_kz_{n,k}b_k\right|^2
\le C\sum_{k\in F}b_k^*b_k. k ∈ F ∑ a k z n , k b k 2 ≤ C k ∈ F ∑ b k ∗ b k . Therefore we get
∥ ∑ k ∈ F a k z n , k b k ∥ 2 2 ≤ C ∑ k ∈ F ∥ b k ∥ 2 2 . \left\|\sum_{k\in F}a_kz_{n,k}b_k\right\|_2^2
\le C\sum_{k\in F}\lVert b_k\rVert_2^2. k ∈ F ∑ a k z n , k b k 2 2 ≤ C k ∈ F ∑ ∥ b k ∥ 2 2 . Since ∑ k ∥ b k ∥ 2 2 < ∞ \sum_k \lVert b_k\rVert_2^2<\infty ∑ k ∥ b k ∥ 2 2 < ∞ x n x_n x n L 2 ( M ) L_2(\mathcal{M}) L 2 ( M ) ∑ k b k ∗ b k \sum_k b_k^*b_k ∑ k b k ∗ b k L 1 ( M ) L_1(\mathcal{M}) L 1 ( M ) x n ∗ x n ≤ C ∑ k ≥ 1 b k ∗ b k x_n^*x_n\leq C\sum_{k\geq1}b_k^*b_k x n ∗ x n ≤ C ∑ k ≥ 1 b k ∗ b k
We now prove Proposition 3.2 .
Proof of Proposition 3.2. The case f = 0 f=0 f = 0 f j = E j f f_j=\mathsf{E}_jf f j = E j f j 0 ∈ Z j_0\in\mathbb{Z} j 0 ∈ Z f j 0 − 1 ≤ t 1 f_{j_0-1}\leq t1 f j 0 − 1 ≤ t 1
∥ E j f ∥ ∞ ≤ 2 j d ∣ supp R d f ∣ ∥ f ∥ ∞ ⟶ 0 ( j → − ∞ ) . \lVert\mathsf{E}_jf\rVert_\infty\leq2^{jd}\lvert\operatorname{supp}_{\mathbb{R}^d}f\rvert\lVert f\rVert_\infty\longrightarrow0\qquad(j\to-\infty). ∥ E j f ∥ ∞ ≤ 2 j d ∣ supp R d f ∣ ∥ f ∥ ∞ ⟶ 0 ( j → − ∞ ) . We first construct the good function g g g Lemma 2.5 , and verify the identities for its dyadic averages needed below. Let q j , p j q_j,p_j q j , p j Lemma 2.5 ,
q j f j q j ≤ t q j , p j f j q j = q j f j p j = 0 , t p j ≤ p j f j p j ≤ 2 d t p j , t ∑ j ≥ j 0 φ ( p j ) ≤ ∥ f ∥ 1 . (23) \begin{aligned}
q_jf_jq_j&\leq tq_j,\qquad p_jf_jq_j=q_jf_jp_j=0,\\
tp_j&\leq p_jf_jp_j\leq2^dtp_j,\qquad t\sum_{j\geq j_0}\varphi(p_j)\leq\lVert f\rVert_1. \tag*{(23)}
\end{aligned} q j f j q j t p j ≤ t q j , p j f j q j = q j f j p j = 0 , ≤ p j f j p j ≤ 2 d t p j , t j ≥ j 0 ∑ φ ( p j ) ≤ ∥ f ∥ 1 . (23) For s ≥ j 0 s\geq j_0 s ≥ j 0
g s = q s f s q s + ∑ j = j 0 s p j f j p j , g j 0 − 1 = f j 0 − 1 . g_s=q_sf_sq_s+\sum_{j=j_0}^{s}p_jf_jp_j,\qquad g_{j_0-1}=f_{j_0-1}. g s = q s f s q s + j = j 0 ∑ s p j f j p j , g j 0 − 1 = f j 0 − 1 . Applying Lemma 2.5 with j 1 = s j_1=s j 1 = s
0 ≤ g s ≤ 2 d t 1 , ∥ g s ∥ 1 = ∥ f ∥ 1 , ∥ g s ∥ 2 2 ≤ 2 d t ∥ f ∥ 1 . 0\leq g_s\leq2^dt1,\qquad\lVert g_s\rVert_1=\lVert f\rVert_1,\qquad\lVert g_s\rVert_2^2\leq2^dt\lVert f\rVert_1. 0 ≤ g s ≤ 2 d t 1 , ∥ g s ∥ 1 = ∥ f ∥ 1 , ∥ g s ∥ 2 2 ≤ 2 d t ∥ f ∥ 1 . Moreover, since q s − 1 = q s + p s q_{s-1}=q_s+p_s q s − 1 = q s + p s p s f s q s = 0 p_sf_sq_s=0 p s f s q s = 0
g s = q s − 1 f s q s − 1 + ∑ j = j 0 s − 1 p j f j p j , g_s=q_{s-1}f_sq_{s-1}+\sum_{j=j_0}^{s-1}p_jf_jp_j, g s = q s − 1 f s q s − 1 + j = j 0 ∑ s − 1 p j f j p j , and hence
E s − 1 g s = q s − 1 f s − 1 q s − 1 + ∑ j = j 0 s − 1 p j f j p j = g s − 1 . \mathsf{E}_{s-1}g_s=q_{s-1}f_{s-1}q_{s-1}+\sum_{j=j_0}^{s-1}p_jf_jp_j=g_{s-1}. E s − 1 g s = q s − 1 f s − 1 q s − 1 + j = j 0 ∑ s − 1 p j f j p j = g s − 1 . Thus ( g s ) (g_s) ( g s ) L 2 L_2 L 2 L 2 ( N ) L_2(\mathcal{N}) L 2 ( N ) g g g
0 ≤ g ≤ 2 d t 1 , E s g = g s ( s ≥ j 0 − 1 ) . 0\leq g\leq2^dt1,\qquad\mathsf{E}_sg=g_s\quad(s\geq j_0-1). 0 ≤ g ≤ 2 d t 1 , E s g = g s ( s ≥ j 0 − 1 ) . Moreover, we have
φ ( g ) = φ ( E j 0 − 1 g ) = φ ( f j 0 − 1 ) = ∥ f ∥ 1 . \varphi(g)=\varphi(\mathsf{E}_{j_0-1}g)=\varphi(f_{j_0-1})=\lVert f\rVert_1. φ ( g ) = φ ( E j 0 − 1 g ) = φ ( f j 0 − 1 ) = ∥ f ∥ 1 . Note that all g s g_s g s Q j 0 − 1 \mathcal{Q}_{j_0-1} Q j 0 − 1 supp R d f \operatorname{supp}_{\mathbb{R}^d}f supp R d f g ∈ D 0 g\in\mathcal{D}_0 g ∈ D 0
Write
p j = ∑ P ∈ Q j 1 P ⊗ p P , f Q = E Q f . p_j=\sum_{P\in\mathcal{Q}_j}1_P\otimes p_P,\qquad f_Q=\mathsf{E}_Qf. p j = P ∈ Q j ∑ 1 P ⊗ p P , f Q = E Q f . For Q ∈ Q s Q\in\mathcal{Q}_s Q ∈ Q s s ≥ j 0 s\geq j_0 s ≥ j 0 q Q q_Q q Q q s q_s q s Q Q Q P ( Q ) \mathcal{P}(Q) P ( Q ) Q Q Q j 0 , … , s j_0,\ldots,s j 0 , … , s Q Q Q E s g = g s \mathsf{E}_sg=g_s E s g = g s
E Q g = q Q f Q q Q + ∑ P ∈ P ( Q ) p P f P p P . (24) \mathsf{E}_Qg=q_Qf_Qq_Q+\sum_{P\in\mathcal{P}(Q)}p_Pf_Pp_P. \tag*{(24)} E Q g = q Q f Q q Q + P ∈ P ( Q ) ∑ p P f P p P . (24) Consequently,
f Q − E Q g = ( 1 − q Q ) f Q ( 1 − q Q ) − ∑ P ∈ P ( Q ) p P f P p P + ( 1 − q Q ) f Q q Q + q Q f Q ( 1 − q Q ) . (25) f_Q-\mathbb{E}_Qg=(1-q_Q)f_Q(1-q_Q)-\sum_{P\in\mathcal{P}(Q)}p_Pf_Pp_P+(1-q_Q)f_Qq_Q+q_Qf_Q(1-q_Q). \tag*{(25)} f Q − E Q g = ( 1 − q Q ) f Q ( 1 − q Q ) − P ∈ P ( Q ) ∑ p P f P p P + ( 1 − q Q ) f Q q Q + q Q f Q ( 1 − q Q ) . (25) We next use the projection ζ \zeta ζ (14) to restrict the terms involving p P p_P p P ( β P ) c (\beta P)^c ( β P ) c
φ ( 1 − ζ ) ≤ β d t ∥ f ∥ 1 , ζ ( x ) p P = p P ζ ( x ) = 0 for almost every x ∈ β P . \varphi(1-\zeta)\leq\frac{\beta^d}{t}\lVert f\rVert_1,\qquad\zeta(x)p_P=p_P\zeta(x)=0\quad\text{for almost every }x\in\beta P. φ ( 1 − ζ ) ≤ t β d ∥ f ∥ 1 , ζ ( x ) p P = p P ζ ( x ) = 0 for almost every x ∈ β P . Put
Π Q ( x ) = ∑ P ∈ P ( Q ) 1 ( β P ) c ( x ) p P . \Pi_Q(x)=\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)^c}(x)p_P. Π Q ( x ) = P ∈ P ( Q ) ∑ 1 ( β P ) c ( x ) p P . The projections in this sum are pairwise orthogonal, so Π Q \Pi_Q Π Q 1 − q Q = ∑ P ∈ P ( Q ) p P 1-q_Q=\sum_{P\in\mathcal{P}(Q)}p_P 1 − q Q = ∑ P ∈ P ( Q ) p P
ζ ( 1 − q Q ) = ζ Π Q , ( 1 − q Q ) ζ = Π Q ζ . (26) \zeta(1-q_Q)=\zeta\Pi_Q,\qquad(1-q_Q)\zeta=\Pi_Q\zeta. \tag*{(26)} ζ ( 1 − q Q ) = ζ Π Q , ( 1 − q Q ) ζ = Π Q ζ . (26) For any projection w ≤ ζ w\leq\zeta w ≤ ζ (26) gives w ( 1 − q Q ) = w Π Q w(1-q_Q)=w\Pi_Q w ( 1 − q Q ) = w Π Q ( 1 − q Q ) w = Π Q w (1-q_Q)w=\Pi_Qw ( 1 − q Q ) w = Π Q w (25) yields
w ( f Q − E Q g ) w = w ( Π Q f Q Π Q − ∑ P ∈ P ( Q ) 1 ( β P ) c p P f P p P ) w + w Π Q f Q q Q w + w q Q f Q Π Q w . w(f_Q-\mathbb{E}_Qg)w=w\left(\Pi_Qf_Q\Pi_Q-\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)^c}p_Pf_Pp_P\right)w+w\Pi_Qf_Qq_Qw+wq_Qf_Q\Pi_Qw. w ( f Q − E Q g ) w = w Π Q f Q Π Q − P ∈ P ( Q ) ∑ 1 ( β P ) c p P f P p P w + w Π Q f Q q Q w + w q Q f Q Π Q w . Multiplying these three terms by b n , Q b_{n,Q} b n , Q
B n = ∑ Q b n , Q w ( Π Q f Q Π Q − ∑ P ∈ P ( Q ) 1 ( β P ) c p P f P p P ) w , B_n=\sum_Qb_{n,Q}w\left(\Pi_Qf_Q\Pi_Q-\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)^c}p_Pf_Pp_P\right)w, B n = Q ∑ b n , Q w Π Q f Q Π Q − P ∈ P ( Q ) ∑ 1 ( β P ) c p P f P p P w , and
B ℓ , n = ∑ Q b n , Q w Π Q f Q q Q w , B r , n = ∑ Q b n , Q w q Q f Q Π Q w . B_{\ell,n}=\sum_Qb_{n,Q}w\Pi_Qf_Qq_Qw,\qquad B_{r,n}=\sum_Qb_{n,Q}wq_Qf_Q\Pi_Qw. B ℓ , n = Q ∑ b n , Q w Π Q f Q q Q w , B r , n = Q ∑ b n , Q w q Q f Q Π Q w . Here and below, Q Q Q ⋃ s ≥ j 0 Q s \bigcup_{s\geq j_0}\mathcal{Q}_s ⋃ s ≥ j 0 Q s s < j 0 s<j_0 s < j 0
E s g = E s E j 0 − 1 g = E s E j 0 − 1 f = E s f . \mathbb{E}_sg=\mathbb{E}_s\mathbb{E}_{j_0-1}g=\mathbb{E}_s\mathbb{E}_{j_0-1}f=\mathbb{E}_sf. E s g = E s E j 0 − 1 g = E s E j 0 − 1 f = E s f . Thus E Q ( f − g ) = 0 \mathbb{E}_Q(f-g)=0 E Q ( f − g ) = 0 Q ∈ Q s Q\in\mathcal{Q}_s Q ∈ Q s s < j 0 s<j_0 s < j 0
w S n ( f ) w = w S n ( g ) w + B n + B ℓ , n + B r , n . w\mathcal{S}_n(f)w=w\mathcal{S}_n(g)w+B_n+B_{\ell,n}+B_{r,n}. w S n ( f ) w = w S n ( g ) w + B n + B ℓ , n + B r , n . Fix λ > 0 \lambda>0 λ > 0 w ≤ ζ w\leq\zeta w ≤ ζ B n B_n B n L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) B ℓ , n B_{\ell,n} B ℓ , n B r , n B_{r,n} B r , n L 2 ( N ) L_2(\mathcal{N}) L 2 ( N )
sup n max { ∥ B n ∥ ∞ , ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ } ≤ λ 4 , \sup_n\max\{\lVert B_n\rVert_\infty,\lVert B_{\ell,n}\rVert_\infty,\lVert B_{r,n}\rVert_\infty\}\leq\frac{\lambda}{4}, n sup max {∥ B n ∥ ∞ , ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ } ≤ 4 λ , with
φ ( 1 − w ) ≤ C d , β ( 1 t + Γ λ + t Γ 2 λ 2 ) ∥ f ∥ 1 . \varphi(1-w)\leq C_{d,\beta}\left(\frac{1}{t}+\frac{\Gamma}{\lambda}+\frac{t\Gamma^2}{\lambda^2}\right)\lVert f\rVert_1. φ ( 1 − w ) ≤ C d , β ( t 1 + λ Γ + λ 2 t Γ 2 ) ∥ f ∥ 1 . We first estimate ∑ Q m Q Π Q \sum_Q m_Q\Pi_Q ∑ Q m Q Π Q (16) over P P P
∑ Q ⊆ P ∫ ( β P ) c m Q ( x ) d x ≤ Γ ∣ P ∣ . \sum_{Q\subseteq P}\int_{(\beta P)^c}m_Q(x)\,dx\leq\Gamma|P|. Q ⊆ P ∑ ∫ ( β P ) c m Q ( x ) d x ≤ Γ∣ P ∣. Thus, we obtain
φ ( ∑ Q m Q Π Q ) = ∑ j ≥ j 0 ∑ Q ∈ Q j τ ( p P ) ∑ Q ⊆ P ∫ ( β P ) c m Q ( x ) d x ≤ Γ ∑ j ≥ j 0 ∑ P ∈ Q j ∣ P ∣ τ ( p P ) = Γ ∑ j ≥ j 0 φ ( p j ) ≤ Γ t ∥ f ∥ 1 , \begin{aligned}
\varphi\left(\sum_Qm_Q\Pi_Q\right)
&=\sum_{j\geq j_0}\sum_{Q\in\mathcal{Q}_j}\tau(p_P)\sum_{Q\subseteq P}\int_{(\beta P)^c}m_Q(x)\,dx\\
&\leq\Gamma\sum_{j\geq j_0}\sum_{P\in\mathcal{Q}_j}|P|\tau(p_P)\\
&=\Gamma\sum_{j\geq j_0}\varphi(p_j)\leq\frac{\Gamma}{t}\|f\|_1,
\end{aligned} φ Q ∑ m Q Π Q = j ≥ j 0 ∑ Q ∈ Q j ∑ τ ( p P ) Q ⊆ P ∑ ∫ ( β P ) c m Q ( x ) d x ≤ Γ j ≥ j 0 ∑ P ∈ Q j ∑ ∣ P ∣ τ ( p P ) = Γ j ≥ j 0 ∑ φ ( p j ) ≤ t Γ ∥ f ∥ 1 , where the last inequality follows from (10) . Set
w 0 = ζ ∧ 1 [ 0 , Γ ] ( ∑ Q m Q Π Q ) . w_0=\zeta\wedge\mathbf{1}_{[0,\Gamma]}\left(\sum_Qm_Q\Pi_Q\right). w 0 = ζ ∧ 1 [ 0 , Γ ] Q ∑ m Q Π Q . For Γ > 0 \Gamma>0 Γ > 0
φ ( 1 − w 0 ) ≤ φ ( 1 − ζ ) + 1 Γ φ ( ∑ Q m Q Π Q ) ≤ β d + 1 t ∥ f ∥ 1 . \varphi(1-w_0)\leq\varphi(1-\zeta)+\frac{1}{\Gamma}\varphi\left(\sum_Qm_Q\Pi_Q\right)\leq\frac{\beta^d+1}{t}\|f\|_1. φ ( 1 − w 0 ) ≤ φ ( 1 − ζ ) + Γ 1 φ Q ∑ m Q Π Q ≤ t β d + 1 ∥ f ∥ 1 . If Γ = 0 \Gamma=0 Γ = 0 w 0 = ζ w_0=\zeta w 0 = ζ
w 0 ( ∑ Q m Q Π Q ) w 0 ≤ Γ w 0 , φ ( 1 − w 0 ) ≤ β d + 1 t ∥ f ∥ 1 . (27) w_0\left(\sum_Qm_Q\Pi_Q\right)w_0\leq\Gamma w_0,\qquad\varphi(1-w_0)\leq\frac{\beta^d+1}{t}\|f\|_1. \tag*{(27)} w 0 Q ∑ m Q Π Q w 0 ≤ Γ w 0 , φ ( 1 − w 0 ) ≤ t β d + 1 ∥ f ∥ 1 . (27) We next estimate the positive terms appearing in B n B_n B n P ( Q ) \mathcal{P}(Q) P ( Q )
τ ( Π Q ( x ) f Π Q ( x ) ) = ∑ P ∈ P ( Q ) 1 ( β P ) c ( x ) τ ( p P f Q ) . \tau(\Pi_Q(x)f\Pi_Q(x))=\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)^c}(x)\tau(p_Pf_Q). τ ( Π Q ( x ) f Π Q ( x )) = P ∈ P ( Q ) ∑ 1 ( β P ) c ( x ) τ ( p P f Q ) . Since
τ ( p P f Q ) = 1 ∣ Q ∣ ∫ Q τ ( p P f ( y ) ) d y ( Q ⊆ P ) , \tau(p_Pf_Q)=\frac{1}{|Q|}\int_Q\tau(p_Pf(y))\,dy\qquad(Q\subseteq P), τ ( p P f Q ) = ∣ Q ∣ 1 ∫ Q τ ( p P f ( y )) d y ( Q ⊆ P ) , we obtain
∑ Q φ ( m Q Π Q f Π Q ) = ∑ j ≥ j 0 ∑ P ∈ Q j ∫ P τ ( p P f ( y ) ) ( ∑ Q ⊆ P y ∈ Q 1 ∣ Q ∣ ∫ ( β P ) c m Q ( x ) d x ) d y ≤ Γ ∑ j ≥ j 0 φ ( p j f ) ≤ Γ ∥ f ∥ 1 , (28) \begin{aligned}
\sum_Q\varphi(m_Q\Pi_Qf\Pi_Q)
&=\sum_{j\geq j_0}\sum_{P\in\mathcal{Q}_j}\int_P\tau(p_Pf(y))\left(\sum_{\substack{Q\subseteq P\\y\in Q}}\frac{1}{|Q|}\int_{(\beta P)^c}m_Q(x)\,dx\right)\,dy\\
&\leq\Gamma\sum_{j\geq j_0}\varphi(p_jf)\leq\Gamma\|f\|_1, \tag*{(28)}
\end{aligned} Q ∑ φ ( m Q Π Q f Π Q ) = j ≥ j 0 ∑ P ∈ Q j ∑ ∫ P τ ( p P f ( y )) Q ⊆ P y ∈ Q ∑ ∣ Q ∣ 1 ∫ ( β P ) c m Q ( x ) d x d y ≤ Γ j ≥ j 0 ∑ φ ( p j f ) ≤ Γ∥ f ∥ 1 , (28) where we used (16) and ∑ j ≥ j 0 p j ≤ 1 \sum_{j\ge j_0}p_j\le1 ∑ j ≥ j 0 p j ≤ 1
∑ Q φ ( m Q ∑ P ∈ P ( Q ) 1 ( β P ) c P p P f P p P ) ≤ Γ ∑ j ≥ j 0 φ ( p j f ) ≤ Γ ∥ f ∥ 1 . (29) \sum_Q\varphi\left(m_Q\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)}c_Pp_Pf_Pp_P\right)\le\Gamma\sum_{j\ge j_0}\varphi(p_jf)\le\Gamma\lVert f\rVert_1. \tag*{(29)} Q ∑ φ m Q P ∈ P ( Q ) ∑ 1 ( β P ) c P p P f P p P ≤ Γ j ≥ j 0 ∑ φ ( p j f ) ≤ Γ ∥ f ∥ 1 . (29) On the other hand, q Q f Q q Q ≤ t q Q q_Qf_Qq_Q\le tq_Q q Q f Q q Q ≤ t q Q (21) give f Q q Q f Q ≤ t f Q f_Qq_Qf_Q\le tf_Q f Q q Q f Q ≤ t f Q
∑ Q ∥ m Q Π Q f Q q Q w 0 ∥ 2 2 ≤ ∑ Q φ ( m Q Π Q f Q q Q f Q Π Q ) ≤ t Γ ∥ f ∥ 1 . (30) \sum_Q\left\lVert\sqrt{m_Q}\Pi_Qf_Qq_Qw_0\right\rVert_2^2\le\sum_Q\varphi(m_Q\Pi_Qf_Qq_Qf_Q\Pi_Q)\le t\Gamma\lVert f\rVert_1. \tag*{(30)} Q ∑ m Q Π Q f Q q Q w 0 2 2 ≤ Q ∑ φ ( m Q Π Q f Q q Q f Q Π Q ) ≤ t Γ ∥ f ∥ 1 . (30) We now define h ∈ L 1 ( N ) + h\in L_1(\mathcal{N})_+ h ∈ L 1 ( N ) +
h = 4 λ ∑ Q m Q w 0 ( Π Q f Q Π Q + ∑ P ∈ P ( Q ) 1 ( β P ) c P p P f P p P ) w 0 + 16 Γ λ 2 ∑ Q m Q w 0 q Q f Q Π Q f Q q Q w 0 . h=\frac{4}{\lambda}\sum_Qm_Qw_0\left(\Pi_Qf_Q\Pi_Q+\sum_{P\in\mathcal{P}(Q)}\mathbf{1}_{(\beta P)}c_Pp_Pf_Pp_P\right)w_0+\frac{16\Gamma}{\lambda^2}\sum_Qm_Qw_0q_Qf_Q\Pi_Qf_Qq_Qw_0. h = λ 4 Q ∑ m Q w 0 Π Q f Q Π Q + P ∈ P ( Q ) ∑ 1 ( β P ) c P p P f P p P w 0 + λ 2 16Γ Q ∑ m Q w 0 q Q f Q Π Q f Q q Q w 0 . By (28) , (29) and (30) , we get
φ ( h ) ≤ ( 8 Γ λ + 16 t Γ 2 λ 2 ) ∥ f ∥ 1 . \varphi(h)\le\left(\frac{8\Gamma}{\lambda}+\frac{16t\Gamma^2}{\lambda^2}\right)\lVert f\rVert_1. φ ( h ) ≤ ( λ 8Γ + λ 2 16 t Γ 2 ) ∥ f ∥ 1 . Take
w = w 0 ∧ 1 [ 0 , 1 ] ( h ) . w=w_0\wedge\mathbf{1}_{[0,1]}(h). w = w 0 ∧ 1 [ 0 , 1 ] ( h ) . Then w ≤ ζ w\le\zeta w ≤ ζ w h w ≤ w whw\le w w h w ≤ w
φ ( 1 − w ) ≤ φ ( 1 − w 0 ) + φ ( h ) ≤ ( β d + 1 t + 8 Γ λ + 16 t Γ 2 λ 2 ) ∥ f ∥ 1 . \varphi(1-w)\le\varphi(1-w_0)+\varphi(h)\le\left(\frac{\beta^d+1}{t}+\frac{8\Gamma}{\lambda}+\frac{16t\Gamma^2}{\lambda^2}\right)\lVert f\rVert_1. φ ( 1 − w ) ≤ φ ( 1 − w 0 ) + φ ( h ) ≤ ( t β d + 1 + λ 8Γ + λ 2 16 t Γ 2 ) ∥ f ∥ 1 . This proves (20) .
Since ∣ b n , Q ∣ ≤ m Q \lvert b_{n,Q}\rvert\le m_Q ∣ b n , Q ∣ ≤ m Q (28) and (29) give absolute convergence of the series defining B n B_n B n L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) Y ≥ 0 Y\ge0 Y ≥ 0 ∣ z ∣ ≤ 1 \lvert z\rvert\le1 ∣ z ∣ ≤ 1
( Y z Y z ˉ Y Y ) ≥ 0. \begin{pmatrix}
Y & zY\\
\bar{z}Y & Y
\end{pmatrix}\ge0. ( Y z ˉ Y z Y Y ) ≥ 0. Applying this to the two positive terms in B n B_n B n b n , Q / m Q b_{n,Q}/m_Q b n , Q / m Q − b n , Q / m Q -b_{n,Q}/m_Q − b n , Q / m Q
\begin{pmatrix}
\frac{\lambda}{4}whw & B_n\\
B_n^* & \frac{\lambda}{4}whw
\end{pmatrix}\ge0. \tag*{(31)}\end{pmatrix}
As w h w ≤ w whw\le w w h w ≤ w ∥ B n ∥ ∞ ≤ λ / 4 \lVert B_n\rVert_\infty\le\lambda/4 ∥ B n ∥ ∞ ≤ λ /4
Furthermore, by (27) , (30) and the fact w ≤ w 0 w\le w_0 w ≤ w 0
∑ Q ( m Q w Π Q ) ( m Q w Π Q ) ∗ ≤ Γ w , ∑ Q ∥ m Q Π Q f Q q Q w ∥ 2 2 ≤ t Γ ∥ f ∥ 1 . \sum_Q(\sqrt{m_Q}w\Pi_Q)(\sqrt{m_Q}w\Pi_Q)^*\le\Gamma w,\qquad\sum_Q\left\lVert\sqrt{m_Q}\Pi_Qf_Qq_Qw\right\rVert_2^2\le t\Gamma\lVert f\rVert_1. Q ∑ ( m Q w Π Q ) ( m Q w Π Q ) ∗ ≤ Γ w , Q ∑ m Q Π Q f Q q Q w 2 2 ≤ t Γ ∥ f ∥ 1 . Apply Lemma 3.3 with
a Q = m Q w Π Q , b Q = m Q Π Q f Q q Q w , z n , Q = b n , Q / m Q . a_Q=\sqrt{m_Q}w\Pi_Q,\qquad b_Q=\sqrt{m_Q}\Pi_Qf_Qq_Qw,\qquad z_{n,Q}=b_{n,Q}/m_Q. a Q = m Q w Π Q , b Q = m Q Π Q f Q q Q w , z n , Q = b n , Q / m Q . Applying the same lemma with z n , Q ‾ \overline{z_{n,Q}} z n , Q B ℓ , n B_{\ell,n} B ℓ , n B r , n B_{r,n} B r , n L 2 ( N ) L_2(\mathcal{N}) L 2 ( N )
B ℓ , n ∗ B ℓ , n , B r , n B r , n ∗ ≤ Γ ∑ Q m Q w q Q f Q Π Q f Q q Q w ≤ λ 2 16 w h w . (32) B_{\ell,n}^*B_{\ell,n},\ B_{r,n}B_{r,n}^*\le\Gamma\sum_Qm_Qwq_Qf_Q\Pi_Qf_Qq_Qw\le\frac{\lambda^2}{16}whw. \tag*{(32)} B ℓ , n ∗ B ℓ , n , B r , n B r , n ∗ ≤ Γ Q ∑ m Q w q Q f Q Π Q f Q q Q w ≤ 16 λ 2 w h w . (32) Thus ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ ≤ λ / 4 \lVert B_{\ell,n}\rVert_{\infty},\lVert B_{r,n}\rVert_{\infty}\leq\lambda/4 ∥ B ℓ , n ∥ ∞ , ∥ B r , n ∥ ∞ ≤ λ /4 (19) . □
A simultaneous dyadic decomposition
In this section, we construct a simultaneous dyadic decomposition of a family of kernels; see Proposition 4.1 . We then express the convolution operators in terms of dyadic averages and verify the coefficient bound required in Theorem 3.1 ; see Proposition 4.4 .
Dyadic expansion and coefficient estimates
In this subsection, we construct a dyadic expansion for a countable family ( K n ) n ≥ 1 ⊂ L 1 , l o c ( R d ∖ { 0 } ) (K_n)_{n\geq1}\subset L_{1,\mathrm{loc}}(\mathbb{R}^{d}\setminus\{0\}) ( K n ) n ≥ 1 ⊂ L 1 , loc ( R d ∖ { 0 }) § 2.3 . For each j ∈ Z j\in\mathbb{Z} j ∈ Z
V j = [ − 2 1 − j , 2 1 − j ) d ∖ [ − 2 − j , 2 − j ) d , B j = { B ∈ Q j : B ⊆ V j } . V_j=[-2^{1-j},2^{1-j})^d\setminus[-2^{-j},2^{-j})^d,\qquad\mathcal{B}_j=\{B\in\mathcal{Q}_j:B\subseteq V_j\}. V j = [ − 2 1 − j , 2 1 − j ) d ∖ [ − 2 − j , 2 − j ) d , B j = { B ∈ Q j : B ⊆ V j } . The 4 d − 2 d 4^d-2^d 4 d − 2 d B j \mathcal{B}_j B j V j V_j V j ⋃ j B j \bigcup_j\mathcal{B}_j ⋃ j B j R d ∖ { 0 } \mathbb{R}^d\setminus\{0\} R d ∖ { 0 } D \mathcal{D} D Q ∈ D Q\in\mathcal{D} Q ∈ D B ∈ ⋃ j B j B\in\bigcup_j\mathcal{B}_j B ∈ ⋃ j B j
On each B ∈ ⋃ j B j B\in\bigcup_j\mathcal{B}_j B ∈ ⋃ j B j a n , Q a_{n,Q} a n , Q B B B Q Q Q
F n , Q = K n − ∑ Q ⊊ P ⊆ B a n , P . F_{n,Q}=K_n-\sum_{Q\subsetneq P\subseteq B}a_{n,P}. F n , Q = K n − Q ⊊ P ⊆ B ∑ a n , P . where the sum is empty for Q = B Q=B Q = B
d n , Q = { E Q F n , Q , ∫ Q ∣ F n , Q − E Q F n , Q ∣ ≤ 1 2 ∫ Q ∣ F n , Q ∣ , 0 , otherwise . d_{n,Q}=
\begin{cases}
\mathbb{E}_QF_{n,Q},&\int_Q\lvert F_{n,Q}-\mathbb{E}_QF_{n,Q}\rvert\leq\frac{1}{2}\int_Q\lvert F_{n,Q}\rvert,\\
0,&\text{otherwise}.
\end{cases} d n , Q = { E Q F n , Q , 0 , ∫ Q ∣ F n , Q − E Q F n , Q ∣ ≤ 2 1 ∫ Q ∣ F n , Q ∣ , otherwise . and define
( a n , Q ) n = { ( d n , Q ) n , ∫ Q sup n ∣ F n , Q − d n , Q ∣ ≤ 1 2 ∫ Q sup n ∣ F n , Q ∣ , ( 0 ) n , otherwise . (33) (a_{n,Q})_n=
\begin{cases}
(d_{n,Q})_n,&\int_Q\sup_n\lvert F_{n,Q}-d_{n,Q}\rvert\leq\frac{1}{2}\int_Q\sup_n\lvert F_{n,Q}\rvert,\\
(0)_n,&\text{otherwise}.
\end{cases} \tag*{(33)} ( a n , Q ) n = { ( d n , Q ) n , ( 0 ) n , ∫ Q sup n ∣ F n , Q − d n , Q ∣ ≤ 2 1 ∫ Q sup n ∣ F n , Q ∣ , otherwise . (33) We impose the following assumptions on ( K n ) n ≥ 1 (K_n)_{n\geq1} ( K n ) n ≥ 1
sup n ∥ K n ∥ ∞ + sup n sup x ≠ y ∣ K n ( x ) − K n ( y ) ∣ ∣ x − y ∣ < ∞ . \sup_n\lVert K_n\rVert_\infty+\sup_n\sup_{x\ne y}\frac{\lvert K_n(x)-K_n(y)\rvert}{\lvert x-y\rvert}<\infty. n sup ∥ K n ∥ ∞ + n sup x = y sup ∣ x − y ∣ ∣ K n ( x ) − K n ( y )∣ < ∞. They satisfy the joint Hörmander condition (51) ,
C D : = sup k ∈ Z sup y ∈ Q 0 ( k ) ∫ ( A Q 0 ( k ) ) c sup n ∣ K n ( x − y ) − K n ( x ) ∣ d x < ∞ C_D:=\sup_{k\in\mathbb{Z}}\sup_{y\in Q_0^{(k)}}\int_{(A Q_0^{(k)})^c}\sup_n\lvert K_n(x-y)-K_n(x)\rvert\,dx<\infty C D := k ∈ Z sup y ∈ Q 0 ( k ) sup ∫ ( A Q 0 ( k ) ) c n sup ∣ K n ( x − y ) − K n ( x )∣ d x < ∞ for some fixed A ≥ 1 A\geq1 A ≥ 1 sup n ∣ K n ∣ \sup_n\lvert K_n\rvert sup n ∣ K n ∣ V j V_j V j
C V : = sup j ∈ Z ∫ V j sup n ∣ K n ( x ) ∣ d x < ∞ . C_V:=\sup_{j\in\mathbb{Z}}\int_{V_j}\sup_n\lvert K_n(x)\rvert\,dx<\infty. C V := j ∈ Z sup ∫ V j n sup ∣ K n ( x )∣ d x < ∞. Proposition 4.1. The coefficients defined above have the following properties.
(i) For every B ∈ ⋃ j B j B \in\bigcup_j \mathcal{B}_j B ∈ ⋃ j B j
∑ Q ⊆ B ∣ Q ∣ ∣ a n , Q ∣ ≤ 2 ∫ B ∣ K n ∣ , ∑ Q ⊆ B ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ∫ B sup n ∣ K n ∣ . (34) \sum_{Q\subseteq B} |Q||a_{n,Q}| \le2\int_B |K_n|,\qquad\sum_{Q\subseteq B} |Q|\sup_n |a_{n,Q}| \le2\int_B \sup_n |K_n|. \tag*{(34)} Q ⊆ B ∑ ∣ Q ∣∣ a n , Q ∣ ≤ 2 ∫ B ∣ K n ∣ , Q ⊆ B ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 ∫ B n sup ∣ K n ∣. (34) Furthermore, for every P ∈ D P\in\mathcal{D} P ∈ D
∑ Q ⊆ P ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 12 ∫ P sup n ∣ K n − E P K n ∣ . (35) \sum_{Q\subseteq P}|Q|\sup_n |a_{n,Q}| \le12\int_P \sup_n |K_n-\mathbb{E}_P K_n|. \tag*{(35)} Q ⊆ P ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 12 ∫ P n sup ∣ K n − E P K n ∣. (35) (ii) For every n ≥ 1 n\ge1 n ≥ 1
K n = ∑ Q ∈ D a n , Q 1 Q almost everywhere on R d ∖ { 0 } . (36) K_n=\sum_{Q\in\mathcal{D}}a_{n,Q}1_Q\quad\text{almost everywhere on }\mathbb{R}^d\setminus\{0\}. \tag*{(36)} K n = Q ∈ D ∑ a n , Q 1 Q almost everywhere on R d ∖ { 0 } . (36) If K n ∈ L 1 ( R d ) K_n\in L_1(\mathbb{R}^d) K n ∈ L 1 ( R d )
∣ K n ∣ ≤ ∑ Q ∈ D ∣ a n , Q ∣ 1 Q a.e. , ∑ Q ∈ D ∣ Q ∣ ∣ a n , Q ∣ ≤ 2 ∥ K n ∥ 1 . (37) |K_n|\le\sum_{Q\in\mathcal{D}}|a_{n,Q}|1_Q\quad\text{a.e.},\qquad\sum_{Q\in\mathcal{D}}|Q||a_{n,Q}|\le2\|K_n\|_1. \tag*{(37)} ∣ K n ∣ ≤ Q ∈ D ∑ ∣ a n , Q ∣ 1 Q a.e. , Q ∈ D ∑ ∣ Q ∣∣ a n , Q ∣ ≤ 2∥ K n ∥ 1 . (37) (iii) For Q ∈ Q Q\in\mathcal{Q} Q ∈ Q ρ Q = ∣ c Q ∣ ∞ + ℓ ( Q ) / 2 \rho_Q=|c_Q|_\infty+\ell(Q)/2 ρ Q = ∣ c Q ∣ ∞ + ℓ ( Q ) /2
E j = { Q ∈ Q : ℓ ( Q ) ≤ 2 − j , ρ Q > 2 − j } . \mathcal{E}_j=\{Q\in\mathcal{Q}:\ell(Q)\le2^{-j},\ \rho_Q>2^{-j}\}. E j = { Q ∈ Q : ℓ ( Q ) ≤ 2 − j , ρ Q > 2 − j } . Then, for every j ∈ Z j\in\mathbb{Z} j ∈ Z
∑ Q ∈ D ∩ E j ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ C d , A ( C V + C D ) . (38) \sum_{Q\in\mathcal{D}\cap\mathcal{E}_j}|Q|\sup_n|a_{n,Q}|\le C_{d,A}(C_V+C_D). \tag*{(38)} Q ∈ D ∩ E j ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ C d , A ( C V + C D ) . (38) Proof. We first obtain the coefficient estimates needed to justify the expansion. By the definition of a n , Q a_{n,Q} a n , Q
∣ Q ∣ ∣ a n , Q ∣ ≤ 2 ( ∫ Q ∣ F n , Q ∣ − ∫ Q ∣ F n , Q − a n , Q ∣ ) , (39) |Q||a_{n,Q}|\le2\left(\int_Q|F_{n,Q}|-\int_Q|F_{n,Q}-a_{n,Q}|\right), \tag*{(39)} ∣ Q ∣∣ a n , Q ∣ ≤ 2 ( ∫ Q ∣ F n , Q ∣ − ∫ Q ∣ F n , Q − a n , Q ∣ ) , (39) ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ( ∫ Q sup n ∣ F n , Q ∣ − ∫ Q sup n ∣ F n , Q − a n , Q ∣ ) . (40) |Q|\sup_n|a_{n,Q}|\le2\left(\int_Q\sup_n|F_{n,Q}|-\int_Q\sup_n|F_{n,Q}-a_{n,Q}|\right). \tag*{(40)} ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 ( ∫ Q n sup ∣ F n , Q ∣ − ∫ Q n sup ∣ F n , Q − a n , Q ∣ ) . (40) Fix B ∈ ⋃ j B j B\in\bigcup_j\mathcal{B}_j B ∈ ⋃ j B j k k k
G k ( B ) = { Q ⊆ B : ℓ ( Q ) = 2 − k ℓ ( B ) } . \mathcal{G}_k(B)=\{Q\subseteq B:\ell(Q)=2^{-k}\ell(B)\}. G k ( B ) = { Q ⊆ B : ℓ ( Q ) = 2 − k ℓ ( B )} . For each child Q ′ Q' Q ′ Q Q Q F n , Q ′ = F n , Q − a n , Q F_{n,Q'}=F_{n,Q}-a_{n,Q} F n , Q ′ = F n , Q − a n , Q
F n , Q ′ = K n − ∑ Q ′ ⊊ P ⊆ B a n , P = K n − a n , Q − ∑ Q ⊊ P ⊆ B a n , P = F n , Q − a n , Q . F_{n,Q'}=K_n-\sum_{Q'\subsetneq P\subseteq B}a_{n,P}=K_n-a_{n,Q}-\sum_{Q\subsetneq P\subseteq B}a_{n,P}=F_{n,Q}-a_{n,Q}. F n , Q ′ = K n − Q ′ ⊊ P ⊆ B ∑ a n , P = K n − a n , Q − Q ⊊ P ⊆ B ∑ a n , P = F n , Q − a n , Q . Consequently,
∑ Q ′ child of Q ∫ Q ′ sup n ∣ F n , Q ′ ∣ = ∫ Q sup n ∣ F n , Q − a n , Q ∣ . \sum_{\substack{Q'\text{ child}\\\text{of }Q}}\int_{Q'}\sup_n|F_{n,Q'}|=\int_Q\sup_n|F_{n,Q}-a_{n,Q}|. Q ′ child of Q ∑ ∫ Q ′ n sup ∣ F n , Q ′ ∣ = ∫ Q n sup ∣ F n , Q − a n , Q ∣. Summing (40) and (39) over generations 0 , … , m 0,\ldots,m 0 , … , m
∑ k = 0 m ∑ Q ∈ G k ( B ) ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ( ∫ B sup n ∣ K n ∣ − ∑ Q ∈ G m ( B ) ∫ Q sup n ∣ F n , Q − a n , Q ∣ ) ≤ 2 ∫ B sup n ∣ K n ∣ , \begin{aligned}
\sum_{k=0}^{m}\sum_{Q\in\mathcal{G}_k(B)} |Q|\sup_n |a_{n,Q}| &\le2\left(\int_B\sup_n |K_n|-\sum_{Q\in\mathcal{G}_m(B)}\int_Q\sup_n |F_{n,Q}-a_{n,Q}|\right)\\
&\le2\int_B\sup_n |K_n|,
\end{aligned} k = 0 ∑ m Q ∈ G k ( B ) ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 ∫ B n sup ∣ K n ∣ − Q ∈ G m ( B ) ∑ ∫ Q n sup ∣ F n , Q − a n , Q ∣ ≤ 2 ∫ B n sup ∣ K n ∣ , and
∑ k = 0 m ∑ Q ∈ G k ( B ) ∣ Q ∣ ∣ a n , Q ∣ ≤ 2 ∫ B ∣ K n ∣ . \sum_{k=0}^{m}\sum_{Q\in\mathcal{G}_k(B)} |Q||a_{n,Q}|\le2\int_B |K_n|. k = 0 ∑ m Q ∈ G k ( B ) ∑ ∣ Q ∣∣ a n , Q ∣ ≤ 2 ∫ B ∣ K n ∣. Letting m → ∞ m\to\infty m → ∞ (34) . Repeating the same summation over the strict descendants of P P P F n , Q = F n , P − a n , P F_{n,Q}=F_{n,P}-a_{n,P} F n , Q = F n , P − a n , P Q ∈ G 1 ( P ) Q\in\mathcal{G}_1(P) Q ∈ G 1 ( P )
∑ Q ⊆ P ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ∫ P sup n ∣ F n , P − a n , P ∣ . (41) \sum_{Q\subseteq P}|Q|\sup_n|a_{n,Q}|\le2\int_P\sup_n|F_{n,P}-a_{n,P}|. \tag*{(41)} Q ⊆ P ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 ∫ P n sup ∣ F n , P − a n , P ∣. (41) We claim that
∫ P sup n ∣ F n , P − a n , P ∣ ≤ 6 ∫ P sup n ∣ K n − E P K n ∣ . (42) \int_P\sup_n|F_{n,P}-a_{n,P}|\le6\int_P\sup_n|K_n-\mathbb{E}_P K_n|. \tag*{(42)} ∫ P n sup ∣ F n , P − a n , P ∣ ≤ 6 ∫ P n sup ∣ K n − E P K n ∣. (42) Together with (41) , this yields
∑ Q ⊆ P ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 12 ∫ P sup n ∣ K n − E P K n ∣ , \sum_{Q\subseteq P}|Q|\sup_n|a_{n,Q}|\le12\int_P\sup_n|K_n-\mathbb{E}_P K_n|, Q ⊆ P ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 12 ∫ P n sup ∣ K n − E P K n ∣ , which is (35) .
To prove the claim, note that F n , P − E P F n , P = K n − E P K n F_{n,P}-\mathbb{E}_P F_{n,P}=K_n-\mathbb{E}_P K_n F n , P − E P F n , P = K n − E P K n d n , P = E P F n , P d_{n,P}=\mathbb{E}_P F_{n,P} d n , P = E P F n , P ∣ F n , P − d n , P ∣ = ∣ K n − E P K n ∣ |F_{n,P}-d_{n,P}|=|K_n-\mathbb{E}_P K_n| ∣ F n , P − d n , P ∣ = ∣ K n − E P K n ∣ d n , P = 0 d_{n,P}=0 d n , P = 0
∫ P ∣ F n , P ∣ < 2 ∫ P ∣ F n , P − E P F n , P ∣ = 2 ∫ P ∣ K n − E P K n ∣ . \int_P|F_{n,P}|<2\int_P|F_{n,P}-\mathbb{E}_P F_{n,P}|=2\int_P|K_n-\mathbb{E}_P K_n|. ∫ P ∣ F n , P ∣ < 2 ∫ P ∣ F n , P − E P F n , P ∣ = 2 ∫ P ∣ K n − E P K n ∣. Consequently,
∣ E P F n , P ∣ ≤ E P ∣ F n , P ∣ ≤ 2 E P ∣ K n − E P K n ∣ , |\mathbb{E}_P F_{n,P}|\le\mathbb{E}_P|F_{n,P}|\le2\mathbb{E}_P|K_n-\mathbb{E}_P K_n|, ∣ E P F n , P ∣ ≤ E P ∣ F n , P ∣ ≤ 2 E P ∣ K n − E P K n ∣ , so in both cases,
∣ F n , P − d n , P ∣ ≤ ∣ K n − E P K n ∣ + 2 E P ∣ K n − E P K n ∣ . |F_{n,P}-d_{n,P}|\le|K_n-\mathbb{E}_P K_n|+2\mathbb{E}_P|K_n-\mathbb{E}_P K_n|. ∣ F n , P − d n , P ∣ ≤ ∣ K n − E P K n ∣ + 2 E P ∣ K n − E P K n ∣. Taking the supremum over n n n P P P
∫ P sup n ∣ F n , P − d n , P ∣ ≤ 3 ∫ P sup n ∣ K n − E P K n ∣ . \int_P\sup_n|F_{n,P}-d_{n,P}|\le3\int_P\sup_n|K_n-\mathbb{E}_P K_n|. ∫ P n sup ∣ F n , P − d n , P ∣ ≤ 3 ∫ P n sup ∣ K n − E P K n ∣. By (33) , either a n , P = d n , P a_{n,P}=d_{n,P} a n , P = d n , P n n n a n , P = 0 a_{n,P}=0 a n , P = 0 n n n
∫ P sup n ∣ F n , P ∣ < 2 ∫ P sup n ∣ F n , P − d n , P ∣ . \int_P\sup_n|F_{n,P}|<2\int_P\sup_n|F_{n,P}-d_{n,P}|. ∫ P n sup ∣ F n , P ∣ < 2 ∫ P n sup ∣ F n , P − d n , P ∣. Thus
∫ P sup n ∣ F n , P − a n , P ∣ ≤ 2 ∫ P sup n ∣ F n , P − d n , P ∣ ≤ 6 ∫ P sup n ∣ K n − E P K n ∣ , \int_P\sup_n|F_{n,P}-a_{n,P}|\le2\int_P\sup_n|F_{n,P}-d_{n,P}|\le6\int_P\sup_n|K_n-\mathbb{E}_P K_n|, ∫ P n sup ∣ F n , P − a n , P ∣ ≤ 2 ∫ P n sup ∣ F n , P − d n , P ∣ ≤ 6 ∫ P n sup ∣ K n − E P K n ∣ , proving the claim.
We now prove (36) . Fix B ∈ ⋃ j B j B \in\bigcup_j \mathcal{B}_j B ∈ ⋃ j B j P ∈ G m ( B ) P \in\mathcal{G}_m(B) P ∈ G m ( B ) x ∈ P x \in P x ∈ P
K n ( x ) − ∑ k = 0 m ∑ Q ∈ G k ( B ) a n , Q 1 Q ( x ) = K n ( x ) − ∑ P ⊆ Q ⊆ B a n , Q = F n , P ( x ) − a n , P . K_n(x)-\sum_{k=0}^{m}\sum_{Q\in\mathcal{G}_k(B)}a_{n,Q}\mathbf{1}_Q(x)=K_n(x)-\sum_{P\subseteq Q\subseteq B}a_{n,Q}=F_{n,P}(x)-a_{n,P}. K n ( x ) − k = 0 ∑ m Q ∈ G k ( B ) ∑ a n , Q 1 Q ( x ) = K n ( x ) − P ⊆ Q ⊆ B ∑ a n , Q = F n , P ( x ) − a n , P . Let C L i p C_{\mathrm{Lip}} C Lip
∣ K n ( x ) − E P K n ∣ ≤ 1 ∣ P ∣ ∫ P ∣ K n ( x ) − K n ( y ) ∣ d y ≤ C L i p d ℓ ( P ) , x ∈ P , \lvert K_n(x)-\mathbb{E}_P K_n\rvert\leq\frac{1}{\lvert P\rvert}\int_P\lvert K_n(x)-K_n(y)\rvert\,\mathrm{d}y\leq C_{\mathrm{Lip}}\sqrt{d}\,\ell(P),\qquad x\in P, ∣ K n ( x ) − E P K n ∣ ≤ ∣ P ∣ 1 ∫ P ∣ K n ( x ) − K n ( y )∣ d y ≤ C Lip d ℓ ( P ) , x ∈ P , we obtain from (42) that
∫ B sup n ∣ K n − ∑ k = 0 m ∑ Q ∈ G k ( B ) a n , Q 1 Q ∣ = ∑ P ∈ G m ( B ) ∫ P sup n ∣ F n , P − a n , P ∣ ≤ 6 ∑ P ∈ G m ( B ) ∫ P sup n ∣ K n − E P K n ∣ ≤ 6 C L i p d 2 − m ℓ ( B ) ∣ B ∣ ⟶ 0. \begin{aligned}
\int_B\sup_n\left\lvert K_n-\sum_{k=0}^{m}\sum_{Q\in\mathcal{G}_k(B)}a_{n,Q}\mathbf{1}_Q\right\rvert
&=\sum_{P\in\mathcal{G}_m(B)}\int_P\sup_n\lvert F_{n,P}-a_{n,P}\rvert\\
&\leq6\sum_{P\in\mathcal{G}_m(B)}\int_P\sup_n\lvert K_n-\mathbb{E}_P K_n\rvert\\
&\leq6C_{\mathrm{Lip}}\sqrt{d}\,2^{-m}\ell(B)\lvert B\rvert\longrightarrow0.
\end{aligned} ∫ B n sup K n − k = 0 ∑ m Q ∈ G k ( B ) ∑ a n , Q 1 Q = P ∈ G m ( B ) ∑ ∫ P n sup ∣ F n , P − a n , P ∣ ≤ 6 P ∈ G m ( B ) ∑ ∫ P n sup ∣ K n − E P K n ∣ ≤ 6 C Lip d 2 − m ℓ ( B ) ∣ B ∣ ⟶ 0. Moreover, by (34) , we have
∫ B ∑ Q ⊆ B sup n ∣ a n , Q ∣ 1 Q = ∑ Q ⊆ B ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ∫ B sup n ∣ K n ∣ < ∞ . \int_B\sum_{Q\subseteq B}\sup_n\lvert a_{n,Q}\rvert\mathbf{1}_Q=\sum_{Q\subseteq B}\lvert Q\rvert\sup_n\lvert a_{n,Q}\rvert\leq2\int_B\sup_n\lvert K_n\rvert<\infty. ∫ B Q ⊆ B ∑ n sup ∣ a n , Q ∣ 1 Q = Q ⊆ B ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 ∫ B n sup ∣ K n ∣ < ∞. Thus the series converge absolutely almost everywhere and in L 1 ( B ) L_1(B) L 1 ( B )
∥ K n − ∑ Q ⊆ B a n , Q 1 Q ∥ L 1 ( B ) = lim m → ∞ ∥ K n − ∑ k = 0 m ∑ Q ∈ G k ( B ) a n , Q 1 Q ∥ L 1 ( B ) = 0. \left\|K_n-\sum_{Q\subseteq B}a_{n,Q}\mathbf{1}_Q\right\|_{L_1(B)}
=\lim_{m\to\infty}\left\|K_n-\sum_{k=0}^{m}\sum_{Q\in\mathcal{G}_k(B)}a_{n,Q}\mathbf{1}_Q\right\|_{L_1(B)}
=0. K n − Q ⊆ B ∑ a n , Q 1 Q L 1 ( B ) = m → ∞ lim K n − k = 0 ∑ m Q ∈ G k ( B ) ∑ a n , Q 1 Q L 1 ( B ) = 0. Since ⋃ j B j \bigcup_j\mathcal{B}_j ⋃ j B j R d ∖ { 0 } \mathbb{R}^d\setminus\{0\} R d ∖ { 0 } (36) .
If K n ∈ L 1 ( R d ) K_n\in L_1(\mathbb{R}^d) K n ∈ L 1 ( R d )
∣ K n ∣ ≤ ∑ Q ∈ D ∣ a n , Q ∣ 1 Q a.e. , \lvert K_n\rvert\leq\sum_{Q\in\mathcal{D}}\lvert a_{n,Q}\rvert\mathbf{1}_Q\quad\text{a.e.}, ∣ K n ∣ ≤ Q ∈ D ∑ ∣ a n , Q ∣ 1 Q a.e. , and summing (34) over B B B
∑ Q ∈ D ∣ Q ∣ ∣ a n , Q ∣ = ∑ B ∈ ⋃ j B j ∑ Q ⊆ B ∣ Q ∣ ∣ a n , Q ∣ ≤ 2 ∑ B ∈ ⋃ j B j ∫ B ∣ K n ∣ = 2 ∥ K n ∥ 1 . \sum_{Q\in\mathcal{D}}\lvert Q\rvert\lvert a_{n,Q}\rvert
=\sum_{B\in\bigcup_j\mathcal{B}_j}\sum_{Q\subseteq B}\lvert Q\rvert\lvert a_{n,Q}\rvert
\leq2\sum_{B\in\bigcup_j\mathcal{B}_j}\int_B\lvert K_n\rvert
=2\lVert K_n\rVert_1. Q ∈ D ∑ ∣ Q ∣ ∣ a n , Q ∣ = B ∈ ⋃ j B j ∑ Q ⊆ B ∑ ∣ Q ∣ ∣ a n , Q ∣ ≤ 2 B ∈ ⋃ j B j ∑ ∫ B ∣ K n ∣ = 2 ∥ K n ∥ 1 . This proves (37) .
It remains to prove (38) . Fix j ∈ Z j\in\mathbb{Z} j ∈ Z J = ⌈ log 2 ( 2 A ) ⌉ J=\lceil\log_2(2A)\rceil J = ⌈ log 2 ( 2 A )⌉ Q ∈ D ∩ E j Q\in\mathcal{D}\cap\mathcal{E}_j Q ∈ D ∩ E j B ∈ B r B\in\mathcal{B}_r B ∈ B r r ≤ j r\leq j r ≤ j r > j r>j r > j B ⊆ [ − 2 − j , 2 − j ) d B\subseteq[-2^{-j},2^{-j})^d B ⊆ [ − 2 − j , 2 − j ) d ρ Q > 2 − j \rho_Q>2^{-j} ρ Q > 2 − j j − J < r ≤ j j-J<r\leq j j − J < r ≤ j (34) , we have
∑ r = j − J + 1 j ∑ B ∈ B r ∑ Q ⊆ B ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 ∑ r = j − J + 1 j ∫ V r sup n ∣ K n ∣ ≤ 2 J C V . \sum_{r=j-J+1}^{j}\sum_{B\in\mathcal{B}_r}\sum_{Q\subseteq B}\lvert Q\rvert\sup_n\lvert a_{n,Q}\rvert
\leq2\sum_{r=j-J+1}^{j}\int_{V_r}\sup_n\lvert K_n\rvert\leq2JCV. r = j − J + 1 ∑ j B ∈ B r ∑ Q ⊆ B ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 r = j − J + 1 ∑ j ∫ V r n sup ∣ K n ∣ ≤ 2 J C V . For r ≤ j − J r\leq j-J r ≤ j − J B ∈ B r B\in\mathcal{B}_r B ∈ B r
B = ⨆ P ∈ Q j − 1 P ⊆ B P . B=\bigsqcup_{\substack{P\in\mathcal{Q}_{j-1}\\P\subseteq B}}P. B = P ∈ Q j − 1 P ⊆ B ⨆ P . Since 2 − r ≥ 2 J − j ≥ A 2 1 − j 2^{-r} \geq2^{J-j} \geq A2^{1-j} 2 − r ≥ 2 J − j ≥ A 2 1 − j P P P
P ⊆ B ⊆ V r ⊆ ( [ − 2 − r , 2 − r ) d ) c ⊆ ( A Q 0 ( j − 2 ) ) c . P \subseteq B \subseteq V_r \subseteq\left([-2^{-r},2^{-r})^d\right)^c \subseteq\left(AQ_0^{(j-2)}\right)^c. P ⊆ B ⊆ V r ⊆ ( [ − 2 − r , 2 − r ) d ) c ⊆ ( A Q 0 ( j − 2 ) ) c . If Q ∈ D ∩ E j Q \in\mathcal{D} \cap\mathcal{E}_j Q ∈ D ∩ E j Q ⊆ B Q \subseteq B Q ⊆ B ℓ ( Q ) ≤ 2 − j < 2 1 − j = ℓ ( P ) \ell(Q) \leq2^{-j} < 2^{1-j} = \ell(P) ℓ ( Q ) ≤ 2 − j < 2 1 − j = ℓ ( P ) Q ⊊ P Q \subsetneq P Q ⊊ P P P P P P P (35) , we obtain
∑ r ≤ j − J ∑ B ∈ B r ∑ Q ∈ D ∩ E j Q ⊆ B ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ ∑ P ∑ Q ⊆ P ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 12 ∑ P ∫ P sup n ∣ K n − E P K n ∣ . \sum_{r\leq j-J}\sum_{B\in\mathcal{B}_r}\sum_{\substack{Q\in\mathcal{D}\cap\mathcal{E}_j\\Q\subseteq B}} |Q|\sup_n|a_{n,Q}|
\leq\sum_P\sum_{Q\subseteq P}|Q|\sup_n|a_{n,Q}|
\leq12\sum_P\int_P\sup_n|K_n-\mathbb{E}_P K_n|. r ≤ j − J ∑ B ∈ B r ∑ Q ∈ D ∩ E j Q ⊆ B ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ P ∑ Q ⊆ P ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 12 P ∑ ∫ P n sup ∣ K n − E P K n ∣. Write D ( z , y ) = sup n ∣ K n ( z − y ) − K n ( z ) ∣ D(z,y)=\sup_n|K_n(z-y)-K_n(z)| D ( z , y ) = sup n ∣ K n ( z − y ) − K n ( z ) ∣
sup n ∣ K n ( z ) − E P K n ∣ ≤ 1 ∣ P ∣ ∫ P D ( z , z − w ) d w , z ∈ P . \sup_n|K_n(z)-\mathbb{E}_P K_n|
\leq\frac{1}{|P|}\int_P D(z,z-w)\,dw,\qquad z\in P. n sup ∣ K n ( z ) − E P K n ∣ ≤ ∣ P ∣ 1 ∫ P D ( z , z − w ) d w , z ∈ P . Since ∣ P ∣ = 2 ( 1 − j ) d |P|=2^{(1-j)d} ∣ P ∣ = 2 ( 1 − j ) d y = z − w y=z-w y = z − w
∑ P ∫ P sup n ∣ K n − E P K n ∣ ≤ 2 ( j − 1 ) d ∑ P ∫ P ∫ P D ( z , z − w ) d w d z = 2 ( j − 1 ) d ∫ ( − 2 1 − j , 2 1 − j ) d ∫ R d D ( z , y ) ∑ P 1 P ( z ) 1 P ( z − y ) d z d y \begin{aligned}
\sum_P\int_P\sup_n|K_n-\mathbb{E}_P K_n|
\leq2^{(j-1)d}\sum_P\int_P\int_P D(z,z-w)\,dw\,dz \\
&=2^{(j-1)d}\int_{(-2^{1-j},2^{1-j})^d}\int_{\mathbb{R}^d}D(z,y)\sum_P\mathbf{1}_P(z)\mathbf{1}_P(z-y)\,dz\,dy
\end{aligned} P ∑ ∫ P n sup ∣ K n − E P K n ∣ ≤ 2 ( j − 1 ) d P ∑ ∫ P ∫ P D ( z , z − w ) d w d z = 2 ( j − 1 ) d ∫ ( − 2 1 − j , 2 1 − j ) d ∫ R d D ( z , y ) P ∑ 1 P ( z ) 1 P ( z − y ) d z d y ≤ 2 ( j − 1 ) d ∫ ( − 2 1 − j , 2 1 − j ) d ∫ ( A Q 0 ( j − 2 ) ) c D ( z , y ) d z d y \leq2^{(j-1)d}\int_{(-2^{1-j},2^{1-j})^d}\int_{(AQ_0^{(j-2)})^c}D(z,y)\,dz\,dy ≤ 2 ( j − 1 ) d ∫ ( − 2 1 − j , 2 1 − j ) d ∫ ( A Q 0 ( j − 2 ) ) c D ( z , y ) d z d y ≤ 2 d C D . \leq2^d C_D. ≤ 2 d C D . Here disjointness gives ∑ P 1 P ( z ) 1 P ( z − y ) ≤ 1 \sum_P\mathbf{1}_P(z)\mathbf{1}_P(z-y)\leq1 ∑ P 1 P ( z ) 1 P ( z − y ) ≤ 1 (51) , since ( − 2 1 − j , 2 1 − j ) d ⊂ Q 0 ( j − 2 ) (-2^{1-j},2^{1-j})^d\subset Q_0^{(j-2)} ( − 2 1 − j , 2 1 − j ) d ⊂ Q 0 ( j − 2 )
∑ Q ∈ D ∩ E j ∣ Q ∣ sup n ∣ a n , Q ∣ ≤ 2 J C V + 12 ⋅ 2 d C D ≤ C d , A ( C V + C D ) , \sum_{Q\in\mathcal{D}\cap\mathcal{E}_j}|Q|\sup_n|a_{n,Q}|
\leq2J C_V+12\cdot2^d C_D\leq C_{d,A}(C_V+C_D), Q ∈ D ∩ E j ∑ ∣ Q ∣ n sup ∣ a n , Q ∣ ≤ 2 J C V + 12 ⋅ 2 d C D ≤ C d , A ( C V + C D ) , which proves (38) . □
Remark 4.2. The conclusions of Proposition 4.1 also hold for a finite family ( K n ) n = 1 N ⊂ L 1 , l o c ( R d ∖ { 0 } ) (K_n)_{n=1}^N\subset L_{1,\mathrm{loc}}(\mathbb{R}^d\setminus\{0\}) ( K n ) n = 1 N ⊂ L 1 , loc ( R d ∖ { 0 }) C D C_D C D C V C_V C V B ∈ ⋃ j B j B\in\bigcup_j\mathcal{B}_j B ∈ ⋃ j B j
∫ B max 1 ≤ n ≤ N ∣ K n ∣ ≤ ∑ n = 1 N ∫ B ∣ K n ∣ < ∞ , \int_B\max_{1\leq n\leq N}|K_n|
\leq\sum_{n=1}^N\int_B|K_n|<\infty, ∫ B 1 ≤ n ≤ N max ∣ K n ∣ ≤ n = 1 ∑ N ∫ B ∣ K n ∣ < ∞ , so the recursion and the coefficient estimates remain valid. For B ∈ B j B\in\mathcal{B}_j B ∈ B j (42) ,
∫ B max 1 ≤ n ≤ N ∣ K n − ∑ ℓ = 0 k ∑ Q ∈ G ℓ ( B ) a n , Q 1 Q ∣ ≤ 6 ∑ n = 1 N ∥ K n − E j + k ( 1 B K n ) ∥ L 1 ( B ) ⟶ 0. \int_B\max_{1\leq n\leq N}\left|K_n-\sum_{\ell=0}^k\sum_{Q\in\mathcal{G}_\ell(B)}a_{n,Q}\mathbf{1}_Q\right|
\leq6\sum_{n=1}^N\|K_n-\mathbb{E}_{j+k}(\mathbf{1}_B K_n)\|_{L_1(B)}
\longrightarrow0. ∫ B 1 ≤ n ≤ N max K n − ℓ = 0 ∑ k Q ∈ G ℓ ( B ) ∑ a n , Q 1 Q ≤ 6 n = 1 ∑ N ∥ K n − E j + k ( 1 B K n ) ∥ L 1 ( B ) ⟶ 0. The identity
a n , Q 1 Q = ( ∣ Q ∣ a n , Q ) 1 Q ∣ Q ∣ a_{n,Q}\mathbf{1}_Q=(|Q|a_{n,Q})\frac{\mathbf{1}_Q}{|Q|} a n , Q 1 Q = ( ∣ Q ∣ a n , Q ) ∣ Q ∣ 1 Q explains the appearance of ∣ Q ∣ a n , Q |Q|a_{n,Q} ∣ Q ∣ a n , Q (37) ; the other conclusions allow ∑ Q ∣ Q ∣ ∣ a n , Q ∣ = ∞ \sum_Q|Q||a_{n,Q}|=\infty ∑ Q ∣ Q ∣∣ a n , Q ∣ = ∞
Related dyadic estimates in terms of local oscillation appear in Lerner [16 ] and in the Banach space valued decomposition of Hänninen–Hytönen [9 Theorem 2.16] . Here the two selection conditions give both the individual estimate in (34) and the common estimate (35) , while retaining the exact expansion for complex-valued kernels.
Finally, if sup n ∥ K n ∥ 1 ≤ A 1 \sup_n \lVert K_n\rVert_1 \le A_1 sup n ∥ K n ∥ 1 ≤ A 1 C V < ∞ C_V < \infty C V < ∞ (51) . To see this, let Q a = [ − a / 2 , a / 2 ) d Q_a=[-a/2,a/2)^d Q a = [ − a /2 , a /2 ) d a a a y ∈ Q a y\in Q_a y ∈ Q a
∣ K n ( z ) ∣ ≤ a − d A 1 + 1 ∣ Q a ∣ ∫ Q a sup m ∣ K m ( z − y ) − K m ( z ) ∣ d y . \lvert K_n(z)\rvert\le a^{-d}A_1+\frac{1}{\lvert Q_a\rvert}\int_{Q_a}\sup_m\lvert K_m(z-y)-K_m(z)\rvert\,dy. ∣ K n ( z )∣ ≤ a − d A 1 + ∣ Q a ∣ 1 ∫ Q a m sup ∣ K m ( z − y ) − K m ( z )∣ d y . Taking the supremum over n n n U ⊂ ( A Q a ) c U\subset(AQ_a)^c U ⊂ ( A Q a ) c
∫ U sup n ∣ K n ∣ ≤ ∣ U ∣ a − d A 1 + C D . \int_U\sup_n\lvert K_n\rvert\le\lvert U\rvert a^{-d}A_1+C_D. ∫ U n sup ∣ K n ∣ ≤ ∣ U ∣ a − d A 1 + C D . For each j j j a a a 2 − j / A < a ≤ 2 1 − j / A 2^{-j}/A<a\le2^{1-j}/A 2 − j / A < a ≤ 2 1 − j / A U = V j U=V_j U = V j V j ⊂ ( A Q a ) c V_j\subset(AQ_a)^c V j ⊂ ( A Q a ) c ∣ V j ∣ a − d ≤ C d , A \lvert V_j\rvert a^{-d}\le C_{d,A} ∣ V j ∣ a − d ≤ C d , A C V ≤ C d , A A 1 + C D C_V\le C_{d,A}A_1+C_D C V ≤ C d , A A 1 + C D (37) and (38) give the two coefficient bounds required for integrable kernels.
Dyadic representations of convolution operators
We use the kernel expansion from Proposition 4.1 to express K n ∗ f K_n*f K n ∗ f E Q f E_Qf E Q f
Write Q ^ \widehat Q Q Q Q Q R R R Q ∈ Q Q\in\mathcal Q Q ∈ Q
X Q R = { x ∈ R d : Q ⊆ x − R , Q ^ ⊈ x − R } . X_Q^R=\{x\in\mathbb{R}^d:Q\subseteq x-R,\ \widehat Q\not\subseteq x-R\}. X Q R = { x ∈ R d : Q ⊆ x − R , Q ⊆ x − R } . Thus x ∈ X Q R x\in X_Q^R x ∈ X Q R Q Q Q x − R x-R x − R
For each x x x x − R x-R x − R x − R x-R x − R
1 R ( x − y ) = ∑ Q ∈ Q 1 X Q R ( x ) 1 Q ( y ) (43) \mathbf{1}_R(x-y)=\sum_{Q\in\mathcal Q}\mathbf{1}_{X_Q^R}(x)\mathbf{1}_Q(y) \tag*{(43)} 1 R ( x − y ) = Q ∈ Q ∑ 1 X Q R ( x ) 1 Q ( y ) (43) for almost every ( x , y ) (x,y) ( x , y )
For any Bochner integrable function f f f
( 1 R ∗ f ) ( x ) = ∑ Q ∈ Q ∣ Q ∣ 1 X Q R ( x ) E Q f (44) (\mathbf{1}_R*f)(x)=\sum_{Q\in\mathcal Q}\lvert Q\rvert\mathbf{1}_{X_Q^R}(x)E_Qf \tag*{(44)} ( 1 R ∗ f ) ( x ) = Q ∈ Q ∑ ∣ Q ∣ 1 X Q R ( x ) E Q f (44) for almost every x x x
If ℓ ( Q ) ≤ ℓ ( R ) \ell(Q)\le\ell(R) ℓ ( Q ) ≤ ℓ ( R ) Q Q Q Q ^ \widehat Q Q ( ℓ ( R ) − ℓ ( Q ) ) d (\ell(R)-\ell(Q))^d ( ℓ ( R ) − ℓ ( Q ) ) d ( ℓ ( R ) − 2 ℓ ( Q ) ) + d (\ell(R)-2\ell(Q))_+^d ( ℓ ( R ) − 2 ℓ ( Q ) ) + d
∣ X Q R ∣ = ( ℓ ( R ) − ℓ ( Q ) ) d − ( ℓ ( R ) − 2 ℓ ( Q ) ) + d ≤ d ℓ ( Q ) ℓ ( R ) d − 1 . (45) \lvert X_Q^R\rvert=(\ell(R)-\ell(Q))^d-(\ell(R)-2\ell(Q))_+^d\le d\ell(Q)\ell(R)^{d-1}. \tag*{(45)} ∣ X Q R ∣ = ( ℓ ( R ) − ℓ ( Q ) ) d − ( ℓ ( R ) − 2 ℓ ( Q ) ) + d ≤ d ℓ ( Q ) ℓ ( R ) d − 1 . (45) If ℓ ( Q ) > ℓ ( R ) \ell(Q)>\ell(R) ℓ ( Q ) > ℓ ( R ) X Q R = ∅ X_Q^R=\varnothing X Q R = ∅
Let ( K n ) n ≥ 1 (K_n)_{n\ge1} ( K n ) n ≥ 1 ( a n , R ) (a_{n,R}) ( a n , R ) Q ∈ Q Q\in\mathcal Q Q ∈ Q
b n , Q ( x ) = ∣ Q ∣ ∑ R ∈ D a n , R 1 X Q R ( x ) , m Q ( x ) = ∣ Q ∣ ∑ R ∈ D sup n ∣ a n , R ∣ 1 X Q R ( x ) . b_{n,Q}(x)=\lvert Q\rvert\sum_{R\in\mathcal D}a_{n,R}\mathbf{1}_{X_Q^R}(x),\qquad m_Q(x)=\lvert Q\rvert\sum_{R\in\mathcal D}\sup_n\lvert a_{n,R}\rvert\mathbf{1}_{X_Q^R}(x). b n , Q ( x ) = ∣ Q ∣ R ∈ D ∑ a n , R 1 X Q R ( x ) , m Q ( x ) = ∣ Q ∣ R ∈ D ∑ n sup ∣ a n , R ∣ 1 X Q R ( x ) . Lemma 4.3. For every Q ∈ Q Q \in\mathcal{Q} Q ∈ Q b n , Q b_{n,Q} b n , Q m Q m_Q m Q L 1 ( R d ) L_1(\mathbb{R}^d) L 1 ( R d ) R d \mathbb{R}^d R d
∣ b n , Q ∣ ≤ m Q , ∥ m Q ∥ ∞ ≤ C d , A ( C V + C D ) , ∥ m Q ∥ 1 ≤ C d , A ( C V + C D ) ∣ Q ∣ . (46) \lvert b_{n,Q}\rvert\leq m_Q,\qquad\lVert m_Q\rVert_\infty\leq C_{d,A}(C_V+C_D),\qquad\lVert m_Q\rVert_1 \leq C_{d,A}(C_V+C_D)\lvert Q\rvert. \tag*{(46)} ∣ b n , Q ∣ ≤ m Q , ∥ m Q ∥ ∞ ≤ C d , A ( C V + C D ) , ∥ m Q ∥ 1 ≤ C d , A ( C V + C D ) ∣ Q ∣ . (46) Proof. Every R ∈ D R \in\mathcal{D} R ∈ D ρ R > ℓ ( R ) \rho_R>\ell(R) ρ R > ℓ ( R ) (38) , we have
∑ R ∈ D ℓ ( R ) = 2 − s ∣ R ∣ sup n ∣ a n , R ∣ ≤ ∑ R ∈ D ∩ E s ∣ R ∣ sup n ∣ a n , R ∣ ≤ C d , A ( C V + C D ) . (47) \sum_{\substack{R\in\mathcal{D}\\ \ell(R)=2^{-s}}} \lvert R\rvert\sup_n \lvert a_{n,R}\rvert
\leq
\sum_{R\in\mathcal{D}\cap E_s} \lvert R\rvert\sup_n \lvert a_{n,R}\rvert
\leq C_{d,A}(C_V+C_D). \tag*{(47)} R ∈ D ℓ ( R ) = 2 − s ∑ ∣ R ∣ n sup ∣ a n , R ∣ ≤ R ∈ D ∩ E s ∑ ∣ R ∣ n sup ∣ a n , R ∣ ≤ C d , A ( C V + C D ) . (47) Fix Q ∈ Q q Q \in\mathcal{Q}_q Q ∈ Q q X Q R = ∅ X_Q^R=\varnothing X Q R = ∅ ℓ ( R ) < ℓ ( Q ) \ell(R)<\ell(Q) ℓ ( R ) < ℓ ( Q )
∑ R ∈ D ∥ ∣ Q ∣ sup n ∣ a n , R ∣ 1 X Q R ∥ ∞ ≤ ∑ s ≤ q 2 − d ( q − s ) ∑ R ∈ D ℓ ( R ) = 2 − s ∣ R ∣ sup n ∣ a n , R ∣ ≤ C d , A ( C V + C D ) ∑ s ≤ q 2 − d ( q − s ) ≤ C d , A ( C V + C D ) . \sum_{R\in\mathcal{D}}
\left\lVert\lvert Q\rvert\sup_n\lvert a_{n,R}\rvert\mathbf{1}_{X_Q^R}\right\rVert_\infty
\leq
\sum_{s\leq q}2^{-d(q-s)}
\sum_{\substack{R\in\mathcal{D}\\ \ell(R)=2^{-s}}}
\lvert R\rvert\sup_n\lvert a_{n,R}\rvert
\leq
C_{d,A}(C_V+C_D)\sum_{s\leq q}2^{-d(q-s)}
\leq C_{d,A}(C_V+C_D). R ∈ D ∑ ∣ Q ∣ n sup ∣ a n , R ∣ 1 X Q R ∞ ≤ s ≤ q ∑ 2 − d ( q − s ) R ∈ D ℓ ( R ) = 2 − s ∑ ∣ R ∣ n sup ∣ a n , R ∣ ≤ C d , A ( C V + C D ) s ≤ q ∑ 2 − d ( q − s ) ≤ C d , A ( C V + C D ) . Similarly, by (45) ,
∑ R ∈ D ∥ ∣ Q ∣ sup n ∣ a n , R ∣ 1 X Q R ∥ 1 ≤ d ∣ Q ∣ ∑ s ≤ q 2 − ( q − s ) ∑ R ∈ D ℓ ( R ) = 2 − s ∣ R ∣ sup n ∣ a n , R ∣ ≤ C d , A ( C V + C D ) ∣ Q ∣ . \sum_{R\in\mathcal{D}}
\left\lVert\lvert Q\rvert\sup_n\lvert a_{n,R}\rvert\mathbf{1}_{X_Q^R}\right\rVert_1
\leq
d\lvert Q\rvert\sum_{s\leq q}2^{-(q-s)}
\sum_{\substack{R\in\mathcal{D}\\ \ell(R)=2^{-s}}}
\lvert R\rvert\sup_n\lvert a_{n,R}\rvert
\leq C_{d,A}(C_V+C_D)\lvert Q\rvert. R ∈ D ∑ ∣ Q ∣ n sup ∣ a n , R ∣ 1 X Q R 1 ≤ d ∣ Q ∣ s ≤ q ∑ 2 − ( q − s ) R ∈ D ℓ ( R ) = 2 − s ∑ ∣ R ∣ n sup ∣ a n , R ∣ ≤ C d , A ( C V + C D ) ∣ Q ∣ . These estimates prove absolute convergence in L 1 L_1 L 1 m Q m_Q m Q ∣ a n , R ∣ ≤ sup m ∣ a m , R ∣ \lvert a_{n,R}\rvert\leq\sup_m\lvert a_{m,R}\rvert ∣ a n , R ∣ ≤ sup m ∣ a m , R ∣ b n , Q b_{n,Q} b n , Q ∣ b n , Q ∣ ≤ m Q \lvert b_{n,Q}\rvert\leq m_Q ∣ b n , Q ∣ ≤ m Q
The next proposition expresses convolution in terms of dyadic averages and controls the coefficients outside enlarged dyadic cubes.
Proposition 4.4. Let X \mathcal{X} X f ∈ L 1 ( R d ; X ) f\in L_1(\mathbb{R}^d;\mathcal{X}) f ∈ L 1 ( R d ; X ) n ≥ 1 n\geq1 n ≥ 1
( K n ∗ f ) ( x ) = ∑ Q ∈ Q b n , Q ( x ) E Q f for almost every x ∈ R d . (48) (K_n*f)(x)=\sum_{Q\in\mathcal{Q}}b_{n,Q}(x)E_Qf
\quad\text{for almost every }x\in\mathbb{R}^d. \tag*{(48)} ( K n ∗ f ) ( x ) = Q ∈ Q ∑ b n , Q ( x ) E Q f for almost every x ∈ R d . (48) For every bounded measurable set E ⊂ R d E\subset\mathbb{R}^d E ⊂ R d
∫ E ∑ Q ∈ Q m Q ( x ) ∥ E Q f ∥ X d x < ∞ . (49) \int_E\sum_{Q\in\mathcal{Q}}m_Q(x)\lVert E_Qf\rVert_{\mathcal{X}}\,dx<\infty. \tag*{(49)} ∫ E Q ∈ Q ∑ m Q ( x ) ∥ E Q f ∥ X d x < ∞. (49) In particular, the series in (48) converges absolutely almost everywhere and in L 1 ( E ; X ) L_1(E;\mathcal{X}) L 1 ( E ; X ) P ∈ Q P\in\mathcal{Q} P ∈ Q β ≥ 3 \beta\geq3 β ≥ 3
∑ Q ⊆ P y ∈ Q 1 ∣ Q ∣ ∫ ( β P ) c m Q ( x ) d x ≤ C d , A ( C V + C D ) for almost every y ∈ P . (50) \sum_{\substack{Q\subseteq P\\ y\in Q}}\frac{1}{\lvert Q\rvert}\int_{(\beta P)^c}m_Q(x)\,dx
\leq C_{d,A}(C_V+C_D)
\quad\text{for almost every }y\in P. \tag*{(50)} Q ⊆ P y ∈ Q ∑ ∣ Q ∣ 1 ∫ ( β P ) c m Q ( x ) d x ≤ C d , A ( C V + C D ) for almost every y ∈ P . (50) Proof. Since V j V_j V j B j \mathcal{B}_j B j R ∈ D R\in\mathcal{D} R ∈ D B ∈ ⋃ k B k B\in\bigcup_k\mathcal{B}_k B ∈ ⋃ k B k
∫ V j ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x ) d x = ∑ R ∈ D sup n ∣ a n , R ∣ ∣ R ∩ V j ∣ = ∑ B ∈ B j ∑ R ∈ D R ⊂ B ∣ R ∣ sup n ∣ a n , R ∣ . \int_{V_j}\sum_{R\in\mathcal{D}}\sup_n\lvert a_{n,R}\rvert\mathbf{1}_R(x)\,dx
=
\sum_{R\in\mathcal{D}}\sup_n\lvert a_{n,R}\rvert\lvert R\cap V_j\rvert
=
\sum_{B\in\mathcal{B}_j}\sum_{\substack{R\in\mathcal{D}\\R\subset B}}\lvert R\rvert\sup_n\lvert a_{n,R}\rvert. ∫ V j R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x ) d x = R ∈ D ∑ n sup ∣ a n , R ∣ ∣ R ∩ V j ∣ = B ∈ B j ∑ R ∈ D R ⊂ B ∑ ∣ R ∣ n sup ∣ a n , R ∣ . Applying (34) on each B ∈ B j B \in\mathcal{B}_j B ∈ B j
∫ V j ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x ) d x ≤ 2 ∑ B ∈ B j ∫ B sup n ∣ K n ( x ) ∣ d x ≤ 2 C V . \int_{V_j} \sum_{R \in\mathcal{D}} \sup_n |a_{n,R}|\mathbf{1}_R(x)\,dx \le2\sum_{B \in\mathcal{B}_j}\int_B \sup_n |K_n(x)|\,dx \le2C_{V}. ∫ V j R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x ) d x ≤ 2 B ∈ B j ∑ ∫ B n sup ∣ K n ( x ) ∣ d x ≤ 2 C V . For M > 0 M>0 M > 0 B ∈ ⋃ j B j B \in\bigcup_j \mathcal{B}_j B ∈ ⋃ j B j ∣ B ∩ [ − M , M ] d ∣ > 0 |B\cap[-M,M]^d|>0 ∣ B ∩ [ − M , M ] d ∣ > 0 [ − 2 M , 2 M ] d [-2M,2M]^d [ − 2 M , 2 M ] d (34) , we have
∫ [ − M , M ] d ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x ) d x ≤ 2 ∫ [ − 2 M , 2 M ] d sup n ∣ K n ∣ ≤ 2 ( 4 M ) d sup n ∥ K n ∥ ∞ . \int_{[-M,M]^d}\sum_{R\in\mathcal{D}}\sup_n |a_{n,R}|\mathbf{1}_R(x)\,dx
\le2\int_{[-2M,2M]^d}\sup_n |K_n|
\le2(4M)^d\sup_n\|K_n\|_\infty. ∫ [ − M , M ] d R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x ) d x ≤ 2 ∫ [ − 2 M , 2 M ] d n sup ∣ K n ∣ ≤ 2 ( 4 M ) d n sup ∥ K n ∥ ∞ . Let E ⊆ [ − M , M ] d E\subseteq[-M,M]^d E ⊆ [ − M , M ] d ∣ y ∣ ∞ ≤ 2 M |y|_\infty\le2M ∣ y ∣ ∞ ≤ 2 M E − y ⊆ [ − 3 M , 3 M ] d E-y\subseteq[-3M,3M]^d E − y ⊆ [ − 3 M , 3 M ] d
∫ E ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 2 ( 12 M ) d sup n ∥ K n ∥ ∞ . \int_E\sum_{R\in\mathcal{D}}\sup_n |a_{n,R}|\mathbf{1}_R(x-y)\,dx
\le2(12M)^d\sup_n\|K_n\|_\infty. ∫ E R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 2 ( 12 M ) d n sup ∥ K n ∥ ∞ . If ∣ y ∣ ∞ > 2 M |y|_\infty>2M ∣ y ∣ ∞ > 2 M z ∈ E − y z\in E-y z ∈ E − y
∣ y ∣ ∞ − M ≤ ∣ z ∣ ∞ ≤ ∣ y ∣ ∞ + M , ∣ y ∣ ∞ + M ∣ y ∣ ∞ − M < 3. |y|_\infty-M\le|z|_\infty\le|y|_\infty+M,\qquad
\frac{|y|_\infty+M}{|y|_\infty-M}<3. ∣ y ∣ ∞ − M ≤ ∣ z ∣ ∞ ≤ ∣ y ∣ ∞ + M , ∣ y ∣ ∞ − M ∣ y ∣ ∞ + M < 3. Thus E − y E-y E − y V j V_j V j
∫ E ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 8 C V . \int_E\sum_{R\in\mathcal{D}}\sup_n |a_{n,R}|\mathbf{1}_R(x-y)\,dx\le8C_V. ∫ E R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 8 C V . Consequently, we have
sup y ∈ R d ∫ E ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 2 ( 12 M ) d sup n ∥ K n ∥ ∞ + 8 C V < ∞ . \sup_{y\in\mathbb{R}^d}\int_E\sum_{R\in\mathcal{D}}\sup_n |a_{n,R}|\mathbf{1}_R(x-y)\,dx
\le2(12M)^d\sup_n\|K_n\|_\infty+8C_V<\infty. y ∈ R d sup ∫ E R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x − y ) d x ≤ 2 ( 12 M ) d n sup ∥ K n ∥ ∞ + 8 C V < ∞. Since ∣ Q ∣ ∥ E Q f ∥ X ≤ ∫ Q ∥ f ( y ) ∥ X d y |Q|\|\mathbb{E}_Qf\|_X\le\int_Q\|f(y)\|_X\,dy ∣ Q ∣∥ E Q f ∥ X ≤ ∫ Q ∥ f ( y ) ∥ X d y (43) implies
∫ E ∑ Q ∈ Q m Q ( x ) ∥ E Q f ∥ X d x ≤ ∫ E ∫ R d ∥ f ( y ) ∥ X ∑ R ∈ D sup n ∣ a n , R ∣ ∑ Q ∈ Q 1 Q R X ( x ) 1 Q ( y ) d y d x = ∫ R d ∥ f ( y ) ∥ X ∫ E ∑ R ∈ D sup n ∣ a n , R ∣ 1 R ( x − y ) d x d y ≤ ( 2 ( 12 M ) d sup n ∥ K n ∥ ∞ + 8 C V ) ∥ f ∥ L 1 ( R d ; X ) < ∞ . \begin{aligned}
\int_E\sum_{Q\in\mathcal{Q}}m_Q(x)\|\mathbb{E}_Qf\|_X\,dx
&\le\int_E\int_{\mathbb{R}^d}\|f(y)\|_X\sum_{R\in\mathcal{D}}\sup_n|a_{n,R}|\sum_{Q\in\mathcal{Q}}\mathbf{1}_{Q_R^X}(x)\mathbf{1}_Q(y)\,dy\,dx\\
&=\int_{\mathbb{R}^d}\|f(y)\|_X\int_E\sum_{R\in\mathcal{D}}\sup_n|a_{n,R}|\mathbf{1}_R(x-y)\,dx\,dy\\
&\le\left(2(12M)^d\sup_n\|K_n\|_\infty+8C_V\right)\|f\|_{L^1(\mathbb{R}^d;X)}<\infty.
\end{aligned} ∫ E Q ∈ Q ∑ m Q ( x ) ∥ E Q f ∥ X d x ≤ ∫ E ∫ R d ∥ f ( y ) ∥ X R ∈ D ∑ n sup ∣ a n , R ∣ Q ∈ Q ∑ 1 Q R X ( x ) 1 Q ( y ) d y d x = ∫ R d ∥ f ( y ) ∥ X ∫ E R ∈ D ∑ n sup ∣ a n , R ∣ 1 R ( x − y ) d x d y ≤ ( 2 ( 12 M ) d n sup ∥ K n ∥ ∞ + 8 C V ) ∥ f ∥ L 1 ( R d ; X ) < ∞. This proves (49) . By (36) and (44) ,
K n ∗ f = ∑ R ∈ D a n , R ( 1 R ∗ f ) = ∑ R ∈ D ∑ Q ∈ Q a n , R ∣ Q ∣ 1 Q R X E Q f = ∑ Q ∈ Q b n , Q E Q f K_n*f=\sum_{R\in\mathcal{D}}a_{n,R}(\mathbf{1}_R*f)
=\sum_{R\in\mathcal{D}}\sum_{Q\in\mathcal{Q}}a_{n,R}|Q|\mathbf{1}_{Q_R^X}\mathbb{E}_Qf
=\sum_{Q\in\mathcal{Q}}b_{n,Q}\mathbb{E}_Qf K n ∗ f = R ∈ D ∑ a n , R ( 1 R ∗ f ) = R ∈ D ∑ Q ∈ Q ∑ a n , R ∣ Q ∣ 1 Q R X E Q f = Q ∈ Q ∑ b n , Q E Q f almost everywhere and in L 1 ( E ; X ) L_1(E;X) L 1 ( E ; X ) (48) .
It remains to prove (50) . Fix P ∈ Q j P\in\mathcal{Q}_j P ∈ Q j β ≥ 3 \beta\ge3 β ≥ 3 y ∈ P y\in P y ∈ P Q ⊆ P Q\subseteq P Q ⊆ P y ∈ Q y\in Q y ∈ Q x ∈ Q R X ∖ β P x\in Q_R^X\setminus\beta P x ∈ Q R X ∖ β P x − y ∈ R x-y\in R x − y ∈ R ∣ x − y ∣ ∞ > 2 − j |x-y|_\infty>2^{-j} ∣ x − y ∣ ∞ > 2 − j ρ R > 2 − j \rho_R>2^{-j} ρ R > 2 − j k ≥ 0 k\ge0 k ≥ 0 Q ∈ Q j + k Q\in\mathcal{Q}_{j+k} Q ∈ Q j + k y ∈ Q ⊆ P y\in Q\subseteq P y ∈ Q ⊆ P ℓ ( Q ) ≤ ℓ ( R ) \ell(Q)\le\ell(R) ℓ ( Q ) ≤ ℓ ( R )
∑ Q ⊆ P , y ∈ Q ℓ ( Q ) ≤ ℓ ( R ) ℓ ( Q ) = ∑ k ≥ 0 2 − j − k ≤ ℓ ( R ) 2 − j − k = 2 min { 2 − j , ℓ ( R ) } . \sum_{\substack{Q\subseteq P,\ y\in Q\\ \ell(Q)\le\ell(R)}}\ell(Q)
=\sum_{\substack{k\ge0\\2^{-j-k}\le\ell(R)}}2^{-j-k}
=2\min\{2^{-j},\ell(R)\}. Q ⊆ P , y ∈ Q ℓ ( Q ) ≤ ℓ ( R ) ∑ ℓ ( Q ) = k ≥ 0 2 − j − k ≤ ℓ ( R ) ∑ 2 − j − k = 2 min { 2 − j , ℓ ( R )} . By (45) and the preceding identity, we have
∑ Q ⊆ P y ∈ Q 1 ∣ Q ∣ ∫ ( β P ) c m Q ( x ) d x = ∑ R ∈ D sup n ∣ a n , R ∣ ∑ Q ⊆ P y ∈ Q ∣ X Q R ∖ β P ∣ ≤ d ∑ R ∈ D ρ R > 2 − j ℓ ( R ) d − 1 sup n ∣ a n , R ∣ ∑ Q ⊆ P , y ∈ Q ℓ ( Q ) ≤ ℓ ( R ) ℓ ( Q ) = 2 d ∑ R ∈ D ρ R > 2 − j ∣ R ∣ sup n ∣ a n , R ∣ min { 1 , 2 − j ℓ ( R ) } . \begin{aligned}
\sum_{\substack{Q\subseteq P\\y\in Q}}\frac{1}{|Q|}\int_{(\beta P)^{c}}m_Q(x)\,dx
=\sum_{R\in\mathcal{D}}\sup_n|a_{n,R}|\sum_{\substack{Q\subseteq P\\y\in Q}}|X_Q^R\setminus\beta P| \\
&\leq d\sum_{\substack{R\in\mathcal{D}\\\rho_R>2^{-j}}}\ell(R)^{d-1}\sup_n|a_{n,R}|\sum_{\substack{Q\subseteq P,\,y\in Q\\\ell(Q)\leq\ell(R)}}\ell(Q) \\
&=2d\sum_{\substack{R\in\mathcal{D}\\\rho_R>2^{-j}}}|R|\sup_n|a_{n,R}|\min\left\{1,\frac{2^{-j}}{\ell(R)}\right\}.
\end{aligned} Q ⊆ P y ∈ Q ∑ ∣ Q ∣ 1 ∫ ( β P ) c m Q ( x ) d x = R ∈ D ∑ n sup ∣ a n , R ∣ Q ⊆ P y ∈ Q ∑ ∣ X Q R ∖ β P ∣ ≤ d R ∈ D ρ R > 2 − j ∑ ℓ ( R ) d − 1 n sup ∣ a n , R ∣ Q ⊆ P , y ∈ Q ℓ ( Q ) ≤ ℓ ( R ) ∑ ℓ ( Q ) = 2 d R ∈ D ρ R > 2 − j ∑ ∣ R ∣ n sup ∣ a n , R ∣ min { 1 , ℓ ( R ) 2 − j } . Splitting the last sum according to ℓ ( R ) ≤ 2 − j \ell(R)\leq2^{-j} ℓ ( R ) ≤ 2 − j ℓ ( R ) = 2 k − j \ell(R)=2^{k-j} ℓ ( R ) = 2 k − j k ≥ 1 k\geq1 k ≥ 1 (38) and (47) , we obtain
∑ R ∈ D ρ R > 2 − j ∣ R ∣ sup n ∣ a n , R ∣ min { 1 , 2 − j ℓ ( R ) } ≤ ∑ R ∈ D ∩ E j ∣ R ∣ sup n ∣ a n , R ∣ + ∑ k ≥ 1 2 − k ∑ ℓ ( R ) = 2 k − j ∣ R ∣ sup n ∣ a n , R ∣ ≤ C d , A ( C V + C D ) ( 1 + ∑ k ≥ 1 2 − k ) . \begin{aligned}
\sum_{\substack{R\in\mathcal{D}\\\rho_R>2^{-j}}}|R|\sup_n|a_{n,R}|\min\left\{1,\frac{2^{-j}}{\ell(R)}\right\}
\leq\sum_{R\in\mathcal{D}\cap E_j}|R|\sup_n|a_{n,R}|+\sum_{k\geq1}2^{-k}\sum_{\ell(R)=2^{k-j}}|R|\sup_n|a_{n,R}| \\
&\leq C_{d,A}(C_V+C_D)\left(1+\sum_{k\geq1}2^{-k}\right).
\end{aligned} R ∈ D ρ R > 2 − j ∑ ∣ R ∣ n sup ∣ a n , R ∣ min { 1 , ℓ ( R ) 2 − j } ≤ R ∈ D ∩ E j ∑ ∣ R ∣ n sup ∣ a n , R ∣ + k ≥ 1 ∑ 2 − k ℓ ( R ) = 2 k − j ∑ ∣ R ∣ n sup ∣ a n , R ∣ ≤ C d , A ( C V + C D ) ( 1 + k ≥ 1 ∑ 2 − k ) . Combining these estimates proves (50) . □
Proofs of the main theorems
In this section, we prove Theorems 1.1 and 1.2 . We first recall an equivalent form of the joint L 1 L_1 L 1 § 4 and § 5 to prove the main theorems.
The joint L 1 L_1 L 1
Let ( K λ ) λ ∈ Λ (K_\lambda)_{\lambda\in\Lambda} ( K λ ) λ ∈ Λ R d ∖ { 0 } \mathbb{R}^d\setminus\{0\} R d ∖ { 0 } joint L 1 L_1 L 1 if there exists a finite constant C H C_H C H
(H) sup y ∈ R d ∫ ∣ x ∣ ≥ 2 ∣ y ∣ sup λ ∈ Λ ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ C H . \sup_{y\in\mathbb{R}^d}\int_{|x|\geq2|y|}\sup_{\lambda\in\Lambda}|K_\lambda(x-y)-K_\lambda(x)|\,dx\leq C_H. sup y ∈ R d ∫ ∣ x ∣ ≥ 2∣ y ∣ sup λ ∈ Λ ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ C H .
For later use, we record an equivalent formulation in terms of centred cubes with dyadic side lengths. For k ∈ Z k\in\mathbb{Z} k ∈ Z
Q 0 ( k ) = [ − 2 − k − 1 , 2 − k − 1 ] d . Q_0^{(k)}=[-2^{-k-1},2^{-k-1}]^d. Q 0 ( k ) = [ − 2 − k − 1 , 2 − k − 1 ] d . Proposition 5.1. Condition (H) holds if and only if there exist constants A ≥ 1 A\geq1 A ≥ 1 and C D < ∞ C_D<\infty C D < ∞ such that
sup k ∈ Z sup y ∈ Q 0 ( k ) ∫ ( A Q 0 ( k ) ) c sup λ ∈ Λ ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ C D . (51) \sup_{k\in\mathbb{Z}}\sup_{y\in Q_0^{(k)}}\int_{(AQ_0^{(k)})^c}\sup_{\lambda\in\Lambda}|K_\lambda(x-y)-K_\lambda(x)|\,dx\leq C_D. \tag*{(51)} k ∈ Z sup y ∈ Q 0 ( k ) sup ∫ ( A Q 0 ( k ) ) c λ ∈ Λ sup ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ C D . (51) Proof. Suppose first that (H) holds. If y ∈ Q 0 ( k ) y\in Q_0^{(k)} y ∈ Q 0 ( k ) x ∉ A Q 0 ( k ) x\notin AQ_0^{(k)} x ∈ / A Q 0 ( k ) A > 2 d A>2\sqrt{d} A > 2 d
∣ x ∣ ≥ A 2 2 − k ≥ d 2 − k ≥ 2 ∣ y ∣ . |x|\geq\frac{A}{2}2^{-k}\geq\sqrt{d}2^{-k}\geq2|y|. ∣ x ∣ ≥ 2 A 2 − k ≥ d 2 − k ≥ 2∣ y ∣. Thus (51) follows with C D ≤ C H C_D \le C_H C D ≤ C H
Conversely, suppose that (51) holds. Fix y ≠ 0 y \ne0 y = 0 m = ⌈ 4 A d ⌉ m = \lceil4A\sqrt{d}\rceil m = ⌈ 4 A d ⌉ y ′ = y / m y' = y/m y ′ = y / m k ∈ Z k \in\mathbb{Z} k ∈ Z 2 ∣ y ∣ ∞ < 2 − k ≤ 4 ∣ y ′ ∣ ∞ 2|y|_{\infty} < 2^{-k} \le4|y'|_{\infty} 2∣ y ∣ ∞ < 2 − k ≤ 4∣ y ′ ∣ ∞ y ′ ∈ Q 0 ( k ) y' \in Q_0^{(k)} y ′ ∈ Q 0 ( k ) z ∈ A Q 0 ( k ) z \in A Q_0^{(k)} z ∈ A Q 0 ( k ) ∣ z ∣ ≤ ∣ y ∣ 2 |z| \le\frac{|y|}{2} ∣ z ∣ ≤ 2 ∣ y ∣ ∣ x ∣ ≥ 2 ∣ y ∣ |x| \ge2|y| ∣ x ∣ ≥ 2∣ y ∣ 0 ≤ j < m 0 \le j < m 0 ≤ j < m
∣ x − j y ′ ∣ ≥ ∣ x ∣ − j ∣ y ′ ∣ > ∣ y ∣ , |x-jy'| \ge|x|-j|y'| > |y|, ∣ x − j y ′ ∣ ≥ ∣ x ∣ − j ∣ y ′ ∣ > ∣ y ∣ , so x − j y ′ ∉ A Q 0 ( k ) x-jy' \notin A Q_0^{(k)} x − j y ′ ∈ / A Q 0 ( k )
∫ ∣ x ∣ ≥ 2 ∣ y ∣ sup λ ∈ Λ ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ ∑ j = 0 m − 1 ∫ ∣ x ∣ ≥ 2 ∣ y ∣ sup λ ∈ Λ ∣ K λ ( x − ( j + 1 ) y ′ ) − K λ ( x − j y ′ ) ∣ d x ≤ m ∫ ( A Q 0 ( k ) ) c sup λ ∈ Λ ∣ K λ ( z − y ′ ) − K λ ( z ) ∣ d z ≤ m C D . \begin{aligned}
\int_{|x|\ge2|y|}\sup_{\lambda\in\Lambda}|K_\lambda(x-y)-K_\lambda(x)|\,dx
\le\sum_{j=0}^{m-1}\int_{|x|\ge2|y|}\sup_{\lambda\in\Lambda}|K_\lambda(x-(j+1)y')-K_\lambda(x-jy')|\,dx \\
&\le m\int_{(A Q_0^{(k)})^c}\sup_{\lambda\in\Lambda}|K_\lambda(z-y')-K_\lambda(z)|\,dz \le mC_D.
\end{aligned} ∫ ∣ x ∣ ≥ 2∣ y ∣ λ ∈ Λ sup ∣ K λ ( x − y ) − K λ ( x ) ∣ d x ≤ j = 0 ∑ m − 1 ∫ ∣ x ∣ ≥ 2∣ y ∣ λ ∈ Λ sup ∣ K λ ( x − ( j + 1 ) y ′ ) − K λ ( x − j y ′ ) ∣ d x ≤ m ∫ ( A Q 0 ( k ) ) c λ ∈ Λ sup ∣ K λ ( z − y ′ ) − K λ ( z ) ∣ d z ≤ m C D . The case y = 0 y=0 y = 0 y y y
Regularization of convolution operators
In this subsection, we regularize the convolution operators in Theorem 1.1 . The resulting kernels are bounded and Lipschitz, so that we can apply the decomposition established in § 5 .
Let ( T n ) n ≥ 1 (T_n)_{n\ge1} ( T n ) n ≥ 1 σ n \sigma_n σ n § 2.2 . Assume that the restriction of σ n \sigma_n σ n R d ∖ { 0 } \mathbb{R}^d \setminus\{0\} R d ∖ { 0 } K n ∈ L 1 , l o c ( R d ∖ { 0 } ) K_n \in L_{1,\mathrm{loc}}(\mathbb{R}^d \setminus\{0\}) K n ∈ L 1 , loc ( R d ∖ { 0 }) f ∈ D f \in\mathcal{D} f ∈ D
T n f ( x ) = ∫ R d K n ( x − y ) f ( y ) d y T_n f(x)=\int_{\mathbb{R}^d}K_n(x-y)f(y)\,dy T n f ( x ) = ∫ R d K n ( x − y ) f ( y ) d y for almost every x ∉ supp R d f x \notin\operatorname{supp}_{\mathbb{R}^d} f x ∈ / supp R d f
Suppose that ( K n ) n (K_n)_n ( K n ) n C H C_H C H 1 < r ≤ ∞ 1<r\le\infty 1 < r ≤ ∞
∥ ( T n f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . \|(T_n f)_n\|_{L_r(\mathbb{N};\ell_\infty)}\le M_r\|f\|_r,\qquad f\in L_r(\mathbb{N}). ∥ ( T n f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . Fix a nonnegative function ψ ∈ C c ∞ ( B ( 0 , 1 / 8 ) ) \psi\in C_c^\infty(B(0,1/8)) ψ ∈ C c ∞ ( B ( 0 , 1/8 )) ∫ ψ = 1 \int\psi=1 ∫ ψ = 1 ε > 0 \varepsilon>0 ε > 0
ψ ε ( x ) = ε − d ψ ( x / ε ) , η ε = ψ ε ∗ ψ ε , \psi_\varepsilon(x)=\varepsilon^{-d}\psi(x/\varepsilon),\qquad\eta_\varepsilon=\psi_\varepsilon* \psi_\varepsilon, ψ ε ( x ) = ε − d ψ ( x / ε ) , η ε = ψ ε ∗ ψ ε , and define
K n , ε ( x ) = ⟨ σ n , η ε ( x − ⋅ ) ⟩ , T n , ε f = T n ( η ε ∗ f ) . K_{n,\varepsilon}(x)=\langle\sigma_n,\eta_\varepsilon(x-\cdot)\rangle,\qquad T_{n,\varepsilon}f=T_n(\eta_\varepsilon*f). K n , ε ( x ) = ⟨ σ n , η ε ( x − ⋅ )⟩ , T n , ε f = T n ( η ε ∗ f ) . Since supp η ε ⊂ B ( 0 , ε / 4 ) \operatorname{supp}\eta_\varepsilon\subset B(0,\varepsilon/4) supp η ε ⊂ B ( 0 , ε /4 )
K n , ε ( x ) = ∫ R d K n ( x − y ) η ε ( y ) d y , ∣ x ∣ > ε / 4. (52) K_{n,\varepsilon}(x)=\int_{\mathbb{R}^d}K_n(x-y)\eta_\varepsilon(y)\,dy,\qquad|x|>\varepsilon/4. \tag*{(52)} K n , ε ( x ) = ∫ R d K n ( x − y ) η ε ( y ) d y , ∣ x ∣ > ε /4. (52) The following estimates follow from standard arguments. We include a short proof to record their uniformity in ε \varepsilon ε
Lemma 5.2. For every ε > 0 \varepsilon>0 ε > 0 ( K n , ε ) n ≥ 1 (K_{n,\varepsilon})_{n\ge1} ( K n , ε ) n ≥ 1
sup n ( ε d ∥ K n , ε ∥ ∞ + ε d + 1 ∥ ∇ K n , ε ∥ ∞ ) ≤ C d , r M r . (53) \sup_n\left(\varepsilon^d\|K_{n,\varepsilon}\|_\infty+\varepsilon^{d+1}\|\nabla K_{n,\varepsilon}\|_\infty\right)\le C_{d,r}M_r. \tag*{(53)} n sup ( ε d ∥ K n , ε ∥ ∞ + ε d + 1 ∥∇ K n , ε ∥ ∞ ) ≤ C d , r M r . (53) Moreover,
sup h ≠ 0 sup n ∫ ∣ x ∣ > 2 ∣ h ∣ ∣ K n , ε ( x − h ) − K n , ε ( x ) ∣ d x ≤ C d , r ( M r + C H ) , (54) \sup_{h\ne0}\sup_n\int_{|x|>2|h|}|K_{n,\varepsilon}(x-h)-K_{n,\varepsilon}(x)|\,dx\le C_{d,r}(M_r+C_H), \tag*{(54)} h = 0 sup n sup ∫ ∣ x ∣ > 2∣ h ∣ ∣ K n , ε ( x − h ) − K n , ε ( x ) ∣ d x ≤ C d , r ( M r + C H ) , (54) and
sup j ∈ Z sup n ∫ V j ∣ K n , ε ( x ) ∣ d x ≤ C d , r ( M r + C H ) . (55) \sup_{j\in\mathbb{Z}}\sup_{n}\int_{V_j}|K_{n,\varepsilon}(x)|\,dx\leq C_{d,r}(M_r+C_H). \tag*{(55)} j ∈ Z sup n sup ∫ V j ∣ K n , ε ( x ) ∣ d x ≤ C d , r ( M r + C H ) . (55) Proof. Choose p ∈ P ( M ) p\in\mathcal{P}(\mathcal{M}) p ∈ P ( M ) 0 < τ ( p ) < ∞ 0<\tau(p)<\infty 0 < τ ( p ) < ∞ 1 < r < ∞ 1<r<\infty 1 < r < ∞ u ∈ C c ∞ ( R d ) u\in C^\infty_c(\mathbb{R}^d) u ∈ C c ∞ ( R d )
∥ sup n ∣ σ n ∗ u ∣ ∥ r = τ ( p ) − 1 / r ∥ ( T n ( u ⊗ p ) ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ u ∥ r , (56) \left\|\sup_n|\sigma_n*u|\right\|_r=\tau(p)^{-1/r}\|(T_n(u\otimes p))_n\|_{L_r(\mathcal{N};\ell_\infty)}\leq M_r\|u\|_r, \tag*{(56)} n sup ∣ σ n ∗ u ∣ r = τ ( p ) − 1/ r ∥ ( T n ( u ⊗ p ) ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ u ∥ r , (56) see [14 Remark 2.3] . For r = ∞ r=\infty r = ∞ u ⊗ p u\otimes p u ⊗ p K n , ε = ( σ n ∗ ψ ε ) ∗ ψ ε K_{n,\varepsilon}=(\sigma_n*\psi_\varepsilon)*\psi_\varepsilon K n , ε = ( σ n ∗ ψ ε ) ∗ ψ ε
∥ K n , ε ∥ ∞ + ε ∥ ∇ K n , ε ∥ ∞ ≤ ∥ σ n ∗ ψ ε ∥ r ( ∥ ψ ε ∥ r ′ + ε ∥ ∇ ψ ε ∥ r ′ ) ≤ C d , r M r ε − d , \|K_{n,\varepsilon}\|_\infty+\varepsilon\|\nabla K_{n,\varepsilon}\|_\infty\leq\|\sigma_n*\psi_\varepsilon\|_r\left(\|\psi_\varepsilon\|_{r'}+\varepsilon\|\nabla\psi_\varepsilon\|_{r'}\right)\leq C_{d,r}M_r\varepsilon^{-d}, ∥ K n , ε ∥ ∞ + ε ∥∇ K n , ε ∥ ∞ ≤ ∥ σ n ∗ ψ ε ∥ r ( ∥ ψ ε ∥ r ′ + ε ∥∇ ψ ε ∥ r ′ ) ≤ C d , r M r ε − d , where 1 / r + 1 / r ′ = 1 1/r+1/r'=1 1/ r + 1/ r ′ = 1 (53) .
For 0 < a < b 0<a<b 0 < a < b t > 0 t>0 t > 0
K n ( x ) = ( σ n ∗ ψ a t ) ( x ) + ∫ R d ψ a t ( y ) ( K n ( x ) − K n ( x − y ) ) d y , a t < ∣ x ∣ < b t . K_n(x)=(\sigma_n*\psi_{at})(x)+\int_{\mathbb{R}^d}\psi_{at}(y)\left(K_n(x)-K_n(x-y)\right)\,dy,\qquad at<|x|<bt. K n ( x ) = ( σ n ∗ ψ a t ) ( x ) + ∫ R d ψ a t ( y ) ( K n ( x ) − K n ( x − y ) ) d y , a t < ∣ x ∣ < b t . Since ∣ x ∣ > 2 ∣ y ∣ |x|>2|y| ∣ x ∣ > 2∣ y ∣ y ∈ supp ψ a t y\in\operatorname{supp}\psi_{at} y ∈ supp ψ a t (56) imply
∫ a t < ∣ x ∣ < b t sup n ∣ K n ( x ) ∣ d x ≤ C d , r , a , b M r + C H . (57) \int_{at<|x|<bt}\sup_n|K_n(x)|\,dx\leq C_{d,r,a,b}M_r+C_H. \tag*{(57)} ∫ a t < ∣ x ∣ < b t n sup ∣ K n ( x ) ∣ d x ≤ C d , r , a , b M r + C H . (57) To prove (54) , fix z ≠ 0 z\neq0 z = 0 (52) , we have
∫ ∣ x ∣ > 4 max { ∣ z ∣ , ε } sup n ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ C H . \int_{|x|>4\max\{|z|,\varepsilon\}}\sup_n|K_{n,\varepsilon}(x-z)-K_{n,\varepsilon}(x)|\,dx\leq C_H. ∫ ∣ x ∣ > 4 m a x { ∣ z ∣ , ε } n sup ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ C H . Here we used ∣ x − y ∣ > 2 ∣ z ∣ |x-y|>2|z| ∣ x − y ∣ > 2∣ z ∣ y ∈ supp η ε y\in\operatorname{supp}\eta_\varepsilon y ∈ supp η ε ∣ z ∣ ≤ ε |z|\leq\varepsilon ∣ z ∣ ≤ ε (56) , we get
∫ 2 ∣ z ∣ < ∣ x ∣ ≤ 4 ε sup n ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ 2 ∣ B ( 0 , 5 ε ) ∣ 1 / r ′ M r ∥ η ε ∥ r ≤ C d , r M r . \int_{2|z|<|x|\leq4\varepsilon}\sup_n|K_{n,\varepsilon}(x-z)-K_{n,\varepsilon}(x)|\,dx\leq2|B(0,5\varepsilon)|^{1/r'}M_r\|\eta_\varepsilon\|_r\leq C_{d,r}M_r. ∫ 2∣ z ∣ < ∣ x ∣ ≤ 4 ε n sup ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ 2∣ B ( 0 , 5 ε ) ∣ 1/ r ′ M r ∥ η ε ∥ r ≤ C d , r M r . If ∣ z ∣ > ε |z|>\varepsilon ∣ z ∣ > ε (52) yields
∫ 2 ∣ z ∣ < ∣ x ∣ ≤ 4 ∣ z ∣ sup n ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ 2 ∫ 3 ∣ z ∣ / 4 < ∣ x ∣ < 21 ∣ z ∣ / 4 sup n ∣ K n ( x ) ∣ d x . \int_{2|z|<|x|\leq4|z|}\sup_n|K_{n,\varepsilon}(x-z)-K_{n,\varepsilon}(x)|\,dx\leq2\int_{3|z|/4<|x|<21|z|/4}\sup_n|K_n(x)|\,dx. ∫ 2∣ z ∣ < ∣ x ∣ ≤ 4∣ z ∣ n sup ∣ K n , ε ( x − z ) − K n , ε ( x ) ∣ d x ≤ 2 ∫ 3∣ z ∣/4 < ∣ x ∣ < 21∣ z ∣/4 n sup ∣ K n ( x ) ∣ d x . Combining the preceding estimates yields (54) .
Finally, applying the argument for (57) to K n , ε K_{n,\varepsilon} K n , ε (54) , we have
∫ a t < ∣ x ∣ < b t sup n ∣ K n , ε ( x ) ∣ d x ≤ C d , r , a , b ( M r + C H ) . \int_{at<|x|<bt}\sup_n|K_{n,\varepsilon}(x)|\,dx\leq C_{d,r,a,b}(M_r+C_H). ∫ a t < ∣ x ∣ < b t n sup ∣ K n , ε ( x ) ∣ d x ≤ C d , r , a , b ( M r + C H ) . Since V j ⊂ { x : 2 − j − 1 < ∣ x ∣ < 3 d 2 − j } V_j\subset\{x:2^{-j-1}<|x|<3\sqrt{d}\,2^{-j}\} V j ⊂ { x : 2 − j − 1 < ∣ x ∣ < 3 d 2 − j } (55) . □
We now verify that K n , ε K_{n,\varepsilon} K n , ε T n , ε T_{n,\varepsilon} T n , ε
Proposition 5.3. Let 1 < r ≤ ∞ 1<r\leq\infty 1 < r ≤ ∞ ε > 0 \varepsilon>0 ε > 0 f ∈ L 1 ( N ) ∩ L r ( N ) f\in L_1(\mathcal{N})\cap L_r(\mathcal{N}) f ∈ L 1 ( N ) ∩ L r ( N )
T n , ε f = K n , ε ∗ f almost everywhere . (58) T_{n,\varepsilon}f=K_{n,\varepsilon}*f\qquad\text{almost everywhere}. \tag*{(58)} T n , ε f = K n , ε ∗ f almost everywhere . (58) Moreover,
∥ ( T n , ε f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . (59) \|(T_{n,\varepsilon}f)_n\|_{L_r(\mathcal{N};\ell_\infty)}\leq M_r\|f\|_r,\qquad f\in L_r(\mathcal{N}). \tag*{(59)} ∥ ( T n , ε f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r , f ∈ L r ( N ) . (59) Proof. For r < ∞ r<\infty r < ∞ f ∈ L 1 ( N ) ∩ L r ( N ) f\in L_1(\mathcal{N})\cap L_r(\mathcal{N}) f ∈ L 1 ( N ) ∩ L r ( N ) r = ∞ r=\infty r = ∞ (56) gives ∣ ⟨ σ n , u ⟩ ∣ ≤ M ∞ ∥ u ∥ ∞ |\langle\sigma_n,u\rangle|\le M_\infty\|u\|_\infty ∣ ⟨ σ n , u ⟩ ∣ ≤ M ∞ ∥ u ∥ ∞ σ n \sigma_n σ n K n d x K_n\,dx K n d x
σ n = K n d x + c n δ 0 , ∥ K n ∥ 1 + ∣ c n ∣ ≤ M ∞ , \sigma_n=K_n\,dx+c_n\delta_0,\qquad\|K_n\|_1+|c_n|\le M_\infty, σ n = K n d x + c n δ 0 , ∥ K n ∥ 1 + ∣ c n ∣ ≤ M ∞ , where δ 0 \delta_0 δ 0 T n f = K n ∗ f + c n f T_nf=K_n*f+c_nf T n f = K n ∗ f + c n f f ∈ N f\in\mathcal{N} f ∈ N ∗ ^* ∗
T n , ε f = ( K n ∗ η ε + c n η ε ) ∗ f = K n , ε ∗ f , T_{n,\varepsilon}f=(K_n*\eta_\varepsilon+c_n\eta_\varepsilon)*f=K_{n,\varepsilon}*f, T n , ε f = ( K n ∗ η ε + c n η ε ) ∗ f = K n , ε ∗ f , which completes the proof. □
The preceding estimates allow us to apply Propositions 4.1 and 4.4 to the regularized kernels, with coefficient bounds independent of ε \varepsilon ε
Proof of Theorem 1.1
Theorem 3.1 gives a weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( T n , ε ) n (T_{n,\varepsilon})_n ( T n , ε ) n ε \varepsilon ε ε → 0 \varepsilon\to0 ε → 0 ( T n ) n (T_n)_n ( T n ) n D \mathcal{D} D T n T_n T n L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) ( 1 , 1 ) (1,1) ( 1 , 1 ) L 1 ( N ) L_1(\mathcal{N}) L 1 ( N )
Proof of Theorem 1.1. If M r = 0 M_r=0 M r = 0 M r > 0 M_r>0 M r > 0 ε > 0 \varepsilon>0 ε > 0
T n , ε f = T n ( η ε ∗ f ) . T_{n,\varepsilon}f=T_n(\eta_\varepsilon*f). T n , ε f = T n ( η ε ∗ f ) . The estimates in Subsection 5.2, together with Propositions 5.1, 4.1 and 4.4, give a dyadic representation of ( T n , ε ) n (T_{n,\varepsilon})_n ( T n , ε ) n
Γ ≤ C d , r ( M r + C H ) , β = 3. \Gamma\le C_{d,r}(M_r+C_H),\qquad\beta=3. Γ ≤ C d , r ( M r + C H ) , β = 3. By Proposition 5.3, this representation agrees with T n , ε T_{n,\varepsilon} T n , ε D 0 \mathcal{D}_0 D 0
∥ ( T n , ε f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r . \|(T_{n,\varepsilon}f)_n\|_{L_r(\mathcal{N};\ell_\infty)}\le M_r\|f\|_r. ∥ ( T n , ε f ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f ∥ r . By Theorem 3.1, we can find a constant A > 0 A>0 A > 0 ε \varepsilon ε A ≤ C d , r ( M r + C H ) A\le C_{d,r}(M_r+C_H) A ≤ C d , r ( M r + C H ) f ∈ D f\in\mathcal{D} f ∈ D λ > 0 \lambda>0 λ > 0 e ∈ P ( N ) e\in\mathcal{P}(\mathcal{N}) e ∈ P ( N )
sup n ∥ e T n , ε ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ A ∥ f ∥ 1 . (60) \sup_n\|eT_{n,\varepsilon}(f)e\|_\infty\le\lambda,\qquad\lambda\varphi(1-e)\le A\|f\|_1. \tag*{(60)} n sup ∥ e T n , ε ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ A ∥ f ∥ 1 . (60) Fix f ∈ D f\in\mathcal{D} f ∈ D λ > 0 \lambda>0 λ > 0 r < ∞ r<\infty r < ∞
∥ ( T n ( f − η ε ∗ f ) ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f − η ε ∗ f ∥ r ⟶ 0 ( ε → 0 ) . \|(T_n(f-\eta_\varepsilon*f))_n\|_{L_r(\mathcal{N};\ell_\infty)}
\le M_r\|f-\eta_\varepsilon*f\|_r\longrightarrow0\qquad(\varepsilon\to0). ∥ ( T n ( f − η ε ∗ f ) ) n ∥ L r ( N ; ℓ ∞ ) ≤ M r ∥ f − η ε ∗ f ∥ r ⟶ 0 ( ε → 0 ) . By Lemma 2.3, for sufficiently small ε \varepsilon ε e 1 e_1 e 1
sup n ∥ e 1 T n ( f − η ε ∗ f ) e 1 ∥ ∞ ≤ λ 2 , φ ( 1 − e 1 ) ≤ A λ ∥ f ∥ 1 . \sup_n\|e_1T_n(f-\eta_\varepsilon*f)e_1\|_\infty\le\frac{\lambda}{2},\qquad
\varphi(1-e_1)\le\frac{A}{\lambda}\|f\|_1. n sup ∥ e 1 T n ( f − η ε ∗ f ) e 1 ∥ ∞ ≤ 2 λ , φ ( 1 − e 1 ) ≤ λ A ∥ f ∥ 1 . By (60) , there is also a projection e 2 e_2 e 2
sup n ∥ e 2 T n , ε ( f ) e 2 ∥ ∞ ≤ λ 2 , φ ( 1 − e 2 ) ≤ 2 A λ ∥ f ∥ 1 . \sup_n\|e_2T_{n,\varepsilon}(f)e_2\|_\infty\le\frac{\lambda}{2},\qquad
\varphi(1-e_2)\le\frac{2A}{\lambda}\|f\|_1. n sup ∥ e 2 T n , ε ( f ) e 2 ∥ ∞ ≤ 2 λ , φ ( 1 − e 2 ) ≤ λ 2 A ∥ f ∥ 1 . Set e = e 1 ∧ e 2 e=e_1\wedge e_2 e = e 1 ∧ e 2
T n f = T n , ε f + T n ( f − η ε ∗ f ) , T_nf=T_{n,\varepsilon}f+T_n(f-\eta_\varepsilon*f), T n f = T n , ε f + T n ( f − η ε ∗ f ) , we obtain
sup n ∥ e T n ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ 3 A ∥ f ∥ 1 . (61) \sup_n\|eT_n(f)e\|_\infty\le\lambda,\qquad\lambda\varphi(1-e)\le3A\|f\|_1. \tag*{(61)} n sup ∥ e T n ( f ) e ∥ ∞ ≤ λ , λ φ ( 1 − e ) ≤ 3 A ∥ f ∥ 1 . (61) For r = ∞ r = \infty r = ∞
sup n ∥ T n ( f − η ε ∗ f ) ∥ ∞ ≤ M ∞ ∥ f − η ε ∗ f ∥ ∞ ⟶ 0 , \sup_n \|T_n(f-\eta_\varepsilon* f)\|_\infty\le M_\infty\|f-\eta_\varepsilon* f\|_\infty\longrightarrow0, n sup ∥ T n ( f − η ε ∗ f ) ∥ ∞ ≤ M ∞ ∥ f − η ε ∗ f ∥ ∞ ⟶ 0 , since f f f T n T_n T n L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) ( 1 , 1 ) (1,1) ( 1 , 1 ) L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) C A CA C A A ≤ C d , r ( M r + C H ) A \le C_{d,r}(M_r+C_H) A ≤ C d , r ( M r + C H )
Proof of Theorem 1.2
We apply Theorem 1.1 to the dyadic truncations and then pass to arbitrary radii using the size condition and the noncommutative Hardy–Littlewood maximal inequality. This reduction follows the argument in [11 Section 3.2] , with T 2 k T_{2^k} T 2 k
Lemma 5.4. Suppose that ∣ K ( x ) ∣ ≤ A ∣ x ∣ − d |K(x)| \le A|x|^{-d} ∣ K ( x ) ∣ ≤ A ∣ x ∣ − d x ≠ 0 x \ne0 x = 0 K K K L 1 L_1 L 1 C H C_H C H K k = K 1 { ∣ x ∣ > 2 k } K_k=K1_{\{|x|>2^k\}} K k = K 1 { ∣ x ∣ > 2 k } k ∈ Z k\in\mathbb{Z} k ∈ Z
sup h ≠ 0 ∫ ∣ x ∣ > 2 ∣ h ∣ sup k ∈ Z ∣ K k ( x − h ) − K k ( x ) ∣ d x ≤ C H + C d A . \sup_{h\ne0}\int_{|x|>2|h|}\sup_{k\in\mathbb{Z}}|K_k(x-h)-K_k(x)|\,\mathrm{d}x\le C_H+C_dA. h = 0 sup ∫ ∣ x ∣ > 2∣ h ∣ k ∈ Z sup ∣ K k ( x − h ) − K k ( x ) ∣ d x ≤ C H + C d A . Proof. For every k ∈ Z k\in\mathbb{Z} k ∈ Z
∣ K k ( x − h ) − K k ( x ) ∣ ≤ ∣ K ( x − h ) − K ( x ) ∣ + A ∣ x ∣ − d ∣ 1 { ∣ x − h ∣ > 2 k } − 1 { ∣ x ∣ > 2 k } ∣ . |K_k(x-h)-K_k(x)|\le|K(x-h)-K(x)|+A|x|^{-d}\left|1_{\{|x-h|>2^k\}}-1_{\{|x|>2^k\}}\right|. ∣ K k ( x − h ) − K k ( x ) ∣ ≤ ∣ K ( x − h ) − K ( x ) ∣ + A ∣ x ∣ − d 1 { ∣ x − h ∣ > 2 k } − 1 { ∣ x ∣ > 2 k } . Then, the Hörmander condition therefore yields
∫ ∣ x ∣ > 2 ∣ h ∣ sup k ∣ K k ( x − h ) − K k ( x ) ∣ d x ≤ C H + C d A ( 1 + ∑ k ∈ Z 2 k > 2 ∣ h ∣ ∣ h ∣ 2 k ) ≤ C H + C d A . \int_{|x|>2|h|}\sup_k|K_k(x-h)-K_k(x)|\,\mathrm{d}x\le C_H+C_dA\left(1+\sum_{\substack{k\in\mathbb{Z}\\2^k>2|h|}}\frac{|h|}{2^k}\right)\le C_H+C_dA. ∫ ∣ x ∣ > 2∣ h ∣ k sup ∣ K k ( x − h ) − K k ( x ) ∣ d x ≤ C H + C d A 1 + k ∈ Z 2 k > 2∣ h ∣ ∑ 2 k ∣ h ∣ ≤ C H + C d A . For f ∈ L 1 ( N ) + f\in L_1(\mathcal{N})_+ f ∈ L 1 ( N ) +
M k f ( x ) = 2 − d ( k + 1 ) ∫ x + [ − 2 k , 2 k ] d f ( y ) d y , k ∈ Z . M_kf(x)=2^{-d(k+1)}\int_{x+[-2^k,2^k]^d}f(y)\,\mathrm{d}y,\qquad k\in\mathbb{Z}. M k f ( x ) = 2 − d ( k + 1 ) ∫ x + [ − 2 k , 2 k ] d f ( y ) d y , k ∈ Z . By the noncommutative Hardy–Littlewood maximal inequality [10 Theorem 4.1] , applied to R d \mathbb{R}^d R d ∣ x − y ∣ ∞ |x-y|_\infty ∣ x − y ∣ ∞ f ∈ L 1 ( N ) + f\in L_1(\mathcal{N})_+ f ∈ L 1 ( N ) + t > 0 t>0 t > 0 e ∈ P ( N ) e\in\mathcal{P}(\mathcal{N}) e ∈ P ( N )
sup k ∈ Z ∥ e M k ( f ) e ∥ ∞ ≤ t , φ ( 1 − e ) ≤ C d t ∥ f ∥ 1 . (62) \sup_{k\in\mathbb{Z}}\|eM_k(f)e\|_\infty\le t,\qquad\varphi(1-e)\le\frac{C_d}{t}\|f\|_1. \tag*{(62)} k ∈ Z sup ∥ e M k ( f ) e ∥ ∞ ≤ t , φ ( 1 − e ) ≤ t C d ∥ f ∥ 1 . (62) Proof of Theorem 1.2. The case A = 0 A=0 A = 0 A > 0 A>0 A > 0 ε = 2 k \varepsilon=2^k ε = 2 k Theorem 1.1 , we obtain a weak type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( T 2 k ) k ∈ Z (T_{2^k})_{k\in\mathbb{Z}} ( T 2 k ) k ∈ Z C d , p 0 ( M p 0 + C H + A ) C_{d,p_0}(M_{p_0}+C_H+A) C d , p 0 ( M p 0 + C H + A ) K k ∈ L ∞ ( R d ) K_k\in L_\infty(\mathbb{R}^d) K k ∈ L ∞ ( R d ) D \mathcal{D} D L 1 L_1 L 1
Fix f ∈ L 1 ( N ) f\in L_1(\mathcal{N}) f ∈ L 1 ( N ) 2 k ≤ ε < 2 k + 1 2^k\le\varepsilon<2^{k+1} 2 k ≤ ε < 2 k + 1 T ε − T 2 k T_\varepsilon-T_{2^k} T ε − T 2 k
∣ K ( x ) ( 1 { ∣ x ∣ > ε } − 1 { ∣ x ∣ > 2 k } ) ∣ ≤ A 2 − k d 1 [ − 2 k + 1 , 2 k + 1 ] d ( x ) . \left|K(x)\left(1_{\{|x|>\varepsilon\}}-1_{\{|x|>2^k\}}\right)\right|\le A2^{-kd}1_{[-2^{k+1},2^{k+1}]^d}(x). K ( x ) ( 1 { ∣ x ∣ > ε } − 1 { ∣ x ∣ > 2 k } ) ≤ A 2 − k d 1 [ − 2 k + 1 , 2 k + 1 ] d ( x ) . Write r ε , k = K ( 1 { ∣ ⋅ ∣ > ε } − 1 { ∣ ⋅ ∣ > 2 k } ) r_{\varepsilon,k}=K\bigl(1_{\{|\cdot|>\varepsilon\}}-1_{\{|\cdot|>2^k\}}\bigr) r ε , k = K ( 1 { ∣ ⋅ ∣ > ε } − 1 { ∣ ⋅ ∣ > 2 k } ) ∣ c ∣ ≤ 1 |c|\le1 ∣ c ∣ ≤ 1
( ∣ f ( y ) ∗ ∣ c f ( y ) c f ( y ) ∗ ∣ f ( y ) ∣ ) ≥ 0. \begin{pmatrix}
|f(y)^*| & cf(y) \\
c f(y)^* & |f(y)|
\end{pmatrix}\ge0. ( ∣ f ( y ) ∗ ∣ c f ( y ) ∗ c f ( y ) ∣ f ( y ) ∣ ) ≥ 0. Since ∣ r ε , k ( y ) ∣ ≤ A 2 − k d 1 [ − 2 k + 1 , 2 k + 1 ] d ( y ) |r_{\varepsilon,k}(y)|\le A2^{-kd}1_{[-2^{k+1},2^{k+1}]^d}(y) ∣ r ε , k ( y ) ∣ ≤ A 2 − k d 1 [ − 2 k + 1 , 2 k + 1 ] d ( y )
A 2 − k d ∫ [ − 2 k + 1 , 2 k + 1 ] d ∣ f ( x − y ) ∣ d y = 4 d A M k + 1 ( ∣ f ∣ ) ( x ) , A2^{-kd}\int_{[-2^{k+1},2^{k+1}]^d}|f(x-y)|\,dy=4^d A M_{k+1}(|f|)(x), A 2 − k d ∫ [ − 2 k + 1 , 2 k + 1 ] d ∣ f ( x − y ) ∣ d y = 4 d A M k + 1 ( ∣ f ∣ ) ( x ) , and the corresponding identity for ∣ f ∗ ∣ |f^*| ∣ f ∗ ∣
\begin{pmatrix}
4^d A M_{k+1}(|f^*|) & (T_\varepsilon-T_{2^k})f \\
((T_\varepsilon-T_{2^k})f)^* & 4^d A M_{k+1}(|f|)
\end{pmatrix}\ge0. \tag*{(63)}\end{pmatrix}
For t > 0 t>0 t > 0 (62) to ∣ f ∗ ∣ |f^*| ∣ f ∗ ∣ ∣ f ∣ |f| ∣ f ∣ t / ( 4 d A ) t/(4^d A) t / ( 4 d A ) e 1 , e 2 e_1,e_2 e 1 , e 2
sup k ∥ e 1 M k ( ∣ f ∗ ∣ ) e 1 ∥ ∞ ≤ t 4 d A , sup k ∥ e 2 M k ( ∣ f ∣ ) e 2 ∥ ∞ ≤ t 4 d A , \sup_k\|e_1M_k(|f^*|)e_1\|_\infty\le\frac{t}{4^d A},\qquad
\sup_k\|e_2M_k(|f|)e_2\|_\infty\le\frac{t}{4^d A}, k sup ∥ e 1 M k ( ∣ f ∗ ∣ ) e 1 ∥ ∞ ≤ 4 d A t , k sup ∥ e 2 M k ( ∣ f ∣ ) e 2 ∥ ∞ ≤ 4 d A t , and
φ ( 1 − e 1 ) + φ ( 1 − e 2 ) ≤ C d A t ∥ f ∥ 1 . \varphi(1-e_1)+\varphi(1-e_2)\le\frac{C_d A}{t}\|f\|_1. φ ( 1 − e 1 ) + φ ( 1 − e 2 ) ≤ t C d A ∥ f ∥ 1 . Set e 3 = e 1 ∧ e 2 e_3=e_1\wedge e_2 e 3 = e 1 ∧ e 2 (63) ,
( 4 d A e 3 M k + 1 ( ∣ f ∗ ∣ ) e 3 e 3 ( T ε − T 2 k ) ( f ) e 3 ( e 3 ( T ε − T 2 k ) ( f ) e 3 ) ∗ 4 d A e 3 M k + 1 ( ∣ f ∣ ) e 3 ) ≥ 0. \begin{pmatrix}
4^d A e_3M_{k+1}(|f^*|)e_3 & e_3(T_\varepsilon-T_{2^k})(f)e_3 \\
\bigl(e_3(T_\varepsilon-T_{2^k})(f)e_3\bigr)^* & 4^d A e_3M_{k+1}(|f|)e_3
\end{pmatrix}\ge0. ( 4 d A e 3 M k + 1 ( ∣ f ∗ ∣ ) e 3 ( e 3 ( T ε − T 2 k ) ( f ) e 3 ) ∗ e 3 ( T ε − T 2 k ) ( f ) e 3 4 d A e 3 M k + 1 ( ∣ f ∣ ) e 3 ) ≥ 0. Since e 3 ≤ e 1 , e 2 e_3\le e_1,e_2 e 3 ≤ e 1 , e 2 t e 3 te_3 t e 3 Lemma 2.7 , we get
sup k ∈ Z sup 2 k ≤ ε < 2 k + 1 ∥ e 3 ( T ε − T 2 k ) ( f ) e 3 ∥ ∞ ≤ t , φ ( 1 − e 3 ) ≤ C d A t ∥ f ∥ 1 . (64) \sup_{k\in\mathbb{Z}}\sup_{2^k\le\varepsilon<2^{k+1}}\|e_3(T_\varepsilon-T_{2^k})(f)e_3\|_\infty\le t,\qquad
\varphi(1-e_3)\le\frac{C_dA}{t}\|f\|_1. \tag*{(64)} k ∈ Z sup 2 k ≤ ε < 2 k + 1 sup ∥ e 3 ( T ε − T 2 k ) ( f ) e 3 ∥ ∞ ≤ t , φ ( 1 − e 3 ) ≤ t C d A ∥ f ∥ 1 . (64) Finally, fix λ > 0 \lambda>0 λ > 0 e 0 e_0 e 0
sup k ∥ e 0 T 2 k ( f ) e 0 ∥ ∞ ≤ λ 2 , φ ( 1 − e 0 ) ≤ C d , p 0 λ ( M p 0 + C H + A ) ∥ f ∥ 1 . \sup_k\|e_0T_{2^k}(f)e_0\|_\infty\le\frac{\lambda}{2},\qquad
\varphi(1-e_0)\le\frac{C_{d,p_0}}{\lambda}(M_{p_0}+C_H+A)\|f\|_1. k sup ∥ e 0 T 2 k ( f ) e 0 ∥ ∞ ≤ 2 λ , φ ( 1 − e 0 ) ≤ λ C d , p 0 ( M p 0 + C H + A ) ∥ f ∥ 1 . Choose e 3 e_3 e 3 (64) with t = λ / 2 t=\lambda/2 t = λ /2 e = e 0 ∧ e 3 e=e_0\wedge e_3 e = e 0 ∧ e 3 2 k ≤ ε < 2 k + 1 2^k\le\varepsilon<2^{k+1} 2 k ≤ ε < 2 k + 1
∥ e T ε ( f ) e ∥ ∞ ≤ ∥ e T 2 k ( f ) e ∥ ∞ + ∥ e ( T ε − T 2 k ) ( f ) e ∥ ∞ ≤ λ . \|eT_\varepsilon(f)e\|_\infty\le\|eT_{2^k}(f)e\|_\infty+\|e(T_\varepsilon-T_{2^k})(f)e\|_\infty\le\lambda. ∥ e T ε ( f ) e ∥ ∞ ≤ ∥ e T 2 k ( f ) e ∥ ∞ + ∥ e ( T ε − T 2 k ) ( f ) e ∥ ∞ ≤ λ . Moreover,
λ φ ( 1 − e ) ≤ λ φ ( 1 − e 0 ) + λ φ ( 1 − e 3 ) ≤ C d , p 0 ( M p 0 + C H + A ) ∥ f ∥ 1 . \lambda\varphi(1-e)\le\lambda\varphi(1-e_0)+\lambda\varphi(1-e_3)\le C_{d,p_0}(M_{p_0}+C_H+A)\|f\|_1. λ φ ( 1 − e ) ≤ λ φ ( 1 − e 0 ) + λ φ ( 1 − e 3 ) ≤ C d , p 0 ( M p 0 + C H + A ) ∥ f ∥ 1 . This proves the theorem. □
Applications I: Maximal inequalities and approximation of the identity
In this section, we use Theorem 1.1 to obtain maximal inequalities for integrable approximation kernels and multiparameter Poisson averages on Lie groups. The examples include kernels with singularities away from the origin that do not satisfy the L 2 L_2 L 2 L 1 L_1 L 1
We use the classical scalar kernel estimates of Zó [30 ] , Stein [28 ] , and Sjögren [27 ] without repeating their proofs. Applying our theorem to the kernels, gives the noncommutative estimates used below.
Integrable kernels and dyadic approximation
We first establish maximal inequalities for integrable kernels. Applied to dilations of a fixed kernel, these estimates will yield bilateral almost uniform convergence for dyadic approximations of the identity.
Corollary 6.1. Let ( K λ ) λ ∈ Λ ⊂ L 1 ( R d ) (K_\lambda)_{\lambda\in\Lambda}\subset L_1(\mathbb{R}^d) ( K λ ) λ ∈ Λ ⊂ L 1 ( R d ) sup λ ∈ Λ ∥ K λ ∥ 1 < ∞ \sup_{\lambda\in\Lambda}\|K_\lambda\|_1<\infty sup λ ∈ Λ ∥ K λ ∥ 1 < ∞ ( T λ ) λ ∈ Λ (T_\lambda)_{\lambda\in\Lambda} ( T λ ) λ ∈ Λ T λ f = K λ ∗ f T_\lambda f=K_\lambda*f T λ f = K λ ∗ f ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\le\infty 1 < p ≤ ∞
Proof. Without loss of generality, we may assume that Λ = N \Lambda=\mathbb{N} Λ = N f ≥ 0 f\geq0 f ≥ 0
sup n ∥ T n f ∥ ∞ ≤ ( sup n ∥ K n ∥ 1 ) ∥ f ∥ ∞ , \sup_n\|T_nf\|_\infty\leq\left(\sup_n\|K_n\|_1\right)\|f\|_\infty, n sup ∥ T n f ∥ ∞ ≤ ( n sup ∥ K n ∥ 1 ) ∥ f ∥ ∞ , Theorem 1.1, with r = ∞ r=\infty r = ∞ ( 1 , 1 ) (1,1) ( 1 , 1 )
Since
∣ ∣ K n ( x − y ) ∣ − ∣ K n ( x ) ∣ ∣ ≤ ∣ K n ( x − y ) − K n ( x ) ∣ , \left||K_n(x-y)|-|K_n(x)|\right|\leq|K_n(x-y)-K_n(x)|, ∣ ∣ K n ( x − y ) ∣ − ∣ K n ( x ) ∣ ∣ ≤ ∣ K n ( x − y ) − K n ( x ) ∣ , Applying Theorem 1.1 and Lemma 2.4 to f ↦ ∣ K n ∣ ∗ f f\mapsto|K_n|*f f ↦ ∣ K n ∣ ∗ f 1 < p < ∞ 1<p<\infty 1 < p < ∞ U ∈ L p ( N ) + U\in L_p(\mathcal{N})_+ U ∈ L p ( N ) +
∣ K n ∣ ∗ f ≤ U , ∥ U ∥ p ≤ C p ∥ f ∥ p . |K_n|*f\leq U,\qquad\|U\|_p\leq C_p\|f\|_p. ∣ K n ∣ ∗ f ≤ U , ∥ U ∥ p ≤ C p ∥ f ∥ p . As
− U ≤ Re ( T n f ) ≤ U , − U ≤ Im ( T n f ) ≤ U , -U\leq\operatorname{Re}(T_nf)\leq U,\qquad-U\leq\operatorname{Im}(T_nf)\leq U, − U ≤ Re ( T n f ) ≤ U , − U ≤ Im ( T n f ) ≤ U , By Remark 2.1, we have
∥ ( T n f ) n ∥ L p ( N ; ℓ ∞ ) ≤ 2 ∥ U ∥ p ≤ 2 C p ∥ f ∥ p . \|(T_nf)_n\|_{L_p(\mathcal{N};\ell_\infty)}\leq2\|U\|_p\leq2C_p\|f\|_p. ∥ ( T n f ) n ∥ L p ( N ; ℓ ∞ ) ≤ 2∥ U ∥ p ≤ 2 C p ∥ f ∥ p . This proves the strong type estimate. □
We give a sufficient condition for the dyadic dilations of an integrable kernel to satisfy (H).
Proposition 6.2. Let K ∈ L 1 ( R d ) K\in L_1(\mathbb{R}^d) K ∈ L 1 ( R d ) A ≥ 1 A\geq1 A ≥ 1 η \eta η [ 0 , 1 ] [0,1] [ 0 , 1 ] η ( 0 ) = 0 \eta(0)=0 η ( 0 ) = 0 ∫ 0 1 η ( s ) d s s < ∞ \int_0^1\eta(s)\frac{\mathrm{d}s}{s}<\infty ∫ 0 1 η ( s ) s d s < ∞
sup y ∈ Q 0 ( k ) ∫ ( A Q 0 ( k ) ) c ∣ K ( x − y ) − K ( x ) ∣ d x ≤ η ( 2 − k ) ( k ≥ 0 ) . (65) \sup_{y\in Q_0^{(k)}}\int_{(AQ_0^{(k)})^c}|K(x-y)-K(x)|\,\mathrm{d}x\leq\eta(2^{-k})\qquad(k\geq0). \tag*{(65)} y ∈ Q 0 ( k ) sup ∫ ( A Q 0 ( k ) ) c ∣ K ( x − y ) − K ( x ) ∣ d x ≤ η ( 2 − k ) ( k ≥ 0 ) . (65) and
sup y ∈ Q 0 ( k ) ∫ ( A Q 0 ( k ) ) c ∣ K ( x − y ) ∣ d x ≤ η ( 2 k ) ( k ≤ 0 ) . (66) \sup_{y\in Q_0^{(k)}}\int_{(AQ_0^{(k)})^c}|K(x-y)|\,\mathrm{d}x\leq\eta(2^k)\qquad(k\leq0). \tag*{(66)} y ∈ Q 0 ( k ) sup ∫ ( A Q 0 ( k ) ) c ∣ K ( x − y ) ∣ d x ≤ η ( 2 k ) ( k ≤ 0 ) . (66) Then the kernels
K j ( x ) = 2 − j d K ( 2 − j x ) , j ∈ Z , K_j(x)=2^{-jd}K(2^{-j}x),\qquad j\in\mathbb{Z}, K j ( x ) = 2 − j d K ( 2 − j x ) , j ∈ Z , satisfy (H).
Proof. Fix k ∈ Z k\in\mathbb{Z} k ∈ Z y ∈ Q 0 ( k ) y\in Q_0^{(k)} y ∈ Q 0 ( k ) x = 2 j z x=2^jz x = 2 j z
∫ ( A Q 0 ( k ) ) c ∣ K j ( x − y ) − K j ( x ) ∣ d x = ∫ ( A Q 0 ( k + j ) ) c ∣ K ( z − 2 − j y ) − K ( z ) ∣ d z . \int_{(AQ_0^{(k)})^c}|K_j(x-y)-K_j(x)|\,\mathrm{d}x
=
\int_{(AQ_0^{(k+j)})^c}|K(z-2^{-j}y)-K(z)|\,\mathrm{d}z. ∫ ( A Q 0 ( k ) ) c ∣ K j ( x − y ) − K j ( x ) ∣ d x = ∫ ( A Q 0 ( k + j ) ) c ∣ K ( z − 2 − j y ) − K ( z ) ∣ d z . By (65) and (66) , we deduce that
∫ ( A Q 0 ( k ) ) c sup j ∈ Z ∣ K j ( x − y ) − K j ( x ) ∣ d x ≤ ∑ j ≥ 1 − k η ( 2 − k − j ) + 2 ∑ j ≤ − k η ( 2 k + j ) ≤ η ( 1 ) + ∫ 0 1 η ( s ) d s s . \int_{(A Q_{0}^{(k)})^c} \sup_{j\in\mathbb{Z}} \lvert K_j(x-y)-K_j(x)\rvert\, dx \leq\sum_{j\geq1-k}\eta(2^{-k-j})+2\sum_{j\leq-k}\eta(2^{k+j})
\leq\eta(1)+\int_0^1 \eta(s)\frac{ds}{s}. ∫ ( A Q 0 ( k ) ) c j ∈ Z sup ∣ K j ( x − y ) − K j ( x )∣ d x ≤ j ≥ 1 − k ∑ η ( 2 − k − j ) + 2 j ≤ − k ∑ η ( 2 k + j ) ≤ η ( 1 ) + ∫ 0 1 η ( s ) s d s . Here we used monotonicity of η \eta η
η ( 2 − r ) log 2 ≤ ∫ 2 − r 2 − r + 1 η ( s ) d s s , r ≥ 1. \eta(2^{-r})\log2\leq\int_{2^{-r}}^{2^{-r+1}}\eta(s)\frac{ds}{s},\qquad r\geq1. η ( 2 − r ) log 2 ≤ ∫ 2 − r 2 − r + 1 η ( s ) s d s , r ≥ 1. Taking the supremum over k k k y y y
For K ∈ L 1 ( R d ) K\in L_1(\mathbb{R}^d) K ∈ L 1 ( R d ) t > 0 t>0 t > 0
K t ( x ) = t − d K ( t − 1 x ) , T t f = K t ∗ f . K_t(x)=t^{-d}K(t^{-1}x),\qquad T_tf=K_t*f. K t ( x ) = t − d K ( t − 1 x ) , T t f = K t ∗ f . The conditions in Proposition 6.2 allow us to apply Corollary 6.1 to dyadic dilations.
Corollary 6.3. Suppose that K K K (65) and (66) . Then ( T 2 j ) j ∈ Z (T_{2^j})_{j\in\mathbb{Z}} ( T 2 j ) j ∈ Z ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞
If, in addition, K K K r ↦ K ( r x ) r\mapsto K(rx) r ↦ K ( r x ) ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) x x x ( T t ) t > 0 (T_t)_{t>0} ( T t ) t > 0
Proof. Since ∥ K 2 j ∥ 1 = ∥ K ∥ 1 \lVert K_{2^j}\rVert_1=\lVert K\rVert_1 ∥ K 2 j ∥ 1 = ∥ K ∥ 1 K ≥ 0 K\geq0 K ≥ 0 f ≥ 0 f\geq0 f ≥ 0 2 j ≤ t < 2 j + 1 2^j\leq t<2^{j+1} 2 j ≤ t < 2 j + 1
0 ≤ K t ≤ 2 d K 2 j + 1 , 0 ≤ T t f ≤ 2 d T 2 j + 1 f . 0\leq K_t\leq2^{d}K_{2^{j+1}},\qquad0\leq T_tf\leq2^dT_{2^{j+1}}f. 0 ≤ K t ≤ 2 d K 2 j + 1 , 0 ≤ T t f ≤ 2 d T 2 j + 1 f . The weak and strong type estimates therefore follow from the dyadic case. □
Remark. 6.4. For comparison, the L 2 L_2 L 2 [2 Theorem A] takes the following form for convolution kernels:
sup Q ∈ Q ∑ m ≥ 1 sup y ∈ Q ( ( 2 m ℓ ( Q ) ) d ∫ 2 m ℓ ( Q ) ≤ ∣ x − c Q ∣ ≤ 2 m + 1 ℓ ( Q ) ∣ K ( x − y ) − K ( x − c Q ) ∣ 2 d x ) 1 / 2 < ∞ . (67) \sup_{Q\in\mathcal{Q}}\sum_{m\geq1}\sup_{y\in Q}\left((2^m\ell(Q))^d\int_{2^m\ell(Q)\leq\lvert x-c_Q\rvert\leq2^{m+1}\ell(Q)}\lvert K(x-y)-K(x-c_Q)\rvert^2\,dx\right)^{1/2}<\infty. \tag*{(67)} Q ∈ Q sup m ≥ 1 ∑ y ∈ Q sup ( ( 2 m ℓ ( Q ) ) d ∫ 2 m ℓ ( Q ) ≤ ∣ x − c Q ∣ ≤ 2 m + 1 ℓ ( Q ) ∣ K ( x − y ) − K ( x − c Q ) ∣ 2 d x ) 1/2 < ∞. (67) The following kernels satisfy (65) and (66) , but fail (67) .
(i) For 0 < ε ≤ 1 / 2 0<\varepsilon\leq1/2 0 < ε ≤ 1/2
K ( x ) = ( 1 − ∣ x ∣ 2 ) ε − 1 1 { ∣ x ∣ < 1 } ( x ) . K(x)=(1-\lvert x\rvert^2)^{\varepsilon-1}\mathbf{1}_{\{\lvert x\rvert<1\}}(x). K ( x ) = ( 1 − ∣ x ∣ 2 ) ε − 1 1 {∣ x ∣ < 1 } ( x ) . In dimension one, this gives K ( x ) = ( 1 − x 2 ) ε − 1 1 ( − 1 , 1 ) ( x ) K(x)=(1-x^2)^{\varepsilon-1}\mathbf{1}_{(-1,1)}(x) K ( x ) = ( 1 − x 2 ) ε − 1 1 ( − 1 , 1 ) ( x )
(ii) For a = ( 1 , 0 , … , 0 ) a=(1,0,\ldots,0) a = ( 1 , 0 , … , 0 ) d / 2 ≤ α < d d/2\leq\alpha<d d /2 ≤ α < d
K ( x ) = ∣ x − a ∣ − α 1 { 0 < ∣ x − a ∣ < 1 / 4 } ( x ) . K(x)=\lvert x-a\rvert^{-\alpha}\mathbf{1}_{\{0<\lvert x-a\rvert<1/4\}}(x). K ( x ) = ∣ x − a ∣ − α 1 { 0 < ∣ x − a ∣ < 1/4 } ( x ) . (iii) For 1 / 2 ≤ α < 1 1/2\leq\alpha<1 1/2 ≤ α < 1
K ( x ) = ∣ x 1 − 1 ∣ − α 1 { 0 < ∣ x 1 − 1 ∣ < 1 / 4 } ∏ i = 2 d 1 { ∣ x i ∣ < 1 / 4 } . K(x)=\lvert x_1-1\rvert^{-\alpha}\mathbf{1}_{\{0<\lvert x_1-1\rvert<1/4\}}\prod_{i=2}^{d}\mathbf{1}_{\{\lvert x_i\rvert<1/4\}}. K ( x ) = ∣ x 1 − 1 ∣ − α 1 { 0 < ∣ x 1 − 1 ∣ < 1/4 } i = 2 ∏ d 1 {∣ x i ∣ < 1/4 } . (iv) In dimension one, for β > 2 \beta> 2 β > 2
K ( x ) = 1 { 0 < ∣ x − 1 ∣ < 1 / 4 } ∣ x − 1 ∣ ( log ( e / ∣ x − 1 ∣ ) ) β , K(x)=\frac{\mathbf{1}_{\{0<|x-1|<1/4\}}}{|x-1|(\log(e/|x-1|))^\beta}, K ( x ) = ∣ x − 1∣ ( log ( e /∣ x − 1∣ ) ) β 1 { 0 < ∣ x − 1∣ < 1/4 } , with K ( 1 ) = 0 K(1)=0 K ( 1 ) = 0
Each kernel has a singularity away from the origin that is not locally square integrable. For sufficiently small h = ( t , 0 , … , 0 ) h=(t,0,\ldots,0) h = ( t , 0 , … , 0 ) t > 0 t>0 t > 0 K ( ⋅ − h ) − K K(\cdot-h)-K K ( ⋅ − h ) − K (67) is infinite.
We now combine the dyadic maximal estimate with the noncommutative Banach principle to obtain bilateral almost uniform convergence.
Corollary 6.5. Let K ∈ L 1 ( R d ) K\in L_1(\mathbb{R}^d) K ∈ L 1 ( R d ) (65) and (66) , and put c K = ∫ R d K c_K=\int_{\mathbb{R}^d}K c K = ∫ R d K f ∈ L 1 ( N ) f\in L_1(\mathcal{N}) f ∈ L 1 ( N )
K 2 − n ∗ f ⟶ c K f in L 1 ( N ) and b . a . u . ( n → ∞ ) . (68) K_{2^{-n}}*f\longrightarrow c_Kf\quad\text{in }L_1(\mathcal{N})\text{ and }b.a.u.\quad(n\to\infty). \tag*{(68)} K 2 − n ∗ f ⟶ c K f in L 1 ( N ) and b . a . u . ( n → ∞ ) . (68) If K K K b . a . u . b.a.u. b . a . u . f ∈ L 1 , l o c ( R d ; L 1 ( M ) ) f\in L_{1,\mathrm{loc}}(\mathbb{R}^d;L_1(\mathcal{M})) f ∈ L 1 , loc ( R d ; L 1 ( M ))
Proof. Firstly, we have
∥ K 2 − n ∗ f − c K f ∥ 1 ≤ ∫ R d ∣ K ( y ) ∣ ∥ f ( ⋅ − 2 − n y ) − f ∥ 1 d y ⟶ 0. \|K_{2^{-n}}*f-c_Kf\|_1\leq\int_{\mathbb{R}^d}|K(y)|\|f(\cdot-2^{-n}y)-f\|_1\,dy\longrightarrow0. ∥ K 2 − n ∗ f − c K f ∥ 1 ≤ ∫ R d ∣ K ( y ) ∣∥ f ( ⋅ − 2 − n y ) − f ∥ 1 d y ⟶ 0. And for g = ∑ i = 1 m u i ⊗ a i ∈ D g=\sum_{i=1}^{m}u_i\otimes a_i\in\mathcal{D} g = ∑ i = 1 m u i ⊗ a i ∈ D
∥ K 2 − n ∗ g − c K g ∥ ∞ ≤ ∑ i = 1 m ∥ a i ∥ ∞ ∫ R d ∣ K ( y ) ∣ ∥ u i ( ⋅ − 2 − n y ) − u i ∥ ∞ d y ⟶ 0. \|K_{2^{-n}}*g-c_Kg\|_\infty\leq\sum_{i=1}^{m}\|a_i\|_\infty\int_{\mathbb{R}^d}|K(y)|\|u_i(\cdot-2^{-n}y)-u_i\|_\infty\,dy\longrightarrow0. ∥ K 2 − n ∗ g − c K g ∥ ∞ ≤ i = 1 ∑ m ∥ a i ∥ ∞ ∫ R d ∣ K ( y ) ∣∥ u i ( ⋅ − 2 − n y ) − u i ∥ ∞ d y ⟶ 0. By Corollary 6.3 , the operators f ↦ K 2 − n ∗ f f\mapsto K_{2^{-n}}*f f ↦ K 2 − n ∗ f n ≥ 0 n\geq0 n ≥ 0 ( 1 , 1 ) (1,1) ( 1 , 1 ) [3 Theorem 3.4] , the set
{ f ∈ L 1 ( N ) : ( K 2 − n ∗ f ) n ≥ 0 converges b.a.u. } \{f\in L_1(\mathcal{N}):(K_{2^{-n}}*f)_{n\geq0}\text{ converges b.a.u.}\} { f ∈ L 1 ( N ) : ( K 2 − n ∗ f ) n ≥ 0 converges b.a.u. } is closed in L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) D \mathcal{D} D L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) c K f c_Kf c K f
Suppose now that supp K ⊆ [ − R 0 , R 0 ] d \operatorname{supp}K\subseteq[-R_0,R_0]^d supp K ⊆ [ − R 0 , R 0 ] d R > 0 R>0 R > 0 n ≥ 0 n\geq0 n ≥ 0
1 [ − R , R ] d ( K 2 − n ∗ f ) = 1 [ − R , R ] d ( K 2 − n ∗ ( 1 [ − R − R 0 , R + R 0 ] d f ) ) . \mathbf{1}_{[-R,R]^d}(K_{2^{-n}}*f)=\mathbf{1}_{[-R,R]^d}\left(K_{2^{-n}}*\left(\mathbf{1}_{[-R-R_0,R+R_0]^d}f\right)\right). 1 [ − R , R ] d ( K 2 − n ∗ f ) = 1 [ − R , R ] d ( K 2 − n ∗ ( 1 [ − R − R 0 , R + R 0 ] d f ) ) . The truncated function belongs to L 1 ( N ) L_1(\mathcal{N}) L 1 ( N ) [ − R , R ] d [-R,R]^d [ − R , R ] d □ \square □
In particular, the compactly supported kernels in Remark 6.4 satisfy both the global and local convergence conclusions.
Monotone L log L L\log L L log L
Throughout this subsection, we work in dimension one and write N = L ∞ ( R ) ⊗ ‾ M \mathcal{N}=L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M} N = L ∞ ( R ) ⊗ M [30 Theorem 3, pp. 116–117] .
Let L : ( 0 , 1 ) → [ 0 , ∞ ) L:(0,1)\to[0,\infty) L : ( 0 , 1 ) → [ 0 , ∞ )
∫ 0 1 L ( t ) log ( 1 + L ( t ) ) d t < ∞ . \int_{0}^{1}L(t)\log(1+L(t))\,\mathrm{d}t<\infty. ∫ 0 1 L ( t ) log ( 1 + L ( t )) d t < ∞. Extend L L L ( 0 , 1 ) (0,1) ( 0 , 1 ) K : R → C K:\mathbb{R}\to\mathbb{C} K : R → C ∣ K ∣ ≤ L |K|\le L ∣ K ∣ ≤ L ( λ n ) n ≥ 1 (\lambda_n)_{n\ge1} ( λ n ) n ≥ 1
K n ( t ) = λ n K ( λ n t ) , T n f = K n ∗ f , c K = ∫ R K ( t ) d t . K_n(t)=\lambda_nK(\lambda_nt),\qquad T_nf=K_n*f,\qquad c_K=\int_{\mathbb{R}}K(t)\,\mathrm{d}t. K n ( t ) = λ n K ( λ n t ) , T n f = K n ∗ f , c K = ∫ R K ( t ) d t . Theorem 6.6. Suppose that λ n ≤ q λ n + 1 \lambda_n\le q\lambda_{n+1} λ n ≤ q λ n + 1 0 < q < 1 0<q<1 0 < q < 1 n ≥ 1 n\ge1 n ≥ 1 ( T n ) n ≥ 1 (T_n)_{n\ge1} ( T n ) n ≥ 1 ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\le\infty 1 < p ≤ ∞ L , p , q L,p,q L , p , q f ∈ L 1 ( N ) f\in L_1(\mathcal{N}) f ∈ L 1 ( N )
T n f ⟶ c K f in L 1 ( N ) and b.a.u. T_nf\longrightarrow c_Kf\quad\text{in }L_1(\mathcal{N})\text{ and b.a.u.} T n f ⟶ c K f in L 1 ( N ) and b.a.u. For f ∈ L 1 , l o c ( R ; L 1 ( M ) ) f\in L_{1,\mathrm{loc}}(\mathbb{R};L_1(\mathcal{M})) f ∈ L 1 , loc ( R ; L 1 ( M ))
Proof. The scalar estimate in the proof of [30 Theorem 3, pp. 116–117] gives
sup y ≠ 0 ∫ ∣ x ∣ ≥ 2 ∣ y ∣ sup n ≥ 1 ∣ g n ( x − y ) − g n ( x ) ∣ d x ≤ C L , q , g n ( t ) = λ n L ( λ n t ) . \sup_{y\ne0}\int_{|x|\ge2|y|}\sup_{n\ge1}|g_n(x-y)-g_n(x)|\,\mathrm{d}x\le C_{L,q},\qquad g_n(t)=\lambda_nL(\lambda_nt). y = 0 sup ∫ ∣ x ∣ ≥ 2∣ y ∣ n ≥ 1 sup ∣ g n ( x − y ) − g n ( x ) ∣ d x ≤ C L , q , g n ( t ) = λ n L ( λ n t ) . Since ∥ g n ∥ 1 = ∥ L ∥ 1 \|g_n\|_1=\|L\|_1 ∥ g n ∥ 1 = ∥ L ∥ 1 S n f = g n ∗ f S_nf=g_n*f S n f = g n ∗ f ∣ K n ∣ ≤ g n |K_n|\le g_n ∣ K n ∣ ≤ g n ( T n ) n (T_n)_n ( T n ) n ( S n ) n (S_n)_n ( S n ) n
On the other hand, since λ n → ∞ \lambda_n\to\infty λ n → ∞
∥ T n f − c K f ∥ 1 ≤ ∫ 0 1 ∣ K ( u ) ∣ ∥ f ( ⋅ − u / λ n ) − f ∥ 1 d u ⟶ 0. \|T_nf-c_Kf\|_1\le\int_{0}^{1}|K(u)|\|f(\mathord{\cdot}-u/\lambda_n)-f\|_1\,\mathrm{d}u\longrightarrow0. ∥ T n f − c K f ∥ 1 ≤ ∫ 0 1 ∣ K ( u ) ∣∥ f ( ⋅ − u / λ n ) − f ∥ 1 d u ⟶ 0. For f ∈ D f\in\mathcal{D} f ∈ D f ∈ L 1 ( N ) f\in L_1(\mathcal{N}) f ∈ L 1 ( N )
Finally, supp K n ⊆ [ 0 , λ n − 1 ] \operatorname{supp}K_n\subseteq[0,\lambda_n^{-1}] supp K n ⊆ [ 0 , λ n − 1 ] n n n I I I f f f [ inf I − λ n − 1 , sup I ] [\inf I-\lambda_n^{-1},\sup I] [ inf I − λ n − 1 , sup I ]
Remark 6.7. The one-dimensional endpoint kernel from Remark 6.4,
K ( x ) = ( 1 − x 2 ) ε − 1 1 ( − 1 , 1 ) ( x ) , 0 < ε < 1 , K(x)=(1-x^2)^{\varepsilon-1}\mathbf{1}_{(-1,1)}(x),\qquad0<\varepsilon<1, K ( x ) = ( 1 − x 2 ) ε − 1 1 ( − 1 , 1 ) ( x ) , 0 < ε < 1 , also gives an application of Theorem 6.6 ; see [28 Chapter II, §5.16] for the classical example. Indeed, the restriction of K K K ( 0 , 1 ) (0,1) ( 0 , 1 )
L ( t ) = ( 1 − t 2 ) ε − 1 , 0 < t < 1 , L(t)=(1-t^2)^{\varepsilon-1},\qquad0<t<1, L ( t ) = ( 1 − t 2 ) ε − 1 , 0 < t < 1 , is nondecreasing and belongs to L log L ( 0 , 1 ) L\log L(0,1) L log L ( 0 , 1 ) L L L ( 0 , 1 ) (0,1) ( 0 , 1 ) K ( x ) = L ( x ) + L ( − x ) K(x)=L(x)+L(-x) K ( x ) = L ( x ) + L ( − x ) Theorem 6.6 and the change of variables x ↦ − x x\mapsto-x x ↦ − x λ n K ( λ n ⋅ ) \lambda_nK(\lambda_n\mathord{\cdot}) λ n K ( λ n ⋅ ) ( λ n ) (\lambda_n) ( λ n )
Nilpotent groups and Poisson kernels
We first prove a logarithmic estimate for displaced interval averages. We then apply this estimate along one-parameter subgroups to obtain maximal inequalities on nilpotent groups and for Poisson integrals on symmetric spaces.
For the interval estimate, let N = L ∞ ( R ) ⊗ M \mathcal{N}=L_{\infty}(\mathbb{R})\otimes\mathcal{M} N = L ∞ ( R ) ⊗ M b ∈ R b\in\mathbb{R} b ∈ R ℓ > 0 \ell>0 ℓ > 0
B k b , ℓ f ( t ) = 1 ℓ ∫ b − ℓ / 2 b + ℓ / 2 f ( t − e − k s ) d s , k ∈ Z . B_{k}^{b,\ell}f(t)=\frac{1}{\ell}\int_{b-\ell/2}^{b+\ell/2}f(t-e^{-k}s)\,ds,\qquad k\in\mathbb{Z}. B k b , ℓ f ( t ) = ℓ 1 ∫ b − ℓ /2 b + ℓ /2 f ( t − e − k s ) d s , k ∈ Z . The following estimate is the noncommutative version of the classical displaced-interval estimate; see [28 Chapter II, §5.10, p. 78] and [4 Lemma 4, pp. 340–341] .
Lemma 6.8. For every 1 < p < ∞ 1<p<\infty 1 < p < ∞ f ∈ L p ( N ) + f\in L_p(\mathcal{N})_+ f ∈ L p ( N ) + g ∈ L p ( N ) + g\in L_p(\mathcal{N})_+ g ∈ L p ( N ) +
B k b , ℓ f ≤ g ( k ∈ Z ) , ∥ g ∥ p ≤ C p ( 1 + log ( 2 + ∣ b ∣ / ℓ ) ) 1 / p ∥ f ∥ p . (69) B_{k}^{b,\ell}f\leq g\quad(k\in\mathbb{Z}),\qquad\lVert g\rVert_p\leq C_p\left(1+\log(2+|b|/\ell)\right)^{1/p}\lVert f\rVert_p. \tag*{(69)} B k b , ℓ f ≤ g ( k ∈ Z ) , ∥ g ∥ p ≤ C p ( 1 + log ( 2 + ∣ b ∣/ ℓ ) ) 1/ p ∥ f ∥ p . (69) Proof. The operators B k b , ℓ B_k^{b,\ell} B k b , ℓ
K k ( x ) = e k ℓ 1 [ e − k ( b − ℓ / 2 ) , e − k ( b + ℓ / 2 ) ] ( x ) . K_k(x)=\frac{e^k}{\ell}\mathbf{1}_{[e^{-k}(b-\ell/2),e^{-k}(b+\ell/2)]}(x). K k ( x ) = ℓ e k 1 [ e − k ( b − ℓ /2 ) , e − k ( b + ℓ /2 )] ( x ) . The interval-kernel argument in [28 Chapter II, §5.10, p. 78] applies also to the scales e − k e^{-k} e − k
sup y ≠ 0 ∫ ∣ x ∣ ≥ 2 ∣ y ∣ sup k ∈ Z ∣ K k ( x − y ) − K k ( x ) ∣ d x ≤ C ( 1 + log ( 2 + ∣ b ∣ / ℓ ) ) . \sup_{y\neq0}\int_{\lvert x\rvert\geq2\lvert y\rvert}\sup_{k\in\mathbb{Z}}\lvert K_k(x-y)-K_k(x)\rvert\,dx\leq C(1+\log(2+|b|/\ell)). y = 0 sup ∫ ∣ x ∣ ≥ 2 ∣ y ∣ k ∈ Z sup ∣ K k ( x − y ) − K k ( x )∣ d x ≤ C ( 1 + log ( 2 + ∣ b ∣/ ℓ )) . Since ∥ K k ∥ 1 = 1 \lVert K_k\rVert_1=1 ∥ K k ∥ 1 = 1 Theorem 1.1 , with r = ∞ r=\infty r = ∞ ( 1 , 1 ) (1,1) ( 1 , 1 ) (69) . □
We now follow the cube decomposition and summation in [27 Proposition 5.1, pp. 325–326] . Let N N N X 1 , … , X d X_1,\ldots,X_d X 1 , … , X d span { X 1 , … , X i } \operatorname{span}\{X_1,\ldots,X_i\} span { X 1 , … , X i } i i i [27 p. 323] , use the canonical coordinates
n = exp ( x 1 X 1 ) ⋯ exp ( x d X d ) , d n = d x 1 ⋯ d x d , n=\exp(x_1X_1)\cdots\exp(x_dX_d),\qquad dn=dx_1\cdots dx_d, n = exp ( x 1 X 1 ) ⋯ exp ( x d X d ) , d n = d x 1 ⋯ d x d , and define
δ h ( x 1 , … , x d ) = ( e h 1 x 1 , … , e h d x d ) , h ∈ Z d . \delta_h(x_1,\ldots,x_d)=(e^{h_1}x_1,\ldots,e^{h_d}x_d),\qquad h\in\mathbb{Z}^d. δ h ( x 1 , … , x d ) = ( e h 1 x 1 , … , e h d x d ) , h ∈ Z d . These coordinate dilations need not be group automorphisms. Put A = L ∞ ( N ) ⊗ M \mathcal{A}=L_{\infty}(N)\otimes\mathcal{M} A = L ∞ ( N ) ⊗ M ∫ N d n ⊗ τ \int_N dn\otimes\tau ∫ N d n ⊗ τ
Let P : N → ( 0 , ∞ ) P:N\to(0,\infty) P : N → ( 0 , ∞ ) 1 / P 1/P 1/ P
P ( n ) ≤ C min { 1 , ∣ x 1 ∣ − η , … , ∣ x d ∣ − η } P(n)\leq C\min\{1,|x_1|^{-\eta},\ldots,|x_d|^{-\eta}\} P ( n ) ≤ C min { 1 , ∣ x 1 ∣ − η , … , ∣ x d ∣ − η } for some C , η > 0 C,\eta>0 C , η > 0 β 0 > 0 \beta_0>0 β 0 > 0 P β ∈ L 1 ( N ) P^\beta\in L_1(N) P β ∈ L 1 ( N ) β > β 0 \beta>\beta_0 β > β 0 β > β 0 \beta>\beta_0 β > β 0
T h F ( g ) = ∫ N F ( g ( δ − h n ) − 1 ) P ( n ) β d n , h ∈ Z d . T_hF(g)=\int_N F\left(g(\delta_{-h}n)^{-1}\right)P(n)^\beta\,dn,\qquad h\in\mathbb{Z}^d. T h F ( g ) = ∫ N F ( g ( δ − h n ) − 1 ) P ( n ) β d n , h ∈ Z d . Since P β ∈ L 1 ( N ) P^\beta\in L_1(N) P β ∈ L 1 ( N ) L p ( A ) L_p(\mathcal{A}) L p ( A ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ ( T h ) h ∈ Z d (T_h)_{h\in\mathbb{Z}^d} ( T h ) h ∈ Z d
Proposition 6.9. For every 1 < p < ∞ 1 < p < \infty 1 < p < ∞ F ∈ L p ( A ) F \in L_p(\mathcal{A}) F ∈ L p ( A )
∥ ( T h F ) h ∈ Z d ∥ L p ( A ; ℓ ∞ ( Z d ) ) ≤ C p , N , P , β ∥ F ∥ p . \left\|(T_hF)_{h\in\mathbb{Z}^d}\right\|_{L_p(\mathcal{A};\ell_\infty(\mathbb{Z}^d))} \leq C_{p,N,P,\beta}\|F\|_p. ∥ ( T h F ) h ∈ Z d ∥ L p ( A ; ℓ ∞ ( Z d )) ≤ C p , N , P , β ∥ F ∥ p . Moreover, if F ≥ 0 F \geq0 F ≥ 0 U ∈ L p ( A ) + U \in L_p(\mathcal{A})_+ U ∈ L p ( A ) +
T h F ≤ U ( h ∈ Z d ) , ∥ U ∥ p ≤ C p , N , P , β ∥ F ∥ p . T_hF \leq U \quad(h\in\mathbb{Z}^d), \qquad\|U\|_p \leq C_{p,N,P,\beta}\|F\|_p. T h F ≤ U ( h ∈ Z d ) , ∥ U ∥ p ≤ C p , N , P , β ∥ F ∥ p . Proof. We first construct a common positive majorant for ( T h F ) h ∈ Z d (T_hF)_{h\in\mathbb{Z}^d} ( T h F ) h ∈ Z d F ≥ 0 F \geq0 F ≥ 0 F F F Lemma 2.7 . For each i i i
α t ( i ) F ( g ) = F ( g exp ( − t X i ) ) , t ∈ R . \alpha_t^{(i)}F(g)=F(g\exp(-tX_i)), \qquad t\in\mathbb{R}. α t ( i ) F ( g ) = F ( g exp ( − t X i )) , t ∈ R . Since N N N R \mathbb{R} R A \mathcal{A} A Lemma 2.10 to this action and using Lemma 6.8, we deduce that for every b ∈ R b\in\mathbb{R} b ∈ R ℓ > 0 \ell>0 ℓ > 0 1 < p < ∞ 1<p<\infty 1 < p < ∞ F ∈ L p ( A ) + F\in L_p(\mathcal{A})_+ F ∈ L p ( A ) + G ∈ L p ( A ) + G\in L_p(\mathcal{A})_+ G ∈ L p ( A ) +
1 ℓ ∫ b − ℓ / 2 b + ℓ / 2 F ( g exp ( − e − k s X i ) ) d s ≤ G ( g ) ( k ∈ Z ) , (70) \frac{1}{\ell}\int_{b-\ell/2}^{b+\ell/2}F(g\exp(-e^{-k}sX_i))\,\mathrm{d}s\leq G(g)\qquad(k\in\mathbb{Z}), \tag*{(70)} ℓ 1 ∫ b − ℓ /2 b + ℓ /2 F ( g exp ( − e − k s X i )) d s ≤ G ( g ) ( k ∈ Z ) , (70) and
∥ G ∥ p ≤ C p ( 1 + log ( 2 + ∣ b ∣ / ℓ ) ) 1 / p ∥ F ∥ p . (71) \|G\|_p\leq C_p\left(1+\log(2+|b|/\ell)\right)^{1/p}\|F\|_p. \tag*{(71)} ∥ G ∥ p ≤ C p ( 1 + log ( 2 + ∣ b ∣/ ℓ ) ) 1/ p ∥ F ∥ p . (71) Without loss of generality, we may assume that P ≤ 1 / 2 P\leq1/2 P ≤ 1/2
E m = { n ∈ N : P ( n ) > 2 − m } , m ≥ 1. E_m=\{n\in\mathbb{N}:P(n)>2^{-m}\},\qquad m\geq1. E m = { n ∈ N : P ( n ) > 2 − m } , m ≥ 1. The scalar construction in [27 p. 325] gives pairwise disjoint coordinate cubes Q j m Q_j^m Q j m 1 ≤ j ≤ J m 1\leq j\leq J_m 1 ≤ j ≤ J m ℓ m = 2 − c m \ell_m=2^{-cm} ℓ m = 2 − c m
E m ⊂ ⋃ j = 1 J m Q j m ⊂ E m + 1 , (72) E_m\subset\bigcup_{j=1}^{J_m}Q_j^m\subset E_{m+1}, \tag*{(72)} E m ⊂ j = 1 ⋃ J m Q j m ⊂ E m + 1 , (72) J m ℓ m d ≤ ∣ E m + 1 ∣ , ∣ ( c Q j m ) i ∣ ≤ 2 C m , J_m\ell_m^d\leq|E_{m+1}|,\qquad|(cQ_j^m)_i|\leq2^{Cm}, J m ℓ m d ≤ ∣ E m + 1 ∣ , ∣ ( c Q j m ) i ∣ ≤ 2 C m , where c , C > 0 c,C>0 c , C > 0 P P P
P β ≤ C β ∑ m ≥ 1 2 − β m 1 E m . (73) P^\beta\leq C_\beta\sum_{m\geq1}2^{-\beta m}\mathbf{1}_{E_m}. \tag*{(73)} P β ≤ C β m ≥ 1 ∑ 2 − β m 1 E m . (73) For fixed m , j m,j m , j
A i , k m , j F ( g ) = 1 ℓ m ∫ ( c Q j m ) i − ℓ m / 2 ( c Q j m ) i + ℓ m / 2 F ( g exp ( − e − k s X i ) ) d s . A_{i,k}^{m,j}F(g)=\frac{1}{\ell_m}\int_{(cQ_j^m)_i-\ell_m/2}^{(cQ_j^m)_i+\ell_m/2}F(g\exp(-e^{-k}sX_i))\,\mathrm{d}s. A i , k m , j F ( g ) = ℓ m 1 ∫ ( c Q j m ) i − ℓ m /2 ( c Q j m ) i + ℓ m /2 F ( g exp ( − e − k s X i )) d s . Apply (70) and (71) with b = ( c Q j m ) i b=(cQ_j^m)_i b = ( c Q j m ) i ℓ = ℓ m \ell=\ell_m ℓ = ℓ m (72) gives
∣ ( c Q j m ) i ∣ ℓ m ≤ 2 ( C + c ) m , \frac{|(cQ_j^m)_i|}{\ell_m}\leq2^{(C+c)m}, ℓ m ∣ ( c Q j m ) i ∣ ≤ 2 ( C + c ) m , we have
C p ( 1 + log ( 2 + ∣ ( c Q j m ) i ∣ ℓ m ) ) 1 / p ≤ C p , N , P ( 1 + m ) 1 / p . C_p\left(1+\log\left(2+\frac{|(cQ_j^m)_i|}{\ell_m}\right)\right)^{1/p}\leq C_{p,N,P}(1+m)^{1/p}. C p ( 1 + log ( 2 + ℓ m ∣ ( c Q j m ) i ∣ ) ) 1/ p ≤ C p , N , P ( 1 + m ) 1/ p . By the coordinate formula for the inverse, we get
1 ℓ m d ∫ Q j m F ( g ( δ − h n ) − 1 ) d n = A d , h d m , j ⋯ A 1 , h 1 m , j F ( g ) . (74) \frac{1}{\ell_m^d}\int_{Q_j^m}F(g(\delta_{-hn})^{-1})\,\mathrm{d}n
=A_{d,h_d}^{m,j}\cdots A_{1,h_1}^{m,j}F(g). \tag*{(74)} ℓ m d 1 ∫ Q j m F ( g ( δ − hn ) − 1 ) d n = A d , h d m , j ⋯ A 1 , h 1 m , j F ( g ) . (74) For F ≥ 0 F \ge0 F ≥ 0 U 0 m , j = F U_0^{m,j}=F U 0 m , j = F (70) , (71) and the preceding bound, we may successively choose U i m , j ∈ L p ( A ) + U_i^{m,j} \in L_p(\mathcal{A})_+ U i m , j ∈ L p ( A ) +
A i , k m , j U i − 1 m , j ≤ U i m , j ( k ∈ Z ) , A_{i,k}^{m,j}U_{i-1}^{m,j}\le U_i^{m,j}\qquad(k\in\mathbb{Z}), A i , k m , j U i − 1 m , j ≤ U i m , j ( k ∈ Z ) , and
∥ U i m , j ∥ p ≤ C p , N , P ( 1 + m ) 1 / p ∥ U i − 1 m , j ∥ p . \lVert U_i^{m,j}\rVert_p\le C_{p,N,P}(1+m)^{1/p}\lVert U_{i-1}^{m,j}\rVert_p. ∥ U i m , j ∥ p ≤ C p , N , P ( 1 + m ) 1/ p ∥ U i − 1 m , j ∥ p . By positivity and (74) ,
1 ℓ m d ∫ Q j m F ( g ( δ − h n ) − 1 ) d n ≤ U d m , j ( g ) ( h ∈ Z d ) . (75) \frac{1}{\ell_m^d}\int_{Q_j^m}F\bigl(g(\delta_{-h}n)^{-1}\bigr)\,dn\le U_d^{m,j}(g)\qquad(h\in\mathbb{Z}^d). \tag*{(75)} ℓ m d 1 ∫ Q j m F ( g ( δ − h n ) − 1 ) d n ≤ U d m , j ( g ) ( h ∈ Z d ) . (75) Moreover, iterating the norm estimate yields
∥ U d m , j ∥ p ≤ C p , N , P ( 1 + m ) d / p ∥ F ∥ p . (76) \lVert U_d^{m,j}\rVert_p\le C_{p,N,P}(1+m)^{d/p}\lVert F\rVert_p. \tag*{(76)} ∥ U d m , j ∥ p ≤ C p , N , P ( 1 + m ) d / p ∥ F ∥ p . (76) Choose β 0 < β ′ < β \beta_0<\beta'<\beta β 0 < β ′ < β ∣ E m + 1 ∣ ≤ 2 β ′ ( m + 1 ) ∥ P β ′ ∥ 1 \lvert E_{m+1}\rvert\le2^{\beta'(m+1)}\lVert P^{\beta'}\rVert_1 ∣ E m + 1 ∣ ≤ 2 β ′ ( m + 1 ) ∥ P β ′ ∥ 1
U = C β ∑ m ≥ 1 2 − β m ℓ m d ∑ j = 1 J m U d m , j . U=C_\beta\sum_{m\ge1}2^{-\beta m}\ell_m^d\sum_{j=1}^{J_m}U_d^{m,j}. U = C β m ≥ 1 ∑ 2 − β m ℓ m d j = 1 ∑ J m U d m , j . By (76) , (72) and the preceding measure estimate,
∥ U ∥ p ≤ C p , N , P , β ∑ m ≥ 1 2 − β m ( 1 + m ) d / p J m ℓ m d ∥ F ∥ p ≤ C p , N , P , β , β ′ ∥ P β ′ ∥ 1 ∑ m ≥ 1 ( 1 + m ) d / p 2 − ( β − β ′ ) m ∥ F ∥ p < ∞ . \begin{aligned}
\lVert U\rVert_p\le C_{p,N,P,\beta}\sum_{m\ge1}2^{-\beta m}(1+m)^{d/p}J_m\ell_m^d\lVert F\rVert_p \\
&\le C_{p,N,P,\beta,\beta'}\lVert P^{\beta'}\rVert_1\sum_{m\ge1}(1+m)^{d/p}2^{-(\beta-\beta')m}\lVert F\rVert_p<\infty.
\end{aligned} ∥ U ∥ p ≤ C p , N , P , β m ≥ 1 ∑ 2 − β m ( 1 + m ) d / p J m ℓ m d ∥ F ∥ p ≤ C p , N , P , β , β ′ ∥ P β ′ ∥ 1 m ≥ 1 ∑ ( 1 + m ) d / p 2 − ( β − β ′ ) m ∥ F ∥ p < ∞. Thus the positive series defining U U U L p ( A ) L_p(\mathcal{A}) L p ( A ) (73) , (72) and (75) yield
T h F ( g ) ≤ C β ∑ m ≥ 1 2 − β m ∑ j = 1 J m ∫ Q j m F ( g ( δ − h n ) − 1 ) d n ≤ U ( g ) . T_hF(g)\le C_\beta\sum_{m\ge1}2^{-\beta m}\sum_{j=1}^{J_m}\int_{Q_j^m}F\bigl(g(\delta_{-h}n)^{-1}\bigr)\,dn\le U(g). T h F ( g ) ≤ C β m ≥ 1 ∑ 2 − β m j = 1 ∑ J m ∫ Q j m F ( g ( δ − h n ) − 1 ) d n ≤ U ( g ) . Hence U U U ( T h F ) h ∈ Z d (T_hF)_{h\in\mathbb{Z}^d} ( T h F ) h ∈ Z d F F F
( T h ( ∣ F ∗ ∣ ) T h F ( T h F ) ∗ T h ( ∣ F ∣ ) ) ≥ 0 \begin{pmatrix}
T_h(\lvert F^*\rvert) & T_hF\\
(T_hF)^* & T_h(\lvert F\rvert)
\end{pmatrix}\ge0 ( T h (∣ F ∗ ∣) ( T h F ) ∗ T h F T h (∣ F ∣) ) ≥ 0 and Lemma 2.7 give the maximal estimate from the positive case, as in the proof of Corollary 6.1. □
We next apply Proposition 6.9 to Poisson integrals on symmetric spaces. We use the boundary coordinates and Poisson formulas from [27 Sections 2–5] .
Let X = G / K X=G/K X = G / K G G G K K K G = K A N G=KAN G = K A N a = Lie ( A ) \mathfrak{a}=\operatorname{Lie}(A) a = Lie ( A ) Σ + \Sigma^+ Σ + Π \Pi Π
2 ρ = ∑ α ∈ Σ + ( dim g α ) α . 2\rho=\sum_{\alpha\in\Sigma^+}(\dim\mathfrak{g}_\alpha)\alpha. 2 ρ = α ∈ Σ + ∑ ( dim g α ) α . where g α \mathfrak{g}_{\alpha} g α H : G → a H:G\to\mathfrak{a} H : G → a M 0 = Z K ( A ) M_{0}=Z_{K}(A) M 0 = Z K ( A ) B = M 0 A N B=M_{0}AN B = M 0 A N S = G / B = K / M 0 S=G/B=K/M_{0} S = G / B = K / M 0 K K K σ \sigma σ π B : G → S \pi_{B}:G\to S π B : G → S
Fix E ⊊ Π E\subsetneq\Pi E ⊊ Π
a ( E ) = ⋂ α ∈ E ker α , A ( E ) = exp a ( E ) . \mathfrak{a}(E)=\bigcap_{\alpha\in E}\ker\alpha,\qquad A(E)=\exp\mathfrak{a}(E). a ( E ) = α ∈ E ⋂ ker α , A ( E ) = exp a ( E ) . and
A ( E ) Z = { a ∈ A ( E ) : α ( log a ) ∈ Z for every α ∈ Π ∖ E } . A(E)_{\mathbb{Z}}=\{a\in A(E):\alpha(\log a)\in\mathbb{Z}\text{ for every }\alpha\in\Pi\setminus E\}. A ( E ) Z = { a ∈ A ( E ) : α ( log a ) ∈ Z for every α ∈ Π ∖ E } . Let N ‾ ( E ) \overline{N}(E) N ( E )
n ‾ ( E ) = ⨁ α ∈ Σ + α ∣ a ( E ) ≠ 0 g − α . \overline{\mathfrak{n}}(E)=\bigoplus_{\substack{\alpha\in\Sigma^{+}\\ \alpha|_{\mathfrak{a}(E)}\ne0}}\mathfrak{g}_{-\alpha}. n ( E ) = α ∈ Σ + α ∣ a ( E ) = 0 ⨁ g − α . Let G E G^{E} G E Z G ( A ( E ) ) Z_{G}(A(E)) Z G ( A ( E )) K E = G E ∩ K K^{E}=G^{E}\cap K K E = G E ∩ K d k dk d k K E K^{E} K E N ‾ ( E ) \overline{N}(E) N ( E ) A ( E ) A(E) A ( E )
Normalize Haar measure d n dn d n N ‾ ( E ) \overline{N}(E) N ( E ) [27 Section 3, p. 318] reads
∫ S f ( b ) d σ ( b ) = ∫ K E ∫ N ‾ ( E ) f ( π B ( n k ) ) e − ⟨ 2 ρ , H ( n k ) ⟩ d n d k . (77) \int_{S}f(b)\,d\sigma(b)=\int_{K^{E}}\int_{\overline{N}(E)}f\bigl(\pi_{B}(nk)\bigr)e^{-\langle2\rho,H(nk)\rangle}\,dn\,dk. \tag*{(77)} ∫ S f ( b ) d σ ( b ) = ∫ K E ∫ N ( E ) f ( π B ( nk ) ) e − ⟨ 2 ρ , H ( nk )⟩ d n d k . (77) With this normalization,
∫ N ‾ ( E ) e − ⟨ 2 ρ , H ( n k ) ⟩ d n = 1 , k ∈ K E . \int_{\overline{N}(E)}e^{-\langle2\rho,H(nk)\rangle}\,dn=1,\qquad k\in K^{E}. ∫ N ( E ) e − ⟨ 2 ρ , H ( nk )⟩ d n = 1 , k ∈ K E . The formula also holds for positive operator-valued functions. On the open boundary chart, the N ‾ ( E ) \overline{N}(E) N ( E ) M 0 ∩ K E M_{0}\cap K^{E} M 0 ∩ K E
Choose the canonical basis above from the negative root spaces, with X i ∈ g − α i X_{i}\in\mathfrak{g}_{-\alpha_{i}} X i ∈ g − α i d n = d x 1 ⋯ d x d dn=dx_{1}\cdots dx_{d} d n = d x 1 ⋯ d x d ( α i ) i = 1 d (\alpha_{i})_{i=1}^{d} ( α i ) i = 1 d a ∈ A ( E ) Z a\in A(E)_{\mathbb{Z}} a ∈ A ( E ) Z
h ( a ) = ( α 1 ( log a ) , … , α d ( log a ) ) ∈ Z d , J ( a ) = e ⟨ 2 ρ , log a ⟩ . h(a)=(\alpha_{1}(\log a),\ldots,\alpha_{d}(\log a))\in\mathbb{Z}^{d},\qquad J(a)=e^{\langle2\rho,\log a\rangle}. h ( a ) = ( α 1 ( log a ) , … , α d ( log a )) ∈ Z d , J ( a ) = e ⟨ 2 ρ , l o g a ⟩ . Define κ a ( n ) = a − 1 n a \kappa_{a}(n)=a^{-1}na κ a ( n ) = a − 1 na
κ a ( n ) = δ h ( a ) n , d ( κ a n ) = e ∑ i α i ( log a ) d n = J ( a ) d n . (78) \kappa_{a}(n)=\delta_{h(a)}n,\qquad d(\kappa_{a}n)=e^{\sum_{i}\alpha_{i}(\log a)}\,dn=J(a)\,dn. \tag*{(78)} κ a ( n ) = δ h ( a ) n , d ( κ a n ) = e ∑ i α i ( l o g a ) d n = J ( a ) d n . (78) The last equality follows because the roots in the subsystem generated by E E E a ( E ) \mathfrak{a}(E) a ( E )
The scalar Poisson kernel
P ( n ) = e − ⟨ 2 ρ , H ( n − 1 ) ⟩ , n ∈ N ‾ ( E ) , P(n)=e^{-\langle2\rho,H(n^{-1})\rangle},\qquad n\in\overline{N}(E), P ( n ) = e − ⟨ 2 ρ , H ( n − 1 )⟩ , n ∈ N ( E ) , satisfies conditions (i)–(iii) of Proposition 6.9 with β 0 = 1 / 2 \beta_{0}=1/2 β 0 = 1/2 [27 Section 5, p. 324] .
Put
A E = L ∞ ( N ‾ ( E ) , d n ) ⊗ M , B S = L ∞ ( S , σ ) ⊗ M , \mathcal{A}_{E}=L_{\infty}(\overline{N}(E),dn)\otimes\mathcal{M},\qquad\mathcal{B}_{S}=L_{\infty}(S,\sigma)\otimes\mathcal{M}, A E = L ∞ ( N ( E ) , d n ) ⊗ M , B S = L ∞ ( S , σ ) ⊗ M , with their product traces. The Poisson integral of a boundary function is
P f ( g ) = ∫ K f ( π B ( g k ) ) d k , g ∈ G , \mathcal{P}f(g)=\int_{K}f\bigl(\pi_{B}(gk)\bigr)\,dk,\qquad g\in G, P f ( g ) = ∫ K f ( π B ( g k ) ) d k , g ∈ G , where d k dk d k K K K
Corollary 6.10. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞
(i) For every β > 1 / 2 \beta> 1/2 β > 1/2 the operators
Q a ( β ) F ( g ) = J ( a ) ∫ N ( E ) ‾ F ( n ) P ( κ a ( n − 1 g ) ) β d n , a ∈ A ( E ) Z , Q_a^{(\beta)}F(g)=J(a)\int_{\overline{N(E)}}F(n)P\bigl(\kappa_a(n^{-1}g)\bigr)^\beta\,dn,\qquad a\in A(E)_{\mathbb Z}, Q a ( β ) F ( g ) = J ( a ) ∫ N ( E ) F ( n ) P ( κ a ( n − 1 g ) ) β d n , a ∈ A ( E ) Z , satisfy
∥ ( Q a ( β ) F ) a ∈ A ( E ) Z ∥ L p ( A E ; ℓ ∞ ) ≤ C p , E , β ∥ F ∥ L p ( A E ) . \left\|\bigl(Q_a^{(\beta)}F\bigr)_{a\in A(E)_{\mathbb Z}}\right\|_{L_p(A_E;\ell_\infty)}
\leq C_{p,E,\beta}\|F\|_{L_p(A_E)}. ( Q a ( β ) F ) a ∈ A ( E ) Z L p ( A E ; ℓ ∞ ) ≤ C p , E , β ∥ F ∥ L p ( A E ) . (ii) Let L , U ⊂ N ‾ ( E ) L,U\subset\overline{N}(E) L , U ⊂ N ( E ) and V ⊂ G E V\subset G^E V ⊂ G E be compact, and let f ∈ L p ( B S ) f\in L_p(B_S) f ∈ L p ( B S ) be supported in π B ( L K E ) \pi_B(LK^E) π B ( L K E ) Set
Π a , n 0 , v f ( g ) = P f ( g a n 0 v ) , I = A ( E ) × U × V . \Pi_{a,n_0,v}f(g)=\mathcal{P}f(gan_0v),\qquad I=A(E)\times U\times V. Π a , n 0 , v f ( g ) = P f ( g a n 0 v ) , I = A ( E ) × U × V . Then
∥ ( Π a , n 0 , v f ) ( a , n 0 , v ) ∈ I ∥ L p ( A E ; ℓ ∞ ( I ) ) ≤ C p , E , L , U , V ∥ f ∥ L p ( B S ) . \left\|\bigl(\Pi_{a,n_0,v}f\bigr)_{(a,n_0,v)\in I}\right\|_{L_p(A_E;\ell_\infty(I))}
\leq C_{p,E,L,U,V}\|f\|_{L_p(B_S)}. ( Π a , n 0 , v f ) ( a , n 0 , v ) ∈ I L p ( A E ; ℓ ∞ ( I )) ≤ C p , E , L , U , V ∥ f ∥ L p ( B S ) . Proof. For (i), put u = κ a ( n − 1 g ) u=\kappa_a(n^{-1}g) u = κ a ( n − 1 g ) (78) ,
n = g ( δ − h ( a ) u ) − 1 , d n = J ( a ) − 1 d u . n=g\bigl(\delta_{-h(a)}u\bigr)^{-1},\qquad dn=J(a)^{-1}\,du. n = g ( δ − h ( a ) u ) − 1 , d n = J ( a ) − 1 d u . Consequently,
Q a ( β ) F ( g ) = ∫ N ( E ) ‾ F ( g ( δ − h ( a ) u ) − 1 ) P ( u ) β d u = T h ( a ) F ( g ) . (79) Q_a^{(\beta)}F(g)=\int_{\overline{N(E)}}F\bigl(g(\delta_{-h(a)}u)^{-1}\bigr)P(u)^\beta\,du=T_{h(a)}F(g). \tag*{(79)} Q a ( β ) F ( g ) = ∫ N ( E ) F ( g ( δ − h ( a ) u ) − 1 ) P ( u ) β d u = T h ( a ) F ( g ) . (79) Proposition 6.9 therefore proves (i).
For (ii), assume that L L L f ≥ 0 f\geq0 f ≥ 0
F ( n , k ) = f ( π B ( n k ) ) , ( n , k ) ∈ N ‾ ( E ) × K E , F(n,k)=f\bigl(\pi_B(nk)\bigr),\qquad(n,k)\in\overline{N}(E)\times K^E, F ( n , k ) = f ( π B ( nk ) ) , ( n , k ) ∈ N ( E ) × K E , and put
A ~ E = L ∞ ( N ‾ ( E ) × K E , d n d k ) ⊗ ‾ M . \widetilde{\mathcal{A}}_E=L_\infty\bigl(\overline{N}(E)\times K^E,dn\,dk\bigr)\overline{\otimes}\mathcal{M}. A E = L ∞ ( N ( E ) × K E , d n d k ) ⊗ M . Since the weight e − ⟨ 2 ρ , H ( n k ) ⟩ e^{-\langle2\rho,H(nk)\rangle} e − ⟨ 2 ρ , H ( nk )⟩ L × K E L\times K^E L × K E c L = min ( n , k ) ∈ L × K E e − ⟨ 2 ρ , H ( n k ) ⟩ > 0 c_L=\min_{(n,k)\in L\times K^E}e^{-\langle2\rho,H(nk)\rangle}>0 c L = min ( n , k ) ∈ L × K E e − ⟨ 2 ρ , H ( nk )⟩ > 0 (77) to τ ( ∣ f ∣ p ) \tau(|f|^p) τ ( ∣ f ∣ p ) F ( n , k ) = 0 F(n,k)=0 F ( n , k ) = 0 n ∉ L n\notin L n ∈ / L
∥ f ∥ L p ( B S ) p = ∫ K E ∫ L τ ( ∣ F ( n , k ) ∣ p ) e − ⟨ 2 ρ , H ( n k ) ⟩ d n d k ≥ c L ∫ K E ∫ L τ ( ∣ F ( n , k ) ∣ p ) d n d k . \|f\|_{L_p(B_S)}^p
=\int_{K^E}\int_L\tau\bigl(|F(n,k)|^p\bigr)e^{-\langle2\rho,H(nk)\rangle}\,dn\,dk
\geq c_L\int_{K^E}\int_L\tau\bigl(|F(n,k)|^p\bigr)\,dn\,dk. ∥ f ∥ L p ( B S ) p = ∫ K E ∫ L τ ( ∣ F ( n , k ) ∣ p ) e − ⟨ 2 ρ , H ( nk )⟩ d n d k ≥ c L ∫ K E ∫ L τ ( ∣ F ( n , k ) ∣ p ) d n d k . Taking p p p
∥ F ∥ L p ( A ~ E ) ≤ c L − 1 / p ∥ f ∥ L p ( B S ) . (80) \|F\|_{L_p(\widetilde{\mathcal{A}}_E)}
\leq c_L^{-1/p}\|f\|_{L_p(B_S)}. \tag*{(80)} ∥ F ∥ L p ( A E ) ≤ c L − 1/ p ∥ f ∥ L p ( B S ) . (80) For k ∈ K E k\in K^E k ∈ K E κ k ( n ) = k − 1 n k \kappa_k(n)=k^{-1}nk κ k ( n ) = k − 1 nk K E K^E K E κ k \kappa_k κ k d n dn d n K E K^E K E A ( E ) A(E) A ( E ) κ k \kappa_k κ k κ a \kappa_a κ a U F ( g , k ) = F ( κ k − 1 ( g ) , k ) UF(g,k)=F(\kappa_k^{-1}(g),k) U F ( g , k ) = F ( κ k − 1 ( g ) , k ) A ~ E \widetilde{\mathcal{A}}_E A E
Apply (i), with β = 1 \beta=1 β = 1 L ∞ ( K E ) ⊗ ‾ M L_\infty(K^E)\overline{\otimes}\mathcal{M} L ∞ ( K E ) ⊗ M
R a F ( g , k ) = U − 1 Q a ( 1 ) U F ( g , k ) = J ( a ) ∫ N ‾ ( E ) F ( n , k ) P ( κ k κ a ( n − 1 g ) ) d n , R_aF(g,k)=\mathcal{U}^{-1}Q_a^{(1)}\mathcal{U}F(g,k)
=J(a)\int_{\overline{N}(E)}F(n,k)P\bigl(\kappa_k\kappa_a(n^{-1}g)\bigr)\,dn, R a F ( g , k ) = U − 1 Q a ( 1 ) U F ( g , k ) = J ( a ) ∫ N ( E ) F ( n , k ) P ( κ k κ a ( n − 1 g ) ) d n , where Q a ( 1 ) Q_a^{(1)} Q a ( 1 ) N ‾ ( E ) \overline{N}(E) N ( E ) W ∈ L p ( A ~ E ) + W\in L_p(\widetilde{\mathcal{A}}_E)_+ W ∈ L p ( A E ) +
R a F ≤ W ( a ∈ A ( E ) Z ) , ∥ W ∥ p ≤ C p , E ∥ F ∥ p . R_aF\leq W\quad(a\in A(E)_{\mathbb Z}),\qquad\|W\|_p\leq C_{p,E}\|F\|_p. R a F ≤ W ( a ∈ A ( E ) Z ) , ∥ W ∥ p ≤ C p , E ∥ F ∥ p . Integration in the compact variable defines a positive contraction
E : L p ( A ~ E ) → L p ( A E ) , E W ( g ) = ∫ K E W ( g , k ) d k . \mathcal{E}:L_p(\widetilde{\mathcal{A}}_E)\to L_p(\mathcal{A}_E),\qquad\mathcal{E}W(g)=\int_{K_E}W(g,k)\,dk. E : L p ( A E ) → L p ( A E ) , E W ( g ) = ∫ K E W ( g , k ) d k . To express the Poisson integral in these coordinates, apply (77) to b ↦ f ( g a ⋅ b ) b\mapsto f(ga\cdot b) b ↦ f ( g a ⋅ b ) n = g a m a − 1 n=gama^{-1} n = g am a − 1 a a a K E K^E K E B B B
π B ( g a m k ) = π B ( n k ) , m = κ a ( g − 1 n ) , d m = J ( a ) d n . \pi_B(gamk)=\pi_B(nk),\qquad m=\kappa_a(g^{-1}n),\qquad dm=J(a)\,dn. π B ( g amk ) = π B ( nk ) , m = κ a ( g − 1 n ) , d m = J ( a ) d n . By the left K K K H H H
e − ⟨ 2 ρ , H ( m k ) ⟩ = e − ⟨ 2 ρ , H ( k − 1 m k ) ⟩ = P ( κ k κ a ( n − 1 g ) ) . e^{-\langle2\rho,H(mk)\rangle}=e^{-\langle2\rho,H(k^{-1}mk)\rangle}=P(\kappa_k\kappa_a(n^{-1}g)). e − ⟨ 2 ρ , H ( mk )⟩ = e − ⟨ 2 ρ , H ( k − 1 mk )⟩ = P ( κ k κ a ( n − 1 g )) . Hence
P f ( g a ) = E R a F ( g ) ≤ E W ( g ) , a ∈ A ( E ) Z . (81) \mathcal{P}f(ga)=\mathcal{E}R_aF(g)\leq\mathcal{E}W(g),\qquad a\in A(E)Z. \tag*{(81)} P f ( g a ) = E R a F ( g ) ≤ E W ( g ) , a ∈ A ( E ) Z . (81) This is the operator-valued form of the compact integration in [27 (4.7), p. 323] .
For an arbitrary a ∈ A ( E ) a\in A(E) a ∈ A ( E ) [ a ] ∈ A ( E ) Z [a]\in A(E)Z [ a ] ∈ A ( E ) Z
α ( log [ a ] ) = ⌊ α ( log a ) ⌋ , α ∈ Π ∖ E . \alpha(\log[a])=\lfloor\alpha(\log a)\rfloor,\qquad\alpha\in\Pi\setminus E. α ( log [ a ]) = ⌊ α ( log a )⌋ , α ∈ Π ∖ E . Then [ a ] − 1 a [a]^{-1}a [ a ] − 1 a A ( E ) A(E) A ( E ) z = [ a ] − 1 a n 0 v z=[a]^{-1}an_0v z = [ a ] − 1 a n 0 v D D D E E E U U U V V V
By the scalar Poisson formula [27 (2.1), p. 316] , the pushforward of σ \sigma σ b ↦ z ⋅ b b\mapsto z\cdot b b ↦ z ⋅ b
d ( z ∗ σ ) d σ ( k M 0 ) = e − ⟨ 2 ρ , H ( z − 1 k ) ⟩ . \frac{d(z_*\sigma)}{d\sigma}(kM_0)=e^{-\langle2\rho,H(z^{-1}k)\rangle}. d σ d ( z ∗ σ ) ( k M 0 ) = e − ⟨ 2 ρ , H ( z − 1 k )⟩ . Thus C D = max z ∈ D , k ∈ K e − ⟨ 2 ρ , H ( z − 1 k ) ⟩ < ∞ C_D=\max_{z\in D,\,k\in K}e^{-\langle2\rho,H(z^{-1}k)\rangle}<\infty C D = max z ∈ D , k ∈ K e − ⟨ 2 ρ , H ( z − 1 k )⟩ < ∞ f ≥ 0 f\geq0 f ≥ 0
P f ( g [ a ] z ) = ∫ S f ( g [ a ] ⋅ b ) d ( z ∗ σ ) ( b ) ≤ C D ∫ S f ( g [ a ] ⋅ b ) d σ ( b ) = C D P f ( g [ a ] ) . \mathcal{P}f(g[a]z)=\int_S f(g[a]\cdot b)\,d(z_*\sigma)(b)\leq C_D\int_S f(g[a]\cdot b)\,d\sigma(b)=C_D\mathcal{P}f(g[a]). P f ( g [ a ] z ) = ∫ S f ( g [ a ] ⋅ b ) d ( z ∗ σ ) ( b ) ≤ C D ∫ S f ( g [ a ] ⋅ b ) d σ ( b ) = C D P f ( g [ a ]) . Combining this with (81) , we obtain
Π a , n 0 , v f ≤ C D E W ( ( a , n 0 , v ) ∈ I ) . \Pi_{a,n_0,v}f\leq C_D\mathcal{E}W\qquad((a,n_0,v)\in\mathcal{I}). Π a , n 0 , v f ≤ C D E W (( a , n 0 , v ) ∈ I ) . Moreover, by (80) ,
∥ C D E W ∥ p ≤ C D C p , E c L − 1 / p ∥ f ∥ L p ( B S ) . \|C_D\mathcal{E}W\|_p\leq C_D C_{p,E}c_L^{-1/p}\|f\|_{L_p(\mathcal{B}_S)}. ∥ C D E W ∥ p ≤ C D C p , E c L − 1/ p ∥ f ∥ L p ( B S ) . This proves the positive case. The estimate for general f f f ∣ f ∗ ∣ |f^*| ∣ f ∗ ∣ ∣ f ∣ |f| ∣ f ∣ Lemma 2.7 .
Applications II: Fourier multipliers, Schur multipliers, and double operator integrals
The applications in this section follow from a maximal inequality for trace-preserving actions. We first establish this inequality and then apply it to Fourier multipliers, Schur multipliers and double operator integrals. For group von Neumann algebras, we also obtain bilateral almost uniform convergence of dyadic Fourier summation methods.
Throughout this section, unless otherwise stated, let ( K λ ) λ ∈ Λ ⊂ L 1 ( R d ) (K_\lambda)_{\lambda\in\Lambda}\subset L_1(\mathbb{R}^d) ( K λ ) λ ∈ Λ ⊂ L 1 ( R d )
sup λ ∈ Λ ∥ K λ ∥ 1 ≤ A 1 < ∞ . (82) \sup_{\lambda\in\Lambda}\|K_\lambda\|_1\leq A_1<\infty. \tag*{(82)} λ ∈ Λ sup ∥ K λ ∥ 1 ≤ A 1 < ∞. (82) and the joint Hörmander condition (H) with constant C H C_H C H
K λ ^ ( ξ ) = ∫ R d K λ ( y ) e − 2 π i ⟨ y , ξ ⟩ d y . \widehat{K_\lambda}(\xi)=\int_{\mathbb{R}^d}K_\lambda(y)e^{-2\pi i\langle y,\xi\rangle}\,dy. K λ ( ξ ) = ∫ R d K λ ( y ) e − 2 π i ⟨ y , ξ ⟩ d y . Let α = ( α y ) y ∈ R d \alpha=(\alpha_y)_{y\in\mathbb{R}^d} α = ( α y ) y ∈ R d ∗ * ∗ M \mathcal{M} M α 0 = id M \alpha_0=\operatorname{id}_{\mathcal{M}} α 0 = id M α y + z = α y α z \alpha_{y+z}=\alpha_y\alpha_z α y + z = α y α z y ↦ α y ( x ) y\mapsto\alpha_y(x) y ↦ α y ( x ) x ∈ M x\in\mathcal{M} x ∈ M
T λ α x = ∫ R d K λ ( y ) α y ( x ) d y . (83) T^\alpha_\lambda x=\int_{\mathbb{R}^d}K_\lambda(y)\alpha_y(x)\,dy. \tag*{(83)} T λ α x = ∫ R d K λ ( y ) α y ( x ) d y . (83) Corollary 7.1. The family ( T λ α ) λ ∈ Λ (T^\alpha_\lambda)_{\lambda\in\Lambda} ( T λ α ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ d d d A 1 A_1 A 1 C H C_H C H p p p
Proof. Apply Corollary 6.1 and Lemma 2.10 to the kernels ∣ K λ ∣ |K_\lambda| ∣ K λ ∣
Fourier multipliers
We apply Corollary 7.1 to the natural trace-preserving actions on group von Neumann algebras, quantum Euclidean spaces and quantum tori.
Group von Neumann algebras
Let G G G VN ( G ) \operatorname{VN}(G) VN ( G ) ( u g ) g ∈ G (u_g)_{g\in G} ( u g ) g ∈ G τ G ( u g ) = 1 { g = e } \tau_G(u_g)=\mathbf{1}_{\{g=e\}} τ G ( u g ) = 1 { g = e } K λ ^ ∘ b \widehat{K_\lambda}\circ b K λ ∘ b b : G → R d b:G\to\mathbb{R}^d b : G → R d x = ∑ g x g u g x=\sum_g x_g u_g x = ∑ g x g u g
T λ b x = ∑ g K λ ^ ( b ( g ) ) x g u g . T^b_\lambda x=\sum_g\widehat{K_\lambda}(b(g))x_g u_g. T λ b x = g ∑ K λ ( b ( g )) x g u g . Corollary 7.2. The family ( T λ b ) λ ∈ Λ (T^b_\lambda)_{\lambda\in\Lambda} ( T λ b ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ L p L_p L p VN ( G ) \operatorname{VN}(G) VN ( G ) G G G b b b
Proof. Since b b b α y ( u g ) = e − 2 π i ⟨ y , b ( g ) ⟩ u g \alpha_y(u_g)=e^{-2\pi i\langle y,b(g)\rangle}u_g α y ( u g ) = e − 2 π i ⟨ y , b ( g )⟩ u g R d \mathbb{R}^d R d VN ( G ) \operatorname{VN}(G) VN ( G ) α y ( x ) = v y x v y ∗ \alpha_y(x)=v_yxv_y^* α y ( x ) = v y x v y ∗
( v y ξ ) ( g ) = e − 2 π i ⟨ y , b ( g ) ⟩ ξ ( g ) , ξ ∈ ℓ 2 ( G ) . (v_y\xi)(g)=e^{-2\pi i\langle y,b(g)\rangle}\xi(g),\qquad\xi\in\ell_2(G). ( v y ξ ) ( g ) = e − 2 π i ⟨ y , b ( g )⟩ ξ ( g ) , ξ ∈ ℓ 2 ( G ) . For a Fourier polynomial x x x
T λ b x = ∫ R d K λ ( y ) α y ( x ) d y . T^b_\lambda x=\int_{\mathbb{R}^d}K_\lambda(y)\alpha_y(x)\,dy. T λ b x = ∫ R d K λ ( y ) α y ( x ) d y . The conclusion follows from Corollary 7.1. □
We next apply this estimate to Fourier summation. Let K ∈ L 1 ( R d ) K\in L_1(\mathbb{R}^d) K ∈ L 1 ( R d ) (65) and (66) , with ∫ K = 1 \int K=1 ∫ K = 1 n ≥ 0 n\geq0 n ≥ 0
T n b x = ∑ g K ^ ( 2 − n b ( g ) ) x g u g T^b_nx=\sum_g\widehat{K}(2^{-n}b(g))x_g u_g T n b x = g ∑ K ( 2 − n b ( g )) x g u g initially on Fourier polynomials.
Corollary 7.3. The family ( T n b ) n ≥ 0 (T^b_n)_{n\geq0} ( T n b ) n ≥ 0 ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ x ∈ L 1 ( VN ( G ) ) x\in L_1(\operatorname{VN}(G)) x ∈ L 1 ( VN ( G ))
T n b x ⟶ x in L 1 ( VN ( G ) ) and b.a.u. T^b_nx\longrightarrow x\quad\text{in }L_1(\operatorname{VN}(G))\text{ and b.a.u.} T n b x ⟶ x in L 1 ( VN ( G )) and b.a.u. Proof. Proposition 6.2 and Corollary 7.2 give the maximal estimates. Since K ^ \widehat{K} K K ^ ( 0 ) = 1 \widehat{K}(0)=1 K ( 0 ) = 1
∥ T n b x − x ∥ ∞ ≤ ∑ g ∣ x g ∣ ∣ K ^ ( 2 − n b ( g ) ) − 1 ∣ ⟶ 0. \|T_n^b x-x\|_\infty\leq\sum_g |x_g|\left|\widehat{K}(2^{-n}b(g))-1\right| \longrightarrow0. ∥ T n b x − x ∥ ∞ ≤ g ∑ ∣ x g ∣ K ( 2 − n b ( g )) − 1 ⟶ 0. The weak type estimate gives bilateral uniform equicontinuity at zero. By the noncommutative Banach principle [3 Theorem 3.4] and density of Fourier polynomials in L 1 ( VN ( G ) ) L_1(\operatorname{VN}(G)) L 1 ( VN ( G )) ( T n b x ) n (T_n^b x)_n ( T n b x ) n x ∈ L 1 ( VN ( G ) ) x\in L_1(\operatorname{VN}(G)) x ∈ L 1 ( VN ( G ))
On the other hand,
T n b x = ∫ R d K ( y ) α 2 − n y ( x ) d y . T_n^b x=\int_{\mathbb{R}^d}K(y)\alpha_{2^{-n}y}(x)\,dy. T n b x = ∫ R d K ( y ) α 2 − n y ( x ) d y . By strong continuity on L 1 L_1 L 1
∥ T n b x − x ∥ 1 ≤ ∫ R d ∣ K ( y ) ∣ ∥ α 2 − n y ( x ) − x ∥ 1 d y ⟶ 0. \|T_n^b x-x\|_1\leq\int_{\mathbb{R}^d}|K(y)|\|\alpha_{2^{-n}y}(x)-x\|_1\,dy\longrightarrow0. ∥ T n b x − x ∥ 1 ≤ ∫ R d ∣ K ( y ) ∣∥ α 2 − n y ( x ) − x ∥ 1 d y ⟶ 0. Quantum Euclidean spaces
Let θ \theta θ d × d d\times d d × d R θ d \mathbb{R}^d_\theta R θ d L 2 ( R d ) L_2(\mathbb{R}^d) L 2 ( R d )
( U ( t ) ξ ) ( r ) = e − i 2 ⟨ t , θ r ⟩ ξ ( r − t ) , t ∈ R d . (U(t)\xi)(r)=e^{-\frac{i}{2}\langle t,\theta r\rangle}\xi(r-t),\qquad t\in\mathbb{R}^d. ( U ( t ) ξ ) ( r ) = e − 2 i ⟨ t , θ r ⟩ ξ ( r − t ) , t ∈ R d . For g ∈ L 1 ( R d ) g\in L_1(\mathbb{R}^d) g ∈ L 1 ( R d ) U ( g ) = ∫ R d g ( t ) U ( t ) d t U(g)=\int_{\mathbb{R}^d}g(t)U(t)\,dt U ( g ) = ∫ R d g ( t ) U ( t ) d t
τ R ( U ( g ) ) = ( 2 π ) d g ( 0 ) , g ∈ S ( R d ) . \tau_{\mathbb{R}}(U(g))=(2\pi)^d g(0),\qquad g\in\mathcal{S}(\mathbb{R}^d). τ R ( U ( g )) = ( 2 π ) d g ( 0 ) , g ∈ S ( R d ) . The operators U ( g ) U(g) U ( g ) g ∈ S ( R d ) g\in\mathcal{S}(\mathbb{R}^d) g ∈ S ( R d ) L p ( R θ d ) L_p(\mathbb{R}^d_\theta) L p ( R θ d ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ [8 MSX20] .
The canonical action
α y ( U ( t ) ) = e − 2 π i ⟨ y , t ⟩ U ( t ) \alpha_y(U(t))=e^{-2\pi i\langle y,t\rangle}U(t) α y ( U ( t )) = e − 2 π i ⟨ y , t ⟩ U ( t ) is weak∗ ^* ∗ ξ ( r ) ↦ e − 2 π i ⟨ y , r ⟩ ξ ( r ) \xi(r)\mapsto e^{-2\pi i\langle y,r\rangle}\xi(r) ξ ( r ) ↦ e − 2 π i ⟨ y , r ⟩ ξ ( r ) L 2 ( R d ) L_2(\mathbb{R}^d) L 2 ( R d )
T λ U ( g ) = U ( K ^ λ g ) . T_\lambda U(g)=U(\widehat{K}_\lambda g). T λ U ( g ) = U ( K λ g ) . Corollary 7.4. The family ( T λ ) λ ∈ Λ (T_\lambda)_{\lambda\in\Lambda} ( T λ ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ L p ( R θ d ) L_p(\mathbb{R}^d_\theta) L p ( R θ d )
Proof. For g ∈ S ( R d ) g\in\mathcal{S}(\mathbb{R}^d) g ∈ S ( R d )
T λ U ( g ) = ∫ R d K λ ( y ) U ( e − 2 π i ⟨ y , ⋅ ⟩ g ) d y = ∫ R d K λ ( y ) α y ( U ( g ) ) d y . T_\lambda U(g)=\int_{\mathbb{R}^d}K_\lambda(y)U(e^{-2\pi i\langle y,\cdot\rangle}g)\,dy=\int_{\mathbb{R}^d}K_\lambda(y)\alpha_y(U(g))\,dy. T λ U ( g ) = ∫ R d K λ ( y ) U ( e − 2 π i ⟨ y , ⋅ ⟩ g ) d y = ∫ R d K λ ( y ) α y ( U ( g )) d y . Then, the conclusion follows from Corollary 7.1. □
Quantum tori
Let θ \theta θ d × d d \times d d × d A θ A_\theta A θ C ∗ C^* C ∗ U 1 , … , U d U_1,\ldots,U_d U 1 , … , U d
U k U j = e 2 π i θ k j U j U k . U_kU_j=e^{2\pi i\theta_{kj}}U_jU_k. U k U j = e 2 π i θ k j U j U k . Its canonical trace is determined by
τ θ ( U m ) = 1 { m = 0 } , U m = U 1 m 1 ⋯ U d m d , m ∈ Z d . \tau_\theta(U^m)=\mathbf{1}_{\{m=0\}}, \qquad U^m=U_1^{m_1}\cdots U_d^{m_d}, \qquad m\in\mathbb{Z}^d. τ θ ( U m ) = 1 { m = 0 } , U m = U 1 m 1 ⋯ U d m d , m ∈ Z d . Write T θ d \mathbb{T}^d_\theta T θ d
α y ( U m ) = e − 2 π i ⟨ y , m ⟩ U m , y ∈ R d , \alpha_y(U^m)=e^{-2\pi i\langle y,m\rangle}U^m,\qquad y\in\mathbb{R}^d, α y ( U m ) = e − 2 π i ⟨ y , m ⟩ U m , y ∈ R d , is weak∗ ^* ∗ [6 ] . For a trigonometric polynomial x = ∑ m x ^ ( m ) U m x=\sum_m\widehat{x}(m)U^m x = ∑ m x ( m ) U m
T λ x = ∑ m K ^ λ ( m ) x ^ ( m ) U m . T_\lambda x=\sum_m\widehat{K}_\lambda(m)\widehat{x}(m)U^m. T λ x = m ∑ K λ ( m ) x ( m ) U m . Corollary 7.5. The family ( T λ ) λ ∈ Λ (T_\lambda)_{\lambda\in\Lambda} ( T λ ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ L p ( T θ d ) L_p(\mathbb{T}^d_\theta) L p ( T θ d )
Proof. For trigonometric polynomials, we have
T λ x = ∑ m x ^ ( m ) ( ∫ R d K λ ( y ) e − 2 π i ⟨ y , m ⟩ d y ) U m = ∫ R d K λ ( y ) α y ( x ) d y . T_\lambda x=\sum_m\widehat{x}(m)\left(\int_{\mathbb{R}^d}K_\lambda(y)e^{-2\pi i\langle y,m\rangle}\,dy\right)U^m=\int_{\mathbb{R}^d}K_\lambda(y)\alpha_y(x)\,dy. T λ x = m ∑ x ( m ) ( ∫ R d K λ ( y ) e − 2 π i ⟨ y , m ⟩ d y ) U m = ∫ R d K λ ( y ) α y ( x ) d y . Then, the conclusion follows from Corollary 7.1. □
Schur multipliers
Let G G G μ \mu μ β : G → R d \beta:G\to\mathbb{R}^d β : G → R d y ∈ R d y\in\mathbb{R}^d y ∈ R d L 2 ( G ) L_2(G) L 2 ( G )
( u y ξ ) ( g ) = e − 2 π i ⟨ y , β ( g − 1 ) ⟩ ξ ( g ) . (u_y\xi)(g)=e^{-2\pi i\langle y,\beta(g^{-1})\rangle}\xi(g). ( u y ξ ) ( g ) = e − 2 π i ⟨ y , β ( g − 1 )⟩ ξ ( g ) . Then α y ( a ) = u y a u y ∗ \alpha_y(a)=u_yau_y^* α y ( a ) = u y a u y ∗ ∗ ^* ∗ B ( L 2 ( G ) ) \mathcal{B}(L_2(G)) B ( L 2 ( G ))
For a Hilbert–Schmidt operator a a a a ( g , h ) a(g,h) a ( g , h )
( S λ a ) ( g , h ) = K ^ λ ( β ( g − 1 ) − β ( h − 1 ) ) a ( g , h ) . (S_\lambda a)(g,h)=\widehat{K}_\lambda\bigl(\beta(g^{-1})-\beta(h^{-1})\bigr)a(g,h). ( S λ a ) ( g , h ) = K λ ( β ( g − 1 ) − β ( h − 1 ) ) a ( g , h ) . Corollary 7.6. The family ( S λ ) λ ∈ Λ (S_\lambda)_{\lambda\in\Lambda} ( S λ ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) S 1 ( L 2 ( G ) ) S_1(L_2(G)) S 1 ( L 2 ( G )) ( p , p ) (p,p) ( p , p ) S p ( L 2 ( G ) ) S_p(L_2(G)) S p ( L 2 ( G )) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞ S ∞ ( L 2 ( G ) ) = B ( L 2 ( G ) ) S_\infty(L_2(G))=\mathcal{B}(L_2(G)) S ∞ ( L 2 ( G )) = B ( L 2 ( G ))
Proof. Since the integral kernel of u y a u y ∗ u_yau_y^* u y a u y ∗ e − 2 π i ⟨ y , β ( g − 1 ) − β ( h − 1 ) ⟩ a ( g , h ) e^{-2\pi i\langle y,\beta(g^{-1})-\beta(h^{-1})\rangle}a(g,h) e − 2 π i ⟨ y , β ( g − 1 ) − β ( h − 1 )⟩ a ( g , h )
S λ a = ∫ R d K λ ( y ) u y a u y ∗ d y S_\lambda a=\int_{\mathbb{R}^d}K_\lambda(y)u_yau_y^*\,dy S λ a = ∫ R d K λ ( y ) u y a u y ∗ d y on the Hilbert–Schmidt class. Then, the conclusion follows from Corollary 7.1. □
In particular, the result applies when β \beta β [2 ] .
Double operator integrals
Double operator integrals are a standard tool in operator perturbation theory and provide a spectral formulation of Schur multipliers. For background, see [23 Section 1.1] and, in the semifinite setting, [7 ] . Here we obtain maximal inequalities for families whose symbols depend on the difference of the spectral variables.
In this subsection, take d = 1 d=1 d = 1 H H H M \mathcal{M} M E H E_H E H x ∈ L 2 ( M ) x\in L_2(\mathcal{M}) x ∈ L 2 ( M )
T λ H x = ∫ R ∫ R K ^ λ ( s − t ) d E H ( s ) x d E H ( t ) . T^H_\lambda x=\int_{\mathbb{R}}\int_{\mathbb{R}}\widehat{K}_\lambda(s-t)\,dE_H(s)\,x\,dE_H(t). T λ H x = ∫ R ∫ R K λ ( s − t ) d E H ( s ) x d E H ( t ) . Therefore, one has
T λ H x = ∫ R K λ ( y ) e − 2 π i y H x e 2 π i y H d y (84) T^H_\lambda x=\int_{\mathbb{R}}K_\lambda(y)e^{-2\pi iyH}xe^{2\pi iyH}\,dy \tag*{(84)} T λ H x = ∫ R K λ ( y ) e − 2 π i y H x e 2 π i y H d y (84) For every 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞ L p ( M ) L_p(\mathcal{M}) L p ( M )
∥ T λ H x ∥ p ≤ ∥ K λ ∥ 1 ∥ x ∥ p . \|T^H_\lambda x\|_p\leq\|K_\lambda\|_1\|x\|_p. ∥ T λ H x ∥ p ≤ ∥ K λ ∥ 1 ∥ x ∥ p . Corollary 7.7. The family ( T λ H ) λ ∈ Λ (T^H_\lambda)_{\lambda\in\Lambda} ( T λ H ) λ ∈ Λ ( 1 , 1 ) (1,1) ( 1 , 1 ) ( p , p ) (p,p) ( p , p ) 1 < p ≤ ∞ 1<p\leq\infty 1 < p ≤ ∞
Proof. The maps α y ( x ) = e − 2 π i y H x e 2 π i y H \alpha_y(x)=e^{-2\pi iyH}xe^{2\pi iyH} α y ( x ) = e − 2 π i y H x e 2 π i y H R \mathbb{R} R M \mathcal{M} M (84) and Corollary 7.1. □
In particular, for a diagonal matrix H = diag ( h 1 , … , h N ) H=\operatorname{diag}(h_1,\ldots,h_N) H = diag ( h 1 , … , h N )
( T λ H x ) i j = K ^ λ ( h i − h j ) x i j . (T^H_\lambda x)_{ij}=\widehat{K}_\lambda(h_i-h_j)x_{ij}. ( T λ H x ) ij = K λ ( h i − h j ) x ij . Thus the maximal estimates are uniform in N N N h 1 , … , h N h_1,\ldots,h_N h 1 , … , h N
Acknowledgements. G. Hong was partially supported by the National Natural Science Foundation of China (Grant Nos. 12325105, W2441002, and W2611005). L. Wang was supported by a grant from the Research Grants Council of the Hong Kong Administrative Region, China (Project No. CityU 11308924). X. Wang was supported by the National Natural Science Foundation of China (Grant Nos. 12601242 and W2611005). B. Xu was supported by the National Natural Science Foundation of China (Grant No. 12501172) and the Fundamental Research Funds for the Central Universities (Grant No. 20720250061).
AI statement. The authors used GPT-6-Astra to assist with their research on this topic. The model also assisted with language polishing and editing. The authors checked and developed the suggested idea, verified the proofs, and take full responsibility for the content of the paper.
References
Guixiang Hong, Institute for Advanced Study in Mathematics, Harbin Institute of Technology, 15000 Harbin, China Email address: gxhong@hit.edu.cn
Liang Wang, Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China Email address: L.Wang@cityu.edu.hk
Xumin Wang, Institute for Advanced Study in Mathematics, Harbin Institute of Technology, 15000 Harbin, China Email address: xumin.wang@hit.edu.cn
Bang Xu, School of Mathematical Sciences, Xiamen University, Xiamen 361005, China Email address: bangxu@xmu.edu.cn
