Introduction
Weighted norm inequalities are a central part of modern harmonic analysis. On the real line, we say a function w : R → R + w:\mathbb{R}\to\mathbb{R}_+ w : R → R + 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ w w w L p L_p L p
∥ f ∥ L p ( w ) : = ( ∫ R ∣ f ( t ) ∣ p w ( t ) d t ) 1 / p . \|f\|_{L_p(w)}:=\left(\int_{\mathbb{R}}|f(t)|^p w(t)\,dt\right)^{1/p}. ∥ f ∥ L p ( w ) := ( ∫ R ∣ f ( t ) ∣ p w ( t ) d t ) 1/ p . For 1 < p < ∞ 1<p<\infty 1 < p < ∞ A p ( R ) A_p(\mathbb{R}) A p ( R )
[ w ] A p = sup I ⊂ R ⟨ w ⟩ I ⟨ w − p ′ / p ⟩ I p / p ′ < ∞ , ⟨ w ⟩ I = 1 ∣ I ∣ ∫ I w ( t ) d t , [w]_{A_p}=\sup_{I\subset\mathbb{R}}\langle w\rangle_I\langle w^{-p'/p}\rangle_I^{p/p'}<\infty,\qquad\langle w\rangle_I=\frac{1}{|I|}\int_I w(t)\,dt, [ w ] A p = I ⊂ R sup ⟨ w ⟩ I ⟨ w − p ′ / p ⟩ I p / p ′ < ∞ , ⟨ w ⟩ I = ∣ I ∣ 1 ∫ I w ( t ) d t , where the supremum is taken over bounded intervals. Muckenhoupt’s theorem in [19 ] identifies this condition with the boundedness of the Hardy–Littlewood maximal operator
M f ( t ) : = sup I ∋ t 1 ∣ I ∣ ∫ I ∣ f ( s ) ∣ d s Mf(t):=\sup_{I\ni t}\frac{1}{|I|}\int_I|f(s)|\,ds M f ( t ) := I ∋ t sup ∣ I ∣ 1 ∫ I ∣ f ( s ) ∣ d s on L p ( w ) L_p(w) L p ( w )
H f ( t ) : = p.v. 1 π ∫ R f ( s ) t − s d s Hf(t):=\operatorname{p.v.}\frac{1}{\pi}\int_{\mathbb{R}}\frac{f(s)}{t-s}\,ds H f ( t ) := p.v. π 1 ∫ R t − s f ( s ) d s and, more generally, of Calderón–Zygmund operators [11 , 3 ] .
The extrapolation of Rubio de Francia in [25 ] explains why weighted norm inequalities persist between exponents. In one of its standard forms, if a sublinear operator S S S
∥ S f ∥ L r ( w ) ≤ Φ ( [ w ] A r ) ∥ f ∥ L r ( w ) \|Sf\|_{L_r(w)}\le\Phi([w]_{A_r})\|f\|_{L_r(w)} ∥ S f ∥ L r ( w ) ≤ Φ ([ w ] A r ) ∥ f ∥ L r ( w ) for all w ∈ A r ( R ) w\in A_r(\mathbb{R}) w ∈ A r ( R ) 1 < r < ∞ 1<r<\infty 1 < r < ∞ 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ A p ( R ) w\in A_p(\mathbb{R}) w ∈ A p ( R ) A p A_p A p
2020 Mathematics Subject Classification. Primary 46L52, 42B25; Secondary 46L51, 42B20.
Key words and phrases. Matrix weights, noncommutative maximal inequalities, factorization of weights, Rubio de Francia extrapolation, dimension-free.
Matrix-weighted theory differs substantially from its scalar counterpart. We say w : R → M d ( C ) w : \mathbb{R} \to M_d(\mathbb{C}) w : R → M d ( C ) f : R → ℓ 2 d f : \mathbb{R} \to\ell_2^d f : R → ℓ 2 d
∥ f ∥ L p ( w ; ℓ 2 d ) = ( ∫ R ∥ w ( t ) 1 / p f ( t ) ∥ ℓ 2 d p d t ) 1 / p . \|f\|_{L_p(w;\ell_2^d)}=\left(\int_{\mathbb{R}}\|w(t)^{1/p}f(t)\|_{\ell_2^d}^p\,dt\right)^{1/p}. ∥ f ∥ L p ( w ; ℓ 2 d ) = ( ∫ R ∥ w ( t ) 1/ p f ( t ) ∥ ℓ 2 d p d t ) 1/ p . For d = 1 d=1 d = 1 d > 1 d>1 d > 1 A 2 A_2 A 2 [29 ] but with the sharp A 2 A_2 A 2 [20 , 5 ] , and Volberg’s geometric formulation of A p ( R ; M d ( C ) ) A_p(\mathbb{R};M_d(\mathbb{C})) A p ( R ; M d ( C )) [30 ] . Further characterizations and maximal function approaches were developed by Nazarov-Treil, Christ-Goldberg and Goldberg in [21 , 2 , 6 ] . In particular, Goldberg’s Hilbert-valued maximal theorem can be stated as follows: the maximal operator
M w f ( t ) : = sup I ∋ t 1 ∣ I ∣ ∫ I ∥ w 1 / p ( t ) w − 1 / p ( s ) f ( s ) ∥ ℓ 2 d d s (1) M_wf(t):=\sup_{I\ni t}\frac{1}{|I|}\int_I\|w^{1/p}(t)w^{-1/p}(s)f(s)\|_{\ell_2^d}\,ds \tag*{(1)} M w f ( t ) := I ∋ t sup ∣ I ∣ 1 ∫ I ∥ w 1/ p ( t ) w − 1/ p ( s ) f ( s ) ∥ ℓ 2 d d s (1) is bounded from L p ( R ; ℓ 2 d ) L_p(\mathbb{R};\ell_2^d) L p ( R ; ℓ 2 d ) L p ( R ) L_p(\mathbb{R}) L p ( R ) A p ( R ; M d ( C ) ) A_p(\mathbb{R};M_d(\mathbb{C})) A p ( R ; M d ( C )) A p A_p A p [12 ] . These results provide a robust matrix-weighted theory, but the scalar Rubio de Francia algorithm does not transfer directly.
The obstruction is simple to state. In the scalar theory, Rubio de Francia’s algorithm repeatedly applies a positive maximal operator and uses pointwise suprema. In the matrix-valued setting there is no canonical vector-valued supremum with the same input and output structure. Goldberg’s maximal function, for instance, characterizes matrix A p A_p A p [1 ] . This approach preserves an input-output structure at the level of convex bodies.
In the present paper, we try to address this problem from the viewpoint of noncommutative analysis. More precisely, the operator space L p ( M ; ℓ ∞ ) L_p(\mathcal{M};\ell_\infty) L p ( M ; ℓ ∞ ) L p L_p L p L p L_p L p L p ( M ; ℓ ∞ ) L_p(\mathcal{M};\ell_\infty) L p ( M ; ℓ ∞ ) L p L_p L p [22 ] and developed by Junge in his proof of the noncommutative Doob maximal inequality [13 ] . It has since become an essential tool in noncommutative martingale and ergodic theory, including the noncommutative Dunford-Schwartz maximal ergodic theorem of Junge and Xu [14 ] and the noncommutative Hardy-Littlewood maximal inequality by Mei [18 ] . More recently, noncommutative maximal norms have been used to prove maximal ergodic inequalities under doubling conditions [9 ] and pointwise convergence results for noncommutative Fourier series [10 ] .
The noncommutative weighted spaces
For any von Neumann algebra M \mathcal{M} M 0 < p < ∞ 0<p<\infty 0 < p < ∞ L p L_p L p M \mathcal{M} M [7 ] . In the semifinite tracial setting, this construction is compatible with the usual noncommutative L p L_p L p [16 , 23 ] . The symmetric realization has proved particularly useful in reduction arguments for noncommutative L p L_p L p [8 ] .
Motivated by this point of view, let ( M , τ ) (\mathcal{M},\tau) ( M , τ ) τ ( 1 ) < ∞ \tau(1)<\infty τ ( 1 ) < ∞ 0 < p < ∞ 0<p<\infty 0 < p < ∞ w ∈ L 0 + ( M ) w\in L_0^+(\mathcal{M}) w ∈ L 0 + ( M )
∥ x ∥ L p ( M , w ) = ∥ w 1 / 2 p x w 1 / 2 p ∥ L p ( M ) . \|x\|_{L_p(\mathcal{M},w)}=\|w^{1/2p}xw^{1/2p}\|_{L_p(\mathcal{M})}. ∥ x ∥ L p ( M , w ) = ∥ w 1/2 p x w 1/2 p ∥ L p ( M ) . We restrict to trace-finite von Neumann algebras so that full-support positive weights, their inverses, and the products all belong to the same algebra of measurable operators, and thus we may define the weighted space L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) L 0 ( M ) L_0(\mathcal{M}) L 0 ( M )
But an exceptional case that we will take care of in this paper is
N = L ∞ ( R ) ⊗ ‾ M d ( C ) , \mathcal{N}=L_{\infty}(\mathbb{R})\overline{\otimes}M_{d}(\mathbb{C}), N = L ∞ ( R ) ⊗ M d ( C ) , with the tensor trace that is semifinite. A matrix weight is, throughout this paper, a strongly measurable, almost everywhere positive-definite, and locally integrable function w : R → M d ( C ) w:\mathbb{R}\to M_{d}(\mathbb{C}) w : R → M d ( C )
∥ f ∥ L p ( N , w ) = ( ∫ R ∥ w ( t ) 1 / 2 p f ( t ) w ( t ) 1 / 2 p ∥ S p d p d t ) 1 / p (2) \|f\|_{L_{p}(\mathcal{N},w)}=\left(\int_{\mathbb{R}}\left\|w(t)^{1/2p}f(t)w(t)^{1/2p}\right\|_{S_{p}^{d}}^{p}\,\mathrm{d}t\right)^{1/p} \tag*{(2)} ∥ f ∥ L p ( N , w ) = ( ∫ R w ( t ) 1/2 p f ( t ) w ( t ) 1/2 p S p d p d t ) 1/ p (2) and the resulting weighted space L p ( N , w ) L_{p}(\mathcal{N},w) L p ( N , w )
The maximal inequality characterization
Our first result identifies the correct matrix A p A_{p} A p A p ( N ) A_{p}(\mathcal{N}) A p ( N ) § 3 , which is inspired by Volberg’s averaged-norm formulation: the matrix weight w w w M d ( C ) M_{d}(\mathbb{C}) M d ( C ) A p ( N ) A_{p}(\mathcal{N}) A p ( N )
Let I \mathcal{I} I R \mathbb{R} R I ∈ I I\in\mathcal{I} I ∈ I
A I f = ( 1 ∣ I ∣ ∫ I f ( s ) d s ) χ I . A_{I}f=\left(\frac{1}{|I|}\int_{I}f(s)\,\mathrm{d}s\right)\chi_{I}. A I f = ( ∣ I ∣ 1 ∫ I f ( s ) d s ) χ I . The weighted noncommutative maximal norm is defined as follows
∥ ( A I f ) I ∈ I ∥ L p ( N , w ; ℓ ∞ ) : = ∥ sup I ∈ I + A I f ∥ L p ( N , w ) : = ∥ sup I ∈ I + w 1 / 2 p ( A I f ) w 1 / 2 p ∥ L p ( N ) . \|(A_{I}f)_{I\in\mathcal{I}}\|_{L_{p}(\mathcal{N},w;\ell_{\infty})}:=\left\|\sup_{I\in\mathcal{I}}^{+}A_{I}f\right\|_{L_{p}(\mathcal{N},w)}:=\left\|\sup_{I\in\mathcal{I}}^{+}w^{1/2p}(A_{I}f)w^{1/2p}\right\|_{L_{p}(\mathcal{N})}. ∥ ( A I f ) I ∈ I ∥ L p ( N , w ; ℓ ∞ ) := I ∈ I sup + A I f L p ( N , w ) := I ∈ I sup + w 1/2 p ( A I f ) w 1/2 p L p ( N ) . Let S N c \mathcal{S}_{\mathcal{N}}^{c} S N c A p ( N ) A_{p}(\mathcal{N}) A p ( N )
Theorem 1.1. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ w w w w ∈ A p ( N ) w\in A_{p}(\mathcal{N}) w ∈ A p ( N )
∥ sup I ∈ I + A I f ∥ L p ( N , w ) ≤ C ∥ f ∥ L p ( N , w ) , f ∈ S N c . (3) \left\|\sup_{I\in\mathcal{I}}^{+}A_{I}f\right\|_{L_{p}(\mathcal{N},w)}\leq C\|f\|_{L_{p}(\mathcal{N},w)},\qquad f\in\mathcal{S}_{\mathcal{N}}^{c}. \tag*{(3)} I ∈ I sup + A I f L p ( N , w ) ≤ C ∥ f ∥ L p ( N , w ) , f ∈ S N c . (3) Moreover, the optimal constant in (3) denoted by C p , d ( w ) C_{p,d}(w) C p , d ( w )
[ w ] A p ( N ) 1 / p ≤ C p , d ( w ) ≤ W p , d [ w ] A p ( N ) 1 / p ′ (4) [w]_{A_{p}(\mathcal{N})}^{1/p}\leq C_{p,d}(w)\leq\mathcal{W}_{p,d}[w]_{A_{p}(\mathcal{N})}^{1/p'} \tag*{(4)} [ w ] A p ( N ) 1/ p ≤ C p , d ( w ) ≤ W p , d [ w ] A p ( N ) 1/ p ′ (4) with the least upper bound W p , d ≤ d 1 / p + ( 1 + p ′ ) 1 / p − 1 / 2 G p , d 2 \mathcal{W}_{p,d}\leq d^{1/p}+(1+p')^{1/p-1/2}\mathcal{G}_{p,d}^{2} W p , d ≤ d 1/ p + ( 1 + p ′ ) 1/ p − 1/2 G p , d 2 G p , d 2 \mathcal{G}_{p,d}^{2} G p , d 2 (29) .
The if part is easy by direct computation. For the only if part, the first observation is that one may relate w w w W W W S 2 d S_{2}^{d} S 2 d
W ( t ) ( A ) = w ( t ) 1 / 2 A w ( t ) 1 / 2 ; W(t)(A)=w(t)^{1/2}Aw(t)^{1/2}; W ( t ) ( A ) = w ( t ) 1/2 A w ( t ) 1/2 ; see Lemma 3.7 . The second observation is that the noncommutative maximal norm is comparable to the vector-valued maximal norm for finite-dimensional matrix-valued functions; see Lemma 3.9. The desired noncommutative weighted maximal inequality then follows from the argument of [12 ] .
The extrapolation theorem
Our second result is a matrix-valued version of Rubio de Francia’s extrapolation theorem. It is formulated in the framework of noncommutative analysis.
Theorem 1.2. Let ( T α ) α ∈ A (T_\alpha)_{\alpha\in A} ( T α ) α ∈ A S N c S^c_{\mathcal{N}} S N c M d ( C ) M_d(\mathbb{C}) M d ( C ) 1 < r < ∞ 1<r<\infty 1 < r < ∞ Φ \Phi Φ
∥ sup α ∈ A + T α ( f ) ∥ L r ( N , w ) ≤ Φ ( [ w ] A r ( N ) ) ∥ f ∥ L r ( N , w ) (5) \left\|\sup_{\alpha\in A}^{+}T_\alpha(f)\right\|_{L_r(\mathcal{N},w)}\leq\Phi([w]_{A_r(\mathcal{N})})\|f\|_{L_r(\mathcal{N},w)} \tag*{(5)} α ∈ A sup + T α ( f ) L r ( N , w ) ≤ Φ ([ w ] A r ( N ) ) ∥ f ∥ L r ( N , w ) (5) for every w ∈ A r ( N ) w\in A_r(\mathcal{N}) w ∈ A r ( N ) f ∈ S N c f\in S^c_{\mathcal{N}} f ∈ S N c 1 < p < ∞ 1<p<\infty 1 < p < ∞ Φ p , r , d \Phi_{p,r,d} Φ p , r , d p p p r r r d d d Φ \Phi Φ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) f ↦ ( T α ( f ) ) α ∈ A f\mapsto(T_\alpha(f))_{\alpha\in A} f ↦ ( T α ( f ) ) α ∈ A L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p ( N , w ; ℓ ∞ ( A ) ) L_p(\mathcal{N},w;\ell_\infty(A)) L p ( N , w ; ℓ ∞ ( A ))
∥ sup α ∈ A + T α ( f ) ∥ L p ( N , w ) ≤ Φ p , r , d ( [ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) . (6) \left\|\sup_{\alpha\in A}^{+}T_\alpha(f)\right\|_{L_p(\mathcal{N},w)}\leq\Phi_{p,r,d}([w]_{A_p(\mathcal{N})})\|f\|_{L_p(\mathcal{N},w)}. \tag*{(6)} α ∈ A sup + T α ( f ) L p ( N , w ) ≤ Φ p , r , d ([ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) . (6) An explicit choice of Φ p , r , d \Phi_{p,r,d} Φ p , r , d (68) .
In Theorem 1.2 , letting T α ≡ T T_\alpha\equiv T T α ≡ T ∥ T ( f ) ∥ L p ( N , w ) \|T(f)\|_{L_p(\mathcal{N},w)} ∥ T ( f ) ∥ L p ( N , w )
Corollary 1.3. Let T T T S N c S^c_{\mathcal{N}} S N c M d ( C ) M_d(\mathbb{C}) M d ( C ) 1 < r < ∞ 1<r<\infty 1 < r < ∞ Φ \Phi Φ
∥ T ( f ) ∥ L r ( N , w ) ≤ Φ ( [ w ] A r ( N ) ) ∥ f ∥ L r ( N , w ) \|T(f)\|_{L_r(\mathcal{N},w)}\leq\Phi([w]_{A_r(\mathcal{N})})\|f\|_{L_r(\mathcal{N},w)} ∥ T ( f ) ∥ L r ( N , w ) ≤ Φ ([ w ] A r ( N ) ) ∥ f ∥ L r ( N , w ) for every w ∈ A r ( N ) w\in A_r(\mathcal{N}) w ∈ A r ( N ) f ∈ S N c f\in S^c_{\mathcal{N}} f ∈ S N c 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) T T T L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w )
∥ T ( f ) ∥ L p ( N , w ) ≤ Φ p , r , d ( [ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) , \|T(f)\|_{L_p(\mathcal{N},w)}\leq\Phi_{p,r,d}([w]_{A_p(\mathcal{N})})\|f\|_{L_p(\mathcal{N},w)}, ∥ T ( f ) ∥ L p ( N , w ) ≤ Φ p , r , d ([ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) , where the increasing function Φ p , r , d \Phi_{p,r,d} Φ p , r , d Theorem 1.2 .
The key ingredient in the proof of Theorem 1.2 is a noncommutative analogue of Rubio de Francia’s abstract extrapolation theorem for trace-finite von Neumann algebras. More precisely, let ( E β ) β (E_\beta)_\beta ( E β ) β ( M , τ ) (\mathcal{M},\tau) ( M , τ ) W p ( ( E β ) β ) W_p((E_\beta)_\beta) W p (( E β ) β ) ( E β ) β (E_\beta)_\beta ( E β ) β L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) Assumption 4.2 , we obtain first a factorization theorem for the weights—Theorem 4.4 , and then an extrapolation result for a family of linear maps—Theorem 4.5 . This is exactly a noncommutative analogue of Rubio de Francia’s celebrated extrapolation theory [25 ] . However, because of the noncommuting products and the absence of a pointwise maximal function, the proof requires new techniques in the noncommutative setting. For instance, we identify the dense subsets of the weighted space L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) Lemma 2.6 , reformulate the condition ∣ w ∣ 2 p ∈ W p ( ( E β ) β ) |w|^{2p}\in W_p((E_\beta)_\beta) ∣ w ∣ 2 p ∈ W p (( E β ) β ) ( w , w ∗ ) (w,w^*) ( w , w ∗ ) Lemma 4.3 , and establish the Hölder type testing estimates for noncommutative maximal norms in Lemmas 2.7 and 2.8 etc.
With slight modifications of the arguments, a similar strategy in § 4 also works for matrix-weighted theory associated to the family ( A I ) I ∈ I (A_I)_{I\in\mathcal{I}} ( A I ) I ∈ I W p ( ( A I ) I ) W_p((A_I)_I) W p (( A I ) I ) A p ( N ) A_p(\mathcal{N}) A p ( N ) Assumption 4.2 is satisfied with a constant depending on the dimension d d d ( N , ∫ ⊗ tr ) (\mathcal{N},\int\otimes\operatorname{tr}) ( N , ∫ ⊗ tr ) Theorem 1.2 in § 5 .
As an application, we establish weighted maximal inequalities for truncations of Calderón-Zygmund operators. A weighted Cotlar inequality and the matrix A 2 A_2 A 2 L 2 L_2 L 2 L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p ( N , w ; ℓ ∞ ) L_p(\mathcal{N},w;\ell_\infty) L p ( N , w ; ℓ ∞ ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N )
Organization of the paper
§ 2 recalls the semifinite scalar and vector-valued noncommutative L p L_p L p § 3 introduces the geometric matrix A p A_p A p § 4 establishes the abstract factorization and extrapolation theorems for trace-finite algebras. § 5 verifies the structural assumptions for interval averages, transfers the factorization to N \mathcal{N} N Theorem 1.2 with explicit quantitative control, and presents an application to the Calderón–Zygmund operators. Finally, § 6 presents several open problems concerning extensions of our extrapolation framework and the quantitative bounds in our results.
Preliminaries on tracial von Neumann algebras
This section fixes our conventions for noncommutative integration and collects the scalar, vector-valued, and weighted tools used later. In particular, we present the weighted realizations and establish some properties for trace-finite von Neumann algebras and the exceptional semifinite von Neumann algebra N = L ∞ ( R ) ⊗ ‾ M d ( C ) \mathcal{N}=L_\infty(\mathbb{R})\mathbin{\overline{\otimes}}M_d(\mathbb{C}) N = L ∞ ( R ) ⊗ M d ( C )
Vector-valued noncommutative L p L_p L p
We recall the facts about noncommutative integration that will be used below; see [26 , 4 , 28 , 23 ] . Let ( M , τ ) (\mathcal{M},\tau) ( M , τ ) τ \tau τ L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) ∗ * ∗ τ \tau τ M \mathcal{M} M x ∈ L 0 ( M ) x\in L_0(\mathcal{M}) x ∈ L 0 ( M ) x = u ∣ x ∣ x=u|x| x = u ∣ x ∣ s ( x ) = s ( ∣ x ∣ ) s(x)=s(|x|) s ( x ) = s ( ∣ x ∣ ) ∣ x ∣ |x| ∣ x ∣
For 0 < p < ∞ 0<p<\infty 0 < p < ∞ L p L_p L p ( M , τ ) (\mathcal{M},\tau) ( M , τ )
L p ( M ) = { x ∈ L 0 ( M ) : τ ( ∣ x ∣ p ) < ∞ } , ∥ x ∥ p = τ ( ∣ x ∣ p ) 1 / p . L_p(\mathcal{M})=\{x\in L_0(\mathcal{M}):\tau(|x|^p)<\infty\},\qquad\|x\|_p=\tau(|x|^p)^{1/p}. L p ( M ) = { x ∈ L 0 ( M ) : τ ( ∣ x ∣ p ) < ∞ } , ∥ x ∥ p = τ ( ∣ x ∣ p ) 1/ p . We set L ∞ ( M ) = M L_\infty(\mathcal{M})=\mathcal{M} L ∞ ( M ) = M p ≥ 1 p\geq1 p ≥ 1 L p ( M ) L_p(\mathcal{M}) L p ( M ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ p ′ p' p ′ p p p L p ( M ) ∗ = L p ′ ( M ) L_p(\mathcal{M})^*=L_{p'}(\mathcal{M}) L p ( M ) ∗ = L p ′ ( M ) 0 < r , p , q ≤ ∞ 0<r,p,q\leq\infty 0 < r , p , q ≤ ∞ 1 / r = 1 / p + 1 / q 1/r=1/p+1/q 1/ r = 1/ p + 1/ q
∥ x y ∥ r ≤ ∥ x ∥ p ∥ y ∥ q . (7) \|xy\|_r\leq\|x\|_p\|y\|_q. \tag*{(7)} ∥ x y ∥ r ≤ ∥ x ∥ p ∥ y ∥ q . (7) The corresponding estimate for products of finitely many factors follows by iteration whenever the reciprocal exponents add correctly.
We shall also use the following standard tracial property. If a j ∈ L p j ( M ) a_j\in L_{p_j}(\mathcal{M}) a j ∈ L p j ( M ) 1 ≤ j ≤ n 1\leq j\leq n 1 ≤ j ≤ n ∑ j = 1 n 1 / p j = 1 \sum_{j=1}^{n}1/p_j=1 ∑ j = 1 n 1/ p j = 1 a 1 ⋯ a n a_1\cdots a_n a 1 ⋯ a n L 1 ( M ) L_1(\mathcal{M}) L 1 ( M )
Given 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞ a , b ∈ L 0 ( M ) a,b\in L_0(\mathcal{M}) a , b ∈ L 0 ( M ) a b ∈ L p ( M ) ab\in L_p(\mathcal{M}) ab ∈ L p ( M ) ∥ a b ∥ p = ∥ a ∥ ∞ ∥ b ∥ p = ∥ a ⋅ ∣ b ∣ ∥ p \|ab\|_p=\|a\|_\infty\|b\|_p=\|a\cdot|b|\|_p ∥ ab ∥ p = ∥ a ∥ ∞ ∥ b ∥ p = ∥ a ⋅ ∣ b ∣ ∥ p
The positive cone of L p ( M ) L_p(\mathcal{M}) L p ( M ) L p + ( M ) L_p^+(\mathcal{M}) L p + ( M ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ x ∈ L p ( M ) x\in L_p(\mathcal{M}) x ∈ L p ( M )
x = x 1 − x 2 + i x 3 − i x 4 , x j ∈ L p + ( M ) , x=x_1-x_2+ix_3-ix_4,\qquad x_j\in L_p^+(\mathcal{M}), x = x 1 − x 2 + i x 3 − i x 4 , x j ∈ L p + ( M ) , with ∥ x j ∥ p ≤ ∥ x ∥ p \|x_j\|_p\leq\|x\|_p ∥ x j ∥ p ≤ ∥ x ∥ p j = 1 , 2 , 3 , 4 j=1,2,3,4 j = 1 , 2 , 3 , 4
Complex interpolation is compatible with the semifinite noncommutative L p L_p L p 1 ≤ p 0 , p 1 ≤ ∞ 1\leq p_0,p_1\leq\infty 1 ≤ p 0 , p 1 ≤ ∞ 0 < θ < 1 0<\theta<1 0 < θ < 1 1 / p = ( 1 − θ ) / p 0 + θ / p 1 1/p=(1-\theta)/p_0+\theta/p_1 1/ p = ( 1 − θ ) / p 0 + θ / p 1
( L p 0 ( M ) , L p 1 ( M ) ) θ = L p ( M ) \bigl(L_{p_0}(\mathcal{M}),L_{p_1}(\mathcal{M})\bigr)_\theta=L_p(\mathcal{M}) ( L p 0 ( M ) , L p 1 ( M ) ) θ = L p ( M ) isometrically.
We next recall the maximal and summation spaces introduced by Pisier [22 ] and developed by Junge [13 ] . Pisier’s construction is formulated for hyperfinite von Neumann algebras, while Junge’s work supplies the general operator-space formulation used in the maximal–summation duality.
Let A \mathcal{A} A 1 ≤ p ≤ ∞ 1 \le p \le\infty 1 ≤ p ≤ ∞ x = ( x α ) α ∈ A ⊂ L p ( M ) x=(x_\alpha)_{\alpha\in\mathcal{A}}\subset L_p(\mathcal{M}) x = ( x α ) α ∈ A ⊂ L p ( M ) L p ( M ; ℓ ∞ ( A ) ) L_p(\mathcal{M};\ell_\infty(\mathcal{A})) L p ( M ; ℓ ∞ ( A ))
x α = a y α b , a , b ∈ L 2 p ( M ) , y α ∈ M , sup α ∈ A ∥ y α ∥ ∞ < ∞ . x_\alpha=ay_\alpha b,\qquad a,b\in L_{2p}(\mathcal{M}),\qquad y_\alpha\in\mathcal{M},\qquad\sup_{\alpha\in\mathcal{A}}\|y_\alpha\|_\infty<\infty. x α = a y α b , a , b ∈ L 2 p ( M ) , y α ∈ M , α ∈ A sup ∥ y α ∥ ∞ < ∞. Its norm is
∥ x ∥ L p ( M ; ℓ ∞ ( A ) ) = inf x α = a y α b ∥ a ∥ 2 p sup α ∈ A ∥ y α ∥ ∞ ∥ b ∥ 2 p . (8) \|x\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{A}))}=\inf_{x_\alpha=ay_\alpha b}\|a\|_{2p}\sup_{\alpha\in\mathcal{A}}\|y_\alpha\|_\infty\|b\|_{2p}. \tag*{(8)} ∥ x ∥ L p ( M ; ℓ ∞ ( A )) = x α = a y α b inf ∥ a ∥ 2 p α ∈ A sup ∥ y α ∥ ∞ ∥ b ∥ 2 p . (8) The infimum is taken over all common factorizations of the preceding form. With this definition, L p ( M ; ℓ ∞ ( A ) ) L_p(\mathcal{M};\ell_\infty(\mathcal{A})) L p ( M ; ℓ ∞ ( A ))
∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A ) ) = sup F ⊂ A , F f i n i t e ∥ ( x α ) α ∈ F ∥ L p ( M ; ℓ ∞ ( F ) ) . (9) \|(x_\alpha)_{\alpha\in\mathcal{A}}\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{A}))}
=
\sup_{\mathcal{F}\subset\mathcal{A},\ \mathcal{F}\ \mathrm{finite}}
\|(x_\alpha)_{\alpha\in\mathcal{F}}\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{F}))}. \tag*{(9)} ∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A )) = F ⊂ A , F finite sup ∥ ( x α ) α ∈ F ∥ L p ( M ; ℓ ∞ ( F )) . (9) Thus an arbitrary family belongs to the space exactly when its finite subfamilies have uniformly bounded maximal norms. See [14 Proposition 2.1(ii) and Remark 2.4] for more details.
For a positive family, the same result gives the order description
∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A ) ) = inf { ∥ a ∥ p : a ∈ L p + ( M ) , x α ≤ a for every α ∈ A } . (10) \|(x_\alpha)_{\alpha\in\mathcal{A}}\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{A}))}
=
\inf\{\|a\|_p:a\in L_p^+(\mathcal{M}),\ x_\alpha\le a\ \text{for every }\alpha\in\mathcal{A}\}. \tag*{(10)} ∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A )) = inf { ∥ a ∥ p : a ∈ L p + ( M ) , x α ≤ a for every α ∈ A } . (10) For this reason, we also use the customary notation
∥ sup α ∈ A + x α ∥ p : = ∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A ) ) . \left\|\sup_{\alpha\in\mathcal{A}}^{+}x_\alpha\right\|_p:=\|(x_\alpha)_{\alpha\in\mathcal{A}}\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{A}))}. α ∈ A sup + x α p := ∥ ( x α ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A )) . where the symbol “sup + \sup^+ sup + M \mathcal{M} M
For the summation space, let A \mathcal{A} A 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ x = ( x α ) α ∈ A x=(x_\alpha)_{\alpha\in\mathcal{A}} x = ( x α ) α ∈ A
∥ x ∥ L p ( M ; ℓ 1 ( A ) ) : = inf { ∥ ∑ k , α a k α ∗ a k α ∥ p 1 / 2 ∥ ∑ k , α b k α ∗ b k α ∥ p 1 / 2 : x α = ∑ k a k α ∗ b k α for every α ∈ A } . (11) \|x\|_{L_p(\mathcal{M};\ell_1(\mathcal{A}))}:=\inf\left\{\left\|\sum_{k,\alpha}a_{k\alpha}^{*}a_{k\alpha}\right\|_p^{1/2}\left\|\sum_{k,\alpha}b_{k\alpha}^{*}b_{k\alpha}\right\|_p^{1/2}:x_\alpha=\sum_k a_{k\alpha}^{*}b_{k\alpha}\ \text{for every }\alpha\in\mathcal{A}\right\}. \tag*{(11)} ∥ x ∥ L p ( M ; ℓ 1 ( A )) := inf ⎩ ⎨ ⎧ k , α ∑ a k α ∗ a k α p 1/2 k , α ∑ b k α ∗ b k α p 1/2 : x α = k ∑ a k α ∗ b k α for every α ∈ A ⎭ ⎬ ⎫ . (11) The infimum is taken over finite families a k α , b k α ∈ L 2 p ( M ) a_{k\alpha},b_{k\alpha}\in L_{2p}(\mathcal{M}) a k α , b k α ∈ L 2 p ( M ) L p ( M ; ℓ 1 ( A ) ) L_p(\mathcal{M};\ell_1(\mathcal{A})) L p ( M ; ℓ 1 ( A )) ∥ ⋅ ∥ L p ( M ; ℓ 1 ( A ) ) \|\cdot\|_{L_p(\mathcal{M};\ell_1(\mathcal{A}))} ∥ ⋅ ∥ L p ( M ; ℓ 1 ( A ))
We record explicitly the maximal–summation duality used below.
Lemma 2.1 ([14 Proposition 2.1(iii)] ). Let 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ p ′ p' p ′ p p p
L p ( M ; ℓ 1 ( A ) ) ∗ = L p ′ ( M ; ℓ ∞ ( A ) ) (12) L_p(\mathcal{M};\ell_1(\mathcal{A}))^*=L_{p'}(\mathcal{M};\ell_\infty(\mathcal{A})) \tag*{(12)} L p ( M ; ℓ 1 ( A ) ) ∗ = L p ′ ( M ; ℓ ∞ ( A )) (12) isometrically via the following duality bracket
⟨ ( x α ) , ( y α ) ⟩ = ∑ α ∈ A τ ( x α y α ) . (13) \langle(x_\alpha),(y_\alpha)\rangle=\sum_{\alpha\in\mathcal{A}}\tau(x_\alpha y_\alpha). \tag*{(13)} ⟨( x α ) , ( y α )⟩ = α ∈ A ∑ τ ( x α y α ) . (13) Throughout the paper, we will omit the index set A \mathcal{A} A
We shall use the following standard facts without reproving their operator-space constructions.
Lemma 2.2 ([22, Chapter 3] and [14 ] ). Let A \mathcal{A} A 1 ≤ p 0 , p 1 ≤ ∞ 1\le p_0,p_1\le\infty 1 ≤ p 0 , p 1 ≤ ∞ 0 < θ < 1 0<\theta<1 0 < θ < 1
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 , then
( L p 0 ( M ; ℓ ∞ ( A ) ) , L p 1 ( M ; ℓ ∞ ( A ) ) ) θ = L p ( M ; ℓ ∞ ( A ) ) \left(L_{p_0}(\mathcal{M};\ell_\infty(\mathcal{A})),L_{p_1}(\mathcal{M};\ell_\infty(\mathcal{A}))\right)_\theta=L_p(\mathcal{M};\ell_\infty(\mathcal{A})) ( L p 0 ( M ; ℓ ∞ ( A )) , L p 1 ( M ; ℓ ∞ ( A )) ) θ = L p ( M ; ℓ ∞ ( A )) isometrically.
Lemma 2.3 ([14 Section 2] ). Let A \mathcal{A} A 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ x = ( x α ) α ∈ A ∈ L p + ( M ; ℓ 1 ( A ) ) x=(x_\alpha)_{\alpha\in\mathcal{A}}\in L_p^+(\mathcal{M};\ell_1(\mathcal{A})) x = ( x α ) α ∈ A ∈ L p + ( M ; ℓ 1 ( A ))
∥ x ∥ L p ( M ; ℓ 1 ( A ) ) = ∥ ∑ α ∈ A x α ∥ p . \|x\|_{L_p(\mathcal{M};\ell_1(\mathcal{A}))}=\left\|\sum_{\alpha\in\mathcal{A}}x_\alpha\right\|_p. ∥ x ∥ L p ( M ; ℓ 1 ( A )) = α ∈ A ∑ x α p . The following result will allow us to test general maximal families against positive summation sequences, which is well-known to experts; see e.g. [14 ] . We include here a proof with a better resulting constant.
Lemma 2.4. Let 1 ≤ p < ∞ 1 \leq p < \infty 1 ≤ p < ∞ x = ( x α ) α x = (x_{\alpha})_{\alpha} x = ( x α ) α L p ( M ; ℓ 1 ) L_p(\mathcal{M};\ell_1) L p ( M ; ℓ 1 ) ε > 0 \varepsilon> 0 ε > 0 x ( 1 ) , … , x ( 4 ) x^{(1)},\ldots,x^{(4)} x ( 1 ) , … , x ( 4 ) λ j = i j \lambda_j = i^j λ j = i j j = 1 , 2 , 3 , 4 j = 1,2,3,4 j = 1 , 2 , 3 , 4
x = ∑ j = 1 4 λ j x ( j ) x = \sum_{j=1}^{4} \lambda_j x^{(j)} x = j = 1 ∑ 4 λ j x ( j ) and
∥ x ( j ) ∥ L p ( M ; ℓ 1 ) ≤ ∥ x ∥ L p ( M ; ℓ 1 ) + ε , j = 1 , 2 , 3 , 4. \left\|x^{(j)}\right\|_{L_p(\mathcal{M};\ell_1)} \leq\|x\|_{L_p(\mathcal{M};\ell_1)}+\varepsilon,\qquad j=1,2,3,4. x ( j ) L p ( M ; ℓ 1 ) ≤ ∥ x ∥ L p ( M ; ℓ 1 ) + ε , j = 1 , 2 , 3 , 4. If x x x
Proof. Firstly, suppose that x = ( x α ) α x = (x_{\alpha})_{\alpha} x = ( x α ) α ε > 0 \varepsilon> 0 ε > 0
x α = ∑ k a k α ∗ b k α = ∑ k b k α ∗ a k α , ∥ ∑ k , α a k α ∗ a k α ∥ p 1 / 2 ∥ ∑ k , α b k α ∗ b k α ∥ p 1 / 2 ≤ ∥ ( x α ) α ∥ L p ( M ; ℓ 1 ) + ε . x_{\alpha} = \sum_k a_{k\alpha}^{*}b_{k\alpha} = \sum_k b_{k\alpha}^{*}a_{k\alpha},\qquad\left\|\sum_{k,\alpha}a_{k\alpha}^{*}a_{k\alpha}\right\|_{p}^{1/2}\left\|\sum_{k,\alpha}b_{k\alpha}^{*}b_{k\alpha}\right\|_{p}^{1/2} \leq\|(x_{\alpha})_{\alpha}\|_{L_p(\mathcal{M};\ell_1)}+\varepsilon. x α = k ∑ a k α ∗ b k α = k ∑ b k α ∗ a k α , k , α ∑ a k α ∗ a k α p 1/2 k , α ∑ b k α ∗ b k α p 1/2 ≤ ∥ ( x α ) α ∥ L p ( M ; ℓ 1 ) + ε . We then rescale the factorization so that
∥ ∑ k , α a k α ∗ a k α ∥ p = ∥ ∑ k , α b k α ∗ b k α ∥ p = ∥ ∑ k , α a k α ∗ a k α ∥ p 1 / 2 ∥ ∑ k , α b k α ∗ b k α ∥ p 1 / 2 . \left\|\sum_{k,\alpha}a_{k\alpha}^{*}a_{k\alpha}\right\|_{p}=\left\|\sum_{k,\alpha}b_{k\alpha}^{*}b_{k\alpha}\right\|_{p}=\left\|\sum_{k,\alpha}a_{k\alpha}^{*}a_{k\alpha}\right\|_{p}^{1/2}\left\|\sum_{k,\alpha}b_{k\alpha}^{*}b_{k\alpha}\right\|_{p}^{1/2}. k , α ∑ a k α ∗ a k α p = k , α ∑ b k α ∗ b k α p = k , α ∑ a k α ∗ a k α p 1/2 k , α ∑ b k α ∗ b k α p 1/2 . Define u = ( u α ) α u=(u_{\alpha})_{\alpha} u = ( u α ) α v = ( v α ) α v=(v_{\alpha})_{\alpha} v = ( v α ) α
u α = 1 4 ∑ k ( a k α + b k α ) ∗ ( a k α + b k α ) , v α = 1 4 ∑ k ( a k α − b k α ) ∗ ( a k α − b k α ) , u_{\alpha}=\frac{1}{4}\sum_k(a_{k\alpha}+b_{k\alpha})^{*}(a_{k\alpha}+b_{k\alpha}),\qquad v_{\alpha}=\frac{1}{4}\sum_k(a_{k\alpha}-b_{k\alpha})^{*}(a_{k\alpha}-b_{k\alpha}), u α = 4 1 k ∑ ( a k α + b k α ) ∗ ( a k α + b k α ) , v α = 4 1 k ∑ ( a k α − b k α ) ∗ ( a k α − b k α ) , so both u u u v v v
u α − v α = x α , u α + v α = 1 2 ∑ k ( a k α ∗ a k α + b k α ∗ b k α ) . u_{\alpha}-v_{\alpha}=x_{\alpha},\qquad u_{\alpha}+v_{\alpha}=\frac{1}{2}\sum_k(a_{k\alpha}^{*}a_{k\alpha}+b_{k\alpha}^{*}b_{k\alpha}). u α − v α = x α , u α + v α = 2 1 k ∑ ( a k α ∗ a k α + b k α ∗ b k α ) . Then by Lemma 2.3
∥ u ∥ L p ( M ; ℓ 1 ) ≤ ∥ u + v ∥ L p ( M ; ℓ 1 ) ≤ 1 2 ∥ ∑ k , α ( a k α ∗ a k α + b k α ∗ b k α ) ∥ p ≤ 1 2 [ ∥ ∑ k , α a k α ∗ a k α ∥ p + ∥ ∑ k , α b k α ∗ b k α ∥ p ] ≤ ∥ ( x α ) α ∥ L p ( M ; ℓ 1 ) + ε . \begin{aligned}
\|u\|_{L_p(\mathcal{M};\ell_1)}\leq\|u+v\|_{L_p(\mathcal{M};\ell_1)}\leq\frac{1}{2}\left\|\sum_{k,\alpha}(a_{k\alpha}^{*}a_{k\alpha}+b_{k\alpha}^{*}b_{k\alpha})\right\|_p \\
&\leq\frac{1}{2}\left[\left\|\sum_{k,\alpha}a_{k\alpha}^{*}a_{k\alpha}\right\|_p+\left\|\sum_{k,\alpha}b_{k\alpha}^{*}b_{k\alpha}\right\|_p\right]\leq\|(x_{\alpha})_{\alpha}\|_{L_p(\mathcal{M};\ell_1)}+\varepsilon.
\end{aligned} ∥ u ∥ L p ( M ; ℓ 1 ) ≤ ∥ u + v ∥ L p ( M ; ℓ 1 ) ≤ 2 1 k , α ∑ ( a k α ∗ a k α + b k α ∗ b k α ) p ≤ 2 1 k , α ∑ a k α ∗ a k α p + k , α ∑ b k α ∗ b k α p ≤ ∥ ( x α ) α ∥ L p ( M ; ℓ 1 ) + ε . Given a general x ∈ L p ( M ; ℓ 1 ) x\in L_p(\mathcal{M};\ell_1) x ∈ L p ( M ; ℓ 1 ) ℜ x \Re x ℜ x ℑ x \Im x ℑ x L p ( M ; ℓ 1 ) L_p(\mathcal{M};\ell_1) L p ( M ; ℓ 1 )
We shall also use the following well-known balanced module Hölder inequality. See, for instance, [22 , 14 ] .
Lemma 2.5. Let 1 ≤ r , p , s ≤ ∞ 1\leq r,p,s\leq\infty 1 ≤ r , p , s ≤ ∞ 1 / r = 1 / p + 1 / s 1/r=1/p+1/s 1/ r = 1/ p + 1/ s a , b ∈ L 2 s ( M ) a,b\in L_{2s}(\mathcal{M}) a , b ∈ L 2 s ( M ) ( x α ) α ∈ L p ( M ; ℓ ∞ ) (x_{\alpha})_{\alpha}\in L_p(\mathcal{M};\ell_{\infty}) ( x α ) α ∈ L p ( M ; ℓ ∞ )
∥ ( a x α b ) α ∥ L r ( M ; ℓ ∞ ) ≤ ∥ a ∥ 2 s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) ∥ b ∥ 2 s . (14) \|(ax_{\alpha}b)_{\alpha}\|_{L_r(\mathcal{M};\ell_{\infty})}\leq\|a\|_{2s}\|(x_{\alpha})_{\alpha}\|_{L_p(\mathcal{M};\ell_{\infty})}\|b\|_{2s}. \tag*{(14)} ∥ ( a x α b ) α ∥ L r ( M ; ℓ ∞ ) ≤ ∥ a ∥ 2 s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) ∥ b ∥ 2 s . (14) In particular, if u ∈ L s + ( M ) u\in L_s^{+}(\mathcal{M}) u ∈ L s + ( M )
∥ ( u 1 / 2 x α u 1 / 2 ) α ∥ L r ( M ; ℓ ∞ ) ≤ ∥ u ∥ s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) . \|(u^{1/2}x_{\alpha}u^{1/2})_{\alpha}\|_{L_r(\mathcal{M};\ell_{\infty})}\leq\|u\|_s\|(x_{\alpha})_{\alpha}\|_{L_p(\mathcal{M};\ell_{\infty})}. ∥ ( u 1/2 x α u 1/2 ) α ∥ L r ( M ; ℓ ∞ ) ≤ ∥ u ∥ s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) . Proof. Take a common factorization
x α = c y α d , c , d ∈ L 2 p ( M ) , sup α ∥ y α ∥ ∞ < ∞ . x_{\alpha}=cy_{\alpha}d,\qquad c,d\in L_{2p}(\mathcal{M}),\qquad\sup_{\alpha}\|y_{\alpha}\|_{\infty}<\infty. x α = c y α d , c , d ∈ L 2 p ( M ) , α sup ∥ y α ∥ ∞ < ∞. Then a x α b = ( a c ) y α ( d b ) ax_{\alpha}b=(ac)y_{\alpha}(db) a x α b = ( a c ) y α ( d b ) ( a x α b ) α (ax_{\alpha}b)_{\alpha} ( a x α b ) α 1 / ( 2 r ) = 1 / ( 2 s ) + 1 / ( 2 p ) 1/(2r)=1/(2s)+1/(2p) 1/ ( 2 r ) = 1/ ( 2 s ) + 1/ ( 2 p )
∥ ( a x α b ) α ∥ L r ( M ; ℓ ∞ ) ≤ inf a x α b = ( a c ) y α ( d b ) ∥ a c ∥ 2 r ( sup α ∥ y α ∥ ∞ ) ∥ d b ∥ 2 r ≤ inf x α = c y α d ∥ a ∥ 2 s ⋅ ∥ c ∥ 2 p ( sup α ∥ y α ∥ ∞ ) ∥ d ∥ 2 p ⋅ ∥ b ∥ 2 s = ∥ a ∥ 2 s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) ∥ b ∥ 2 s , \begin{aligned}
\|(ax_{\alpha}b)_{\alpha}\|_{L_r(\mathcal{M};\ell_\infty)}
&\leq\inf_{ax_{\alpha}b=(ac)y_{\alpha}(db)} \|ac\|_{2r}\left(\sup_{\alpha}\|y_{\alpha}\|_{\infty}\right)\|db\|_{2r}\\
&\leq\inf_{x_{\alpha}=cy_{\alpha}d}\|a\|_{2s}\cdot\|c\|_{2p}\left(\sup_{\alpha}\|y_{\alpha}\|_{\infty}\right)\|d\|_{2p}\cdot\|b\|_{2s}\\
&=\|a\|_{2s}\|(x_{\alpha})_{\alpha}\|_{L_p(\mathcal{M};\ell_\infty)}\|b\|_{2s},
\end{aligned} ∥ ( a x α b ) α ∥ L r ( M ; ℓ ∞ ) ≤ a x α b = ( a c ) y α ( d b ) inf ∥ a c ∥ 2 r ( α sup ∥ y α ∥ ∞ ) ∥ d b ∥ 2 r ≤ x α = c y α d inf ∥ a ∥ 2 s ⋅ ∥ c ∥ 2 p ( α sup ∥ y α ∥ ∞ ) ∥ d ∥ 2 p ⋅ ∥ b ∥ 2 s = ∥ a ∥ 2 s ∥ ( x α ) α ∥ L p ( M ; ℓ ∞ ) ∥ b ∥ 2 s , which gives the estimate as desired. □
Weighted L p L_p L p
In this subsection, the noncommutative measure space ( M , τ ) (\mathcal{M},\tau) ( M , τ ) w ∈ L 0 + ( M ) w\in L_0^+(\mathcal{M}) w ∈ L 0 + ( M ) w w w w − 1 w^{-1} w − 1 ( M , τ ) (\mathcal{M},\tau) ( M , τ ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ x ∈ L 0 ( M ) x\in L_0(\mathcal{M}) x ∈ L 0 ( M ) L p L_p L p
∥ x ∥ L p ( M , w ) = ∥ w 1 / 2 p x w 1 / 2 p ∥ p , (15) \|x\|_{L_p(\mathcal{M},w)}=\|w^{1/2p}xw^{1/2p}\|_p, \tag*{(15)} ∥ x ∥ L p ( M , w ) = ∥ w 1/2 p x w 1/2 p ∥ p , (15) and the weighted L p L_p L p L p L_p L p
L p ( M , w ) : = { x ∈ L 0 ( M ) : ∥ x ∥ L p ( M , w ) < ∞ } . L_p(\mathcal{M},w):=\{x\in L_0(\mathcal{M}):\|x\|_{L_p(\mathcal{M},w)}<\infty\}. L p ( M , w ) := { x ∈ L 0 ( M ) : ∥ x ∥ L p ( M , w ) < ∞ } . The map
J p , w : L p ( M , w ) ⟶ L p ( M ) x ⟼ w 1 / 2 p x w 1 / 2 p \begin{aligned}
J_{p,w}:L_p(\mathcal{M},w)&\longrightarrow L_p(\mathcal{M})\\
x&\longmapsto w^{1/2p}xw^{1/2p}
\end{aligned} J p , w : L p ( M , w ) x ⟶ L p ( M ) ⟼ w 1/2 p x w 1/2 p is an onto isometry. Indeed, it is an isometry by definition. If y ∈ L p ( M ) y\in L_p(\mathcal{M}) y ∈ L p ( M ) x = w − 1 / 2 p y w − 1 / 2 p x=w^{-1/2p}yw^{-1/2p} x = w − 1/2 p y w − 1/2 p L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) J p , w ( x ) = y J_{p,w}(x)=y J p , w ( x ) = y L p ( M ) L_p(\mathcal{M}) L p ( M ) L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w )
e n ( w ) = 1 [ 1 / n , n ] ( w ) , C ( w ) = ⋃ n ≥ 1 e n ( w ) M e n ( w ) . e_n(w)=1_{[1/n,n]}(w),\qquad\mathcal{C}(w)=\bigcup_{n\geq1}e_n(w)\mathcal{M}e_n(w). e n ( w ) = 1 [ 1/ n , n ] ( w ) , C ( w ) = n ≥ 1 ⋃ e n ( w ) M e n ( w ) . Lemma 2.6. For every 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ C ( w ) \mathcal{C}(w) C ( w ) L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) M ∩ L p ( M , w ) ⊂ L p ( M , w ) \mathcal{M}\cap L_p(\mathcal{M},w)\subset L_p(\mathcal{M},w) M ∩ L p ( M , w ) ⊂ L p ( M , w )
Proof. Given a ∈ L p ( M , w ) a\in L_p(\mathcal{M},w) a ∈ L p ( M , w )
inf z ∈ C ( w ) ∥ J p , w ( z − a ) ∥ p = 0. (16) \inf_{z\in\mathcal{C}(w)}\|J_{p,w}(z-a)\|_p=0. \tag*{(16)} z ∈ C ( w ) inf ∥ J p , w ( z − a ) ∥ p = 0. (16) Firstly, we show that J p , w ( C ( w ) ) = C ( w ) J_{p,w}(\mathcal{C}(w))=\mathcal{C}(w) J p , w ( C ( w )) = C ( w ) x ∈ C ( w ) x\in\mathcal{C}(w) x ∈ C ( w ) m ∈ N + m\in\mathbb{N}_+ m ∈ N + y ∈ M y\in\mathcal{M} y ∈ M x = e m ( w ) y e m ( w ) x=e_m(w)ye_m(w) x = e m ( w ) y e m ( w )
J p , w ( x ) = e m ( w ) w 1 / 2 p y w 1 / 2 p e m ( w ) = e m ( w ) ( e m ( w ) w 1 / 2 p y w 1 / 2 p e m ( w ) ) e m ( w ) ∈ C ( w ) . J_{p,w}(x)=e_m(w)w^{1/2p}yw^{1/2p}e_m(w)=e_m(w)\big(e_m(w)w^{1/2p}yw^{1/2p}e_m(w)\big)e_m(w)\in\mathcal{C}(w). J p , w ( x ) = e m ( w ) w 1/2 p y w 1/2 p e m ( w ) = e m ( w ) ( e m ( w ) w 1/2 p y w 1/2 p e m ( w ) ) e m ( w ) ∈ C ( w ) . On the other hand, we know that e m ( w ) w − 1 / 2 p y w − 1 / 2 p e m ( w ) ∈ C ( w ) e_m(w)w^{-1/2p}yw^{-1/2p}e_m(w)\in\mathcal{C}(w) e m ( w ) w − 1/2 p y w − 1/2 p e m ( w ) ∈ C ( w )
J p , w ( e m ( w ) w − 1 / 2 p y w − 1 / 2 p e m ( w ) ) = x , J_{p,w}\big(e_m(w)w^{-1/2p}yw^{-1/2p}e_m(w)\big)=x, J p , w ( e m ( w ) w − 1/2 p y w − 1/2 p e m ( w ) ) = x , which proves the claim. Then we have C ( w ) ⊂ L p ( M , w ) \mathcal{C}(w)\subset L_p(\mathcal{M},w) C ( w ) ⊂ L p ( M , w )
Thus the equation (16) is equivalent to
inf z ∈ C ( w ) ∥ z − w 1 / 2 p a w 1 / 2 p ∥ p = 0 , \inf_{z\in\mathcal{C}(w)}\|z-w^{1/2p}aw^{1/2p}\|_p=0, z ∈ C ( w ) inf ∥ z − w 1/2 p a w 1/2 p ∥ p = 0 , which is equivalent to the density of C ( w ) \mathcal{C}(w) C ( w ) L p ( M ) L_p(\mathcal{M}) L p ( M ) e n ( w ) ↑ 1 e_n(w)\uparrow1 e n ( w ) ↑ 1 x ∈ L p ( M ) x\in L_p(\mathcal{M}) x ∈ L p ( M )
e n ( w ) x e n ( w ) ⟶ x in L p ( M ) , (17) e_n(w)xe_n(w)\longrightarrow x\qquad\text{in }L_p(\mathcal{M}), \tag*{(17)} e n ( w ) x e n ( w ) ⟶ x in L p ( M ) , (17) which is the standard projection approximation in the noncommutative L p L_p L p M ⊂ L p ( M ) \mathcal{M}\subset L_p(\mathcal{M}) M ⊂ L p ( M )
inf z ∈ M ∥ e n ( w ) ( z − x ) e n ( w ) ∥ p ≤ inf z ∈ M ∥ z − x ∥ p = 0 , (18) \inf_{z\in\mathcal{M}}\|e_n(w)(z-x)e_n(w)\|_p\leq\inf_{z\in\mathcal{M}}\|z-x\|_p=0, \tag*{(18)} z ∈ M inf ∥ e n ( w ) ( z − x ) e n ( w ) ∥ p ≤ z ∈ M inf ∥ z − x ∥ p = 0 , (18) which gives the density as desired. Finally, C ( w ) ⊂ M \mathcal{C}(w) \subset\mathcal{M} C ( w ) ⊂ M M ∩ L p ( M , w ) \mathcal{M} \cap L_p(\mathcal{M},w) M ∩ L p ( M , w ) L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w )
Now we are able to define the vector-valued weighted noncommutative L p L_p L p 1 ≤ p ≤ ∞ 1 \leq p \leq\infty 1 ≤ p ≤ ∞ A \mathcal{A} A L p ( M , w ; ℓ ∞ ( A ) ) L_p(\mathcal{M},w;\ell_\infty(\mathcal{A})) L p ( M , w ; ℓ ∞ ( A )) ( x α ) α ∈ A ⊂ L 0 ( M ) (x_\alpha)_{\alpha\in\mathcal{A}} \subset L_0(\mathcal{M}) ( x α ) α ∈ A ⊂ L 0 ( M )
∥ ( x α ) α ∈ A ∥ L p ( M , w ; ℓ ∞ ( A ) ) : = ∥ ( w 1 / 2 p x α w 1 / 2 p ) α ∈ A ∥ L p ( M ; ℓ ∞ ( A ) ) < ∞ . (19) \left\|(x_\alpha)_{\alpha\in\mathcal{A}}\right\|_{L_p(\mathcal{M},w;\ell_\infty(\mathcal{A}))}:=\left\|\left(w^{1/2p}x_\alpha w^{1/2p}\right)_{\alpha\in\mathcal{A}}\right\|_{L_p(\mathcal{M};\ell_\infty(\mathcal{A}))}<\infty. \tag*{(19)} ∥ ( x α ) α ∈ A ∥ L p ( M , w ; ℓ ∞ ( A )) := ( w 1/2 p x α w 1/2 p ) α ∈ A L p ( M ; ℓ ∞ ( A )) < ∞. (19) Next, we record the scalar and maximal Hölder type testing estimates that will be used in later sections.
Lemma 2.7. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ ( x α ) α ∈ L p ( M ; ℓ ∞ ) (x_\alpha)_\alpha\in L_p(\mathcal{M};\ell_\infty) ( x α ) α ∈ L p ( M ; ℓ ∞ ) x ∈ L p ( M ) x\in L_p(\mathcal{M}) x ∈ L p ( M )
(i) If 1 < p < r < ∞ 1<p<r<\infty 1 < p < r < ∞ 1 / p = 1 / r + 1 / s 1/p=1/r+1/s 1/ p = 1/ r + 1/ s ε > 0 \varepsilon>0 ε > 0 u ∈ L s + ( M ) u\in L_s^+(\mathcal{M}) u ∈ L s + ( M )
∥ u − 1 / 2 x u − 1 / 2 ∥ r ∥ u ∥ s ≤ 2 ∥ x ∥ p + ε . (20) \left\|u^{-1/2}xu^{-1/2}\right\|_r\|u\|_s\leq2\|x\|_p+\varepsilon. \tag*{(20)} u − 1/2 x u − 1/2 r ∥ u ∥ s ≤ 2∥ x ∥ p + ε . (20) Moreover, if x x x
(ii) If 1 < r < p < ∞ 1<r<p<\infty 1 < r < p < ∞ 1 / r = 1 / p + 1 / s 1/r=1/p+1/s 1/ r = 1/ p + 1/ s
sup u ∈ L s + ( M ) s ( u ) = 1 ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ∥ u ∥ s ≤ ∥ sup α + x α ∥ p ≤ 4 sup u ∈ L s + ( M ) s ( u ) = 1 ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ∥ u ∥ s . (21) \sup_{\substack{u\in L_s^+(\mathcal{M})\\s(u)=1}}\frac{\left\|\sup_\alpha^+u^{1/2}x_\alpha u^{1/2}\right\|_r}{\|u\|_s}
\leq\left\|\sup_\alpha^+x_\alpha\right\|_p
\leq4\sup_{\substack{u\in L_s^+(\mathcal{M})\\s(u)=1}}\frac{\left\|\sup_\alpha^+u^{1/2}x_\alpha u^{1/2}\right\|_r}{\|u\|_s}. \tag*{(21)} u ∈ L s + ( M ) s ( u ) = 1 sup ∥ u ∥ s sup α + u 1/2 x α u 1/2 r ≤ α sup + x α p ≤ 4 u ∈ L s + ( M ) s ( u ) = 1 sup ∥ u ∥ s sup α + u 1/2 x α u 1/2 r . (21) Proof. For (i), let x = q ∣ x ∣ x=q|x| x = q ∣ x ∣ x x x
u δ = ∣ x ∣ p / s + q ∣ x ∣ p / s q ∗ + δ 1. u_\delta=|x|^{p/s}+q|x|^{p/s}q^*+\delta1. u δ = ∣ x ∣ p / s + q ∣ x ∣ p / s q ∗ + δ 1. Trace-finiteness makes u δ u_\delta u δ L s + ( M ) L_s^+(\mathcal{M}) L s + ( M ) ∣ x ∣ p / s ≤ u δ |x|^{p/s}\leq u_\delta ∣ x ∣ p / s ≤ u δ q ∣ x ∣ p / s q ∗ ≤ u δ q|x|^{p/s}q^*\leq u_\delta q ∣ x ∣ p / s q ∗ ≤ u δ
u δ − 1 / 2 x u δ − 1 / 2 = ( u δ − 1 / 2 q ∣ x ∣ p / 2 s ) ∣ x ∣ p / r ( ∣ x ∣ p / 2 s u δ − 1 / 2 ) u_\delta^{-1/2}xu_\delta^{-1/2}
=\left(u_\delta^{-1/2}q|x|^{p/2s}\right)|x|^{p/r}\left(|x|^{p/2s}u_\delta^{-1/2}\right) u δ − 1/2 x u δ − 1/2 = ( u δ − 1/2 q ∣ x ∣ p /2 s ) ∣ x ∣ p / r ( ∣ x ∣ p /2 s u δ − 1/2 ) contractions. Therefore
∥ u δ − 1 / 2 x u δ − 1 / 2 ∥ r ≤ ∥ x ∥ p p / r , lim sup δ ↓ 0 ∥ u δ ∥ s ≤ 2 ∥ x ∥ p p / s . \left\|u_\delta^{-1/2}xu_\delta^{-1/2}\right\|_r\leq\|x\|_p^{p/r},\qquad\limsup_{\delta\downarrow0}\|u_\delta\|_s\leq2\|x\|_p^{p/s}. u δ − 1/2 x u δ − 1/2 r ≤ ∥ x ∥ p p / r , δ ↓ 0 lim sup ∥ u δ ∥ s ≤ 2∥ x ∥ p p / s . The identity p / r + p / s = 1 p/r+p/s=1 p / r + p / s = 1 x x x u δ = x p / s + δ 1 u_\delta=x^{p/s}+\delta1 u δ = x p / s + δ 1
The first inequality in (ii) follows immediately from Lemma 2.5. To prove the reverse inequality, assume that the supremum on its right is finite. Lemma 2.1 gives that
∥ sup α + x α ∥ p = sup ( y α ) α : ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 ∣ ∑ α τ ( y α x α ) ∣ . \left\|\sup_\alpha^+x_\alpha\right\|_p
=\sup_{\substack{(y_\alpha)_\alpha:\|(y_\alpha)_\alpha\|_{L_{p'}(\mathcal{M};\ell_1)}=1}}
\left|\sum_\alpha\tau(y_\alpha x_\alpha)\right|. α sup + x α p = ( y α ) α : ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 sup α ∑ τ ( y α x α ) . Using the decomposition in Lemma 2.4, we obtain
∥ sup α + x α ∥ p ≤ sup ( y α ) α : ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 4 ∑ j = 1 4 ∣ ∑ α τ ( y α ( j ) x α ) ∣ ≤ 4 sup ( y α ) α ∈ L p ′ + ( M ; ℓ 1 ) ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 ∣ ∑ α τ ( y α x α ) ∣ . (22) \left\|\sup_\alpha^+x_\alpha\right\|_p
\leq
\sup_{\substack{(y_\alpha)_\alpha:\|(y_\alpha)_\alpha\|_{L_{p'}(\mathcal{M};\ell_1)}=1}}
\frac{4}{\sum_{j=1}^{4}}\left|\sum_\alpha\tau(y_\alpha^{(j)}x_\alpha)\right|
\leq4
\sup_{\substack{(y_\alpha)_\alpha\in L_{p'}^+(\mathcal{M};\ell_1)\\\|(y_\alpha)_\alpha\|_{L_{p'}(\mathcal{M};\ell_1)}=1}}
\left|\sum_\alpha\tau(y_\alpha x_\alpha)\right|. \tag*{(22)} α sup + x α p ≤ ( y α ) α : ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 sup ∑ j = 1 4 4 α ∑ τ ( y α ( j ) x α ) ≤ 4 ( y α ) α ∈ L p ′ + ( M ; ℓ 1 ) ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 sup α ∑ τ ( y α x α ) . (22) Given ( y α ) α ∈ L p ′ + ( M ; ℓ 1 ) (y_\alpha)_\alpha\in L_{p'}^+(\mathcal{M};\ell_1) ( y α ) α ∈ L p ′ + ( M ; ℓ 1 ) ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 \|(y_\alpha)_\alpha\|_{L_{p'}(\mathcal{M};\ell_1)}=1 ∥ ( y α ) α ∥ L p ′ ( M ; ℓ 1 ) = 1 y ~ = ∑ α y α \widetilde{y}=\sum_\alpha y_\alpha y = ∑ α y α ε > 0 \varepsilon>0 ε > 0 u ∈ L s + ( M ) u\in L_s^+(\mathcal{M}) u ∈ L s + ( M )
∥ u − 1 / 2 y ~ u − 1 / 2 ∥ r ′ ∥ u ∥ s ≤ ∥ y ~ ∥ p ′ + ε = 1 + ε . \left\|u^{-1/2}\widetilde{y}u^{-1/2}\right\|_{r'}\|u\|_s\leq\|\widetilde{y}\|_{p'}+\varepsilon=1+\varepsilon. u − 1/2 y u − 1/2 r ′ ∥ u ∥ s ≤ ∥ y ∥ p ′ + ε = 1 + ε . Hölder’s inequality and maximal–summation duality give
∣ ∑ α τ ( y α x α ) ∣ = ∣ ∑ α τ ( u − 1 / 2 y α u − 1 / 2 u 1 / 2 x α u 1 / 2 ) ∣ ≤ ∥ ( u − 1 / 2 y α u − 1 / 2 ) α ∥ L r ′ ( M ; ℓ 1 ) ⋅ ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ≤ ( 1 + ε ) ⋅ ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ∥ u ∥ s ≤ ( 1 + ε ) ⋅ sup u ∈ L s + ( M ) s ( u ) = 1 ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ∥ u ∥ s , \begin{aligned}
\left|\sum_{\alpha}\tau(y_\alpha x_\alpha)\right|=\left|\sum_{\alpha}\tau(u^{-1/2}y_\alpha u^{-1/2}u^{1/2}x_\alpha u^{1/2})\right| \\
&\leq\|(u^{-1/2}y_\alpha u^{-1/2})_\alpha\|_{L_{r'}(\mathcal{M};\ell_1)}\cdot\|\sup_\alpha^{+}u^{1/2}x_\alpha u^{1/2}\|_r \\
&\leq(1+\varepsilon)\cdot\frac{\|\sup_\alpha^{+}u^{1/2}x_\alpha u^{1/2}\|_r}{\|u\|_s} \\
&\leq(1+\varepsilon)\cdot\sup_{\substack{u\in L_s^{+}(\mathcal{M})\\s(u)=1}}\frac{\|\sup_\alpha^{+}u^{1/2}x_\alpha u^{1/2}\|_r}{\|u\|_s},
\end{aligned} α ∑ τ ( y α x α ) = α ∑ τ ( u − 1/2 y α u − 1/2 u 1/2 x α u 1/2 ) ≤ ∥ ( u − 1/2 y α u − 1/2 ) α ∥ L r ′ ( M ; ℓ 1 ) ⋅ ∥ α sup + u 1/2 x α u 1/2 ∥ r ≤ ( 1 + ε ) ⋅ ∥ u ∥ s ∥ sup α + u 1/2 x α u 1/2 ∥ r ≤ ( 1 + ε ) ⋅ u ∈ L s + ( M ) s ( u ) = 1 sup ∥ u ∥ s ∥ sup α + u 1/2 x α u 1/2 ∥ r , which holds for any ε > 0 \varepsilon>0 ε > 0
∣ ∑ α τ ( y α x α ) ∣ ≤ sup u ∈ L s + ( M ) s ( u ) = 1 ∥ sup α + u 1 / 2 x α u 1 / 2 ∥ r ∥ u ∥ s . (23) \left|\sum_{\alpha}\tau(y_\alpha x_\alpha)\right|\leq\sup_{\substack{u\in L_s^{+}(\mathcal{M})\\s(u)=1}}\frac{\|\sup_\alpha^{+}u^{1/2}x_\alpha u^{1/2}\|_r}{\|u\|_s}. \tag*{(23)} α ∑ τ ( y α x α ) ≤ u ∈ L s + ( M ) s ( u ) = 1 sup ∥ u ∥ s ∥ sup α + u 1/2 x α u 1/2 ∥ r . (23) Combining (22) and (23) , we get the conclusion. □
The following lemma establishes a monotonicity property of noncommutative maximal norms.
Lemma 2.8. Let 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ 0 < u ≤ v 0<u\leq v 0 < u ≤ v L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) ( x α ) α ∈ A ⊂ L 0 ( M ) (x_\alpha)_{\alpha\in A}\subset L_0(\mathcal{M}) ( x α ) α ∈ A ⊂ L 0 ( M ) ( v 1 / 2 x α v 1 / 2 ) α ∈ A ∈ L p ( M ; ℓ ∞ ( A ) ) (v^{1/2}x_\alpha v^{1/2})_{\alpha\in A}\in L_p(\mathcal{M};\ell_\infty(A)) ( v 1/2 x α v 1/2 ) α ∈ A ∈ L p ( M ; ℓ ∞ ( A ))
∥ sup α + u 1 / 2 x α u 1 / 2 ∥ p ≤ ∥ sup α + v 1 / 2 x α v 1 / 2 ∥ p . \|\sup_\alpha^{+}u^{1/2}x_\alpha u^{1/2}\|_p\leq\|\sup_\alpha^{+}v^{1/2}x_\alpha v^{1/2}\|_p. ∥ α sup + u 1/2 x α u 1/2 ∥ p ≤ ∥ α sup + v 1/2 x α v 1/2 ∥ p . Moreover, v − 1 ≤ u − 1 v^{-1}\leq u^{-1} v − 1 ≤ u − 1 ( u − 1 / 2 x α u − 1 / 2 ) α ∈ A ∈ L p ( M ; ℓ ∞ ( A ) ) (u^{-1/2}x_\alpha u^{-1/2})_{\alpha\in A}\in L_p(\mathcal{M};\ell_\infty(A)) ( u − 1/2 x α u − 1/2 ) α ∈ A ∈ L p ( M ; ℓ ∞ ( A ))
∥ sup α + v − 1 / 2 x α v − 1 / 2 ∥ p ≤ ∥ sup α + u − 1 / 2 x α u − 1 / 2 ∥ p . \|\sup_\alpha^{+}v^{-1/2}x_\alpha v^{-1/2}\|_p\leq\|\sup_\alpha^{+}u^{-1/2}x_\alpha u^{-1/2}\|_p. ∥ α sup + v − 1/2 x α v − 1/2 ∥ p ≤ ∥ α sup + u − 1/2 x α u − 1/2 ∥ p . Proof. The operator order u ≤ v u\leq v u ≤ v c ∈ M c\in\mathcal{M} c ∈ M u 1 / 2 = c v 1 / 2 u^{1/2}=cv^{1/2} u 1/2 = c v 1/2 L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) u 1 / 2 x α u 1 / 2 = c ( v 1 / 2 x α v 1 / 2 ) c ∗ u^{1/2}x_\alpha u^{1/2}=c(v^{1/2}x_\alpha v^{1/2})c^* u 1/2 x α u 1/2 = c ( v 1/2 x α v 1/2 ) c ∗ Lemma 2.5 , with s = ∞ s=\infty s = ∞ a = c a=c a = c b = c ∗ b=c^* b = c ∗
Full support of v v v u u u 0 < v − 1 ≤ u − 1 0<v^{-1}\leq u^{-1} 0 < v − 1 ≤ u − 1 ( v − 1 , u − 1 ) (v^{-1},u^{-1}) ( v − 1 , u − 1 )
Weighted L p L_p L p N \mathcal{N} N
Even though we do not know how to properly define the weighted L p L_p L p
N = L ∞ ( R ) ⊗ ‾ M d ( C ) , \mathcal{N}=L_\infty(\mathbb{R})\overline{\otimes}M_d(\mathbb{C}), N = L ∞ ( R ) ⊗ M d ( C ) , equipped with the canonical tensor trace. In this case, it is well-known that
L p ( N ) = L p ( R ; S p d ) (24) L_p(\mathcal{N})=L_p(\mathbb{R};S_p^d) \tag*{(24)} L p ( N ) = L p ( R ; S p d ) (24) isometrically for 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞
A matrix weight is, throughout this paper, a strongly measurable function w : R → M d ( C ) w:\mathbb{R}\to M_d(\mathbb{C}) w : R → M d ( C ) 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ f f f L p L_p L p
∥ f ∥ L p ( N , w ) = ( ∫ R ∥ w ( t ) 1 / 2 p f ( t ) w ( t ) 1 / 2 p ∥ S p d p d t ) 1 / p , (25) \|f\|_{L_p(\mathcal{N},w)}=\left(\int_{\mathbb{R}}\|w(t)^{1/2p}f(t)w(t)^{1/2p}\|_{S_p^d}^{p}\,dt\right)^{1/p}, \tag*{(25)} ∥ f ∥ L p ( N , w ) = ( ∫ R ∥ w ( t ) 1/2 p f ( t ) w ( t ) 1/2 p ∥ S p d p d t ) 1/ p , (25) and the related weighted L p L_p L p L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p L_p L p
J p , w : L p ( N , w ) ⟶ L p ( N ) J_{p,w}:L_p(\mathcal{N},w)\longrightarrow L_p(\mathcal{N}) J p , w : L p ( N , w ) ⟶ L p ( N ) f ⟼ w 1 / 2 p f w 1 / 2 p f\longmapsto w^{1/2p}fw^{1/2p} f ⟼ w 1/2 p f w 1/2 p is an onto isometry. Thus L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) w w w w ∉ L 0 ( N ) w\notin L_0(\mathcal{N}) w ∈ / L 0 ( N ) L p ( N , w ) ⊄ L 0 ( N ) L_p(\mathcal{N},w)\not\subset L_0(\mathcal{N}) L p ( N , w ) ⊂ L 0 ( N ) w 1 / 2 p f w 1 / 2 p w^{1/2p}fw^{1/2p} w 1/2 p f w 1/2 p f ∈ L p ( N , w ) f\in L_p(\mathcal{N},w) f ∈ L p ( N , w ) w 1 / 2 p f w 1 / 2 p w^{1/2p}fw^{1/2p} w 1/2 p f w 1/2 p L p ( N ) ⊂ L 0 ( N ) L_p(\mathcal{N})\subset L_0(\mathcal{N}) L p ( N ) ⊂ L 0 ( N ) L p ( N ) L_p(\mathcal{N}) L p ( N ) w w w
Define S N c \mathcal{S}_{\mathcal{N}}^c S N c N \mathcal{N} N
S N c = ⋃ m ≥ 1 L ∞ ( − m , m ) ⊗ ‾ M d ( C ) . \mathcal{S}_{\mathcal{N}}^c=\bigcup_{m\geq1}L_\infty(-m,m)\mathbin{\overline{\otimes}}M_d(\mathbb{C}). S N c = m ≥ 1 ⋃ L ∞ ( − m , m ) ⊗ M d ( C ) . One can verify that S N c ⊂ L p ( N , w ) \mathcal{S}_{\mathcal{N}}^c\subset L_p(\mathcal{N},w) S N c ⊂ L p ( N , w ) w w w L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w )
Lemma 2.9. For 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ S N c \mathcal{S}_{\mathcal{N}}^c S N c L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w )
Proof. Given f ∈ L p ( N , w ) f\in L_p(\mathcal{N},w) f ∈ L p ( N , w )
E m , k = ( − m , m ) ∩ { t : ∥ f ( t ) ∥ S d ∞ ≤ k } , f m , k = χ E m , k f . E_{m,k}=(-m,m)\cap\{t:\|f(t)\|_{S_d^\infty}\leq k\},\qquad f_{m,k}=\chi_{E_{m,k}}f. E m , k = ( − m , m ) ∩ { t : ∥ f ( t ) ∥ S d ∞ ≤ k } , f m , k = χ E m , k f . Then f m , k ∈ S N c f_{m,k}\in\mathcal{S}_{\mathcal{N}}^c f m , k ∈ S N c
∥ f − f m , k ∥ L p ( N , w ) = ∥ J p , w ( f − f m , k ) ∥ p = ∥ χ R ∖ E m , k J p , w f ∥ p ⟶ 0 \|f-f_{m,k}\|_{L_p(\mathcal{N},w)}
=\|J_{p,w}(f-f_{m,k})\|_p
=\|\chi_{\mathbb{R}\setminus E_{m,k}}J_{p,w}f\|_p
\longrightarrow0 ∥ f − f m , k ∥ L p ( N , w ) = ∥ J p , w ( f − f m , k ) ∥ p = ∥ χ R ∖ E m , k J p , w f ∥ p ⟶ 0 when m , k → + ∞ m,k\to+\infty m , k → + ∞ E m , k E_{m,k} E m , k R \mathbb{R} R
For 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ A \mathcal{A} A L p ( N , w ; ℓ ∞ ( A ) ) L_p(\mathcal{N},w;\ell_\infty(\mathcal{A})) L p ( N , w ; ℓ ∞ ( A )) ( f α ) α ∈ A (f_\alpha)_{\alpha\in\mathcal{A}} ( f α ) α ∈ A
∥ ( f α ) α ∈ A ∥ L p ( N , w ; ℓ ∞ ( A ) ) : = ∥ ( w 1 / 2 p f α w 1 / 2 p ) α ∈ A ∥ L p ( N ; ℓ ∞ ( A ) ) . (26) \|(f_\alpha)_{\alpha\in\mathcal{A}}\|_{L_p(\mathcal{N},w;\ell_\infty(\mathcal{A}))}
:=\|(w^{1/2p}f_\alpha w^{1/2p})_{\alpha\in\mathcal{A}}\|_{L_p(\mathcal{N};\ell_\infty(\mathcal{A}))}. \tag*{(26)} ∥ ( f α ) α ∈ A ∥ L p ( N , w ; ℓ ∞ ( A )) := ∥ ( w 1/2 p f α w 1/2 p ) α ∈ A ∥ L p ( N ; ℓ ∞ ( A )) . (26) A strongly measurable matrix-valued function w w w invertible (resp. positive ) when w ( t ) w(t) w ( t ) t ∈ R t\in\mathbb{R} t ∈ R
Lemma 2.10. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ ( f α ) α ∈ L p ( N ; ℓ ∞ ) (f_\alpha)_\alpha\in L_p(\mathcal{N};\ell_\infty) ( f α ) α ∈ L p ( N ; ℓ ∞ ) f ∈ L p ( N ) f\in L_p(\mathcal{N}) f ∈ L p ( N )
(i) If 1 < p < r < ∞ 1<p<r<\infty 1 < p < r < ∞ 1 / p = 1 / r + 1 / s 1/p=1/r+1/s 1/ p = 1/ r + 1/ s ε > 0 \varepsilon>0 ε > 0 u ∈ L s + ( N ) u\in L_s^+(\mathcal{N}) u ∈ L s + ( N )
∥ u − 1 / 2 f u − 1 / 2 ∥ r ∥ u ∥ s ≤ 2 ∥ f ∥ p + ε , (27) \|u^{-1/2}fu^{-1/2}\|_r\|u\|_s\leq2\|f\|_p+\varepsilon, \tag*{(27)} ∥ u − 1/2 f u − 1/2 ∥ r ∥ u ∥ s ≤ 2∥ f ∥ p + ε , (27) where the inverse and multiplication should be understood pointwise. Moreover, if f f f 2 2 2 1 1 1
(ii) If 1 < r < p < ∞ 1<r<p<\infty 1 < r < p < ∞ 1 / r = 1 / p + 1 / s 1/r=1/p+1/s 1/ r = 1/ p + 1/ s
sup u ∈ L s + ( N ) s ( u ) = 1 ∥ sup α + u 1 / 2 f α u 1 / 2 ∥ r ∥ u ∥ s ≤ ∥ sup α + f α ∥ p ≤ 4 sup u ∈ L s + ( N ) s ( u ) = 1 ∥ sup α + u 1 / 2 f α u 1 / 2 ∥ r ∥ u ∥ s . (28) \sup_{\substack{u\in L_s^+(\mathcal{N})\\s(u)=1}}
\frac{\|\sup_\alpha^+u^{1/2}f_\alpha u^{1/2}\|_r}{\|u\|_s}
\leq\|\sup_\alpha^+f_\alpha\|_p
\leq4
\sup_{\substack{u\in L_s^+(\mathcal{N})\\s(u)=1}}
\frac{\|\sup_\alpha^+u^{1/2}f_\alpha u^{1/2}\|_r}{\|u\|_s}. \tag*{(28)} u ∈ L s + ( N ) s ( u ) = 1 sup ∥ u ∥ s ∥ sup α + u 1/2 f α u 1/2 ∥ r ≤ ∥ α sup + f α ∥ p ≤ 4 u ∈ L s + ( N ) s ( u ) = 1 sup ∥ u ∥ s ∥ sup α + u 1/2 f α u 1/2 ∥ r . (28) Proof. The only change from the proof of Lemma 2.7 is that in (i) a small multiple of 1 1 1 N \mathcal{N} N δ e − ∣ t ∣ 1 d \delta e^{-|t|}1_d δ e − ∣ t ∣ 1 d
Lemma 2.11. Let 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ 0 < u ≤ v 0<u\leq v 0 < u ≤ v positive invertible strongly measurable functions, and let ( f α ) α ∈ A (f_\alpha)_{\alpha\in\mathcal{A}} ( f α ) α ∈ A ( v 1 / 2 f α v 1 / 2 ) α ∈ A ∈ L p ( N ; ℓ ∞ ( A ) ) (v^{1/2}f_\alpha v^{1/2})_{\alpha\in\mathcal{A}}\in L_p(\mathcal{N};\ell_\infty(\mathcal{A})) ( v 1/2 f α v 1/2 ) α ∈ A ∈ L p ( N ; ℓ ∞ ( A ))
∥ sup α + u 1 / 2 f α u 1 / 2 ∥ p ≤ ∥ sup α + v 1 / 2 f α v 1 / 2 ∥ p . \|\sup_\alpha^+u^{1/2}f_\alpha u^{1/2}\|_p
\leq
\|\sup_\alpha^+v^{1/2}f_\alpha v^{1/2}\|_p. ∥ α sup + u 1/2 f α u 1/2 ∥ p ≤ ∥ α sup + v 1/2 f α v 1/2 ∥ p . Moreover, v − 1 ≤ u − 1 v^{-1}\leq u^{-1} v − 1 ≤ u − 1 ( u − 1 / 2 f α u − 1 / 2 ) α ∈ A ∈ L p ( N ; ℓ ∞ ( A ) ) (u^{-1/2}f_\alpha u^{-1/2})_{\alpha\in\mathcal{A}}\in L_p(\mathcal{N};\ell_\infty(\mathcal{A})) ( u − 1/2 f α u − 1/2 ) α ∈ A ∈ L p ( N ; ℓ ∞ ( A ))
∥ sup α + v − 1 / 2 f α v − 1 / 2 ∥ p ≤ ∥ sup α + u − 1 / 2 f α u − 1 / 2 ∥ p . \|\sup_\alpha^+v^{-1/2}f_\alpha v^{-1/2}\|_p
\leq
\|\sup_\alpha^+u^{-1/2}f_\alpha u^{-1/2}\|_p. ∥ α sup + v − 1/2 f α v − 1/2 ∥ p ≤ ∥ α sup + u − 1/2 f α u − 1/2 ∥ p . Proof. Compared with the proof of Lemma 2.8 , the only difference is that 0 < u ( t ) ≤ v ( t ) 0<u(t)\leq v(t) 0 < u ( t ) ≤ v ( t ) φ ( t ) = u ( t ) 1 / 2 v ( t ) − 1 / 2 \varphi(t)=u(t)^{1/2}v(t)^{-1/2} φ ( t ) = u ( t ) 1/2 v ( t ) − 1/2 u ( t ) 1 / 2 = φ ( t ) v ( t ) 1 / 2 u(t)^{1/2}=\varphi(t)v(t)^{1/2} u ( t ) 1/2 = φ ( t ) v ( t ) 1/2 φ \varphi φ
To summarize, the unweighted scalar and vector-valued results of this section hold for arbitrary semifinite traces. The affiliated-operator weighted arguments in Subsection 2.2 are presently restricted to trace-finite von Neumann algebras only. But the pointwise weighted realization and Lemma 2.10 apply directly to the concrete algebra N \mathcal{N} N
A new matrix weight class and its maximal characterization
The aim of this section is to present a new type of matrix weights, and their characterizations via noncommutative weighted Hardy–Littlewood maximal inequalities. For this purpose, we will relate our new A p A_p A p [30 ] .
Volberg's matrix weights
Let us recall the definition of the matrix A p A_p A p w : R → M d ( C ) w:\mathbb{R}\to M_d(\mathbb{C}) w : R → M d ( C ) t ∈ R t\in\mathbb{R} t ∈ R 1 < p < ∞ 1<p<\infty 1 < p < ∞
ρ w , p , t ( u ) : = ∥ w 1 / p ( t ) u ∥ ℓ 2 d , u ∈ ℓ 2 d , \rho_{w,p,t}(u):=\left\|w^{1/p}(t)u\right\|_{\ell^d_2},\qquad u\in\ell^d_2, ρ w , p , t ( u ) := w 1/ p ( t ) u ℓ 2 d , u ∈ ℓ 2 d , and its dual norm by
ρ w , p , t ∗ ( u ) : = ( ρ w , p , t ) ∗ ( u ) = sup 0 ≠ v ∈ ℓ 2 d ∣ ⟨ u , v ⟩ ℓ 2 d ∣ ρ w , p , t ( v ) . \rho^*_{w,p,t}(u):=(\rho_{w,p,t})^*(u)=\sup_{0\ne v\in\ell^d_2}\frac{|\langle u,v\rangle_{\ell^d_2}|}{\rho_{w,p,t}(v)}. ρ w , p , t ∗ ( u ) := ( ρ w , p , t ) ∗ ( u ) = 0 = v ∈ ℓ 2 d sup ρ w , p , t ( v ) ∣ ⟨ u , v ⟩ ℓ 2 d ∣ . It is easy to check ρ w , p , t ∗ ( u ) = ∥ w − 1 / p ( t ) u ∥ ℓ 2 d \rho^*_{w,p,t}(u)=\left\|w^{-1/p}(t)u\right\|_{\ell^{d}_{2}} ρ w , p , t ∗ ( u ) = w − 1/ p ( t ) u ℓ 2 d
Let I \mathcal{I} I R \mathbb{R} R I ∈ I I\in\mathcal{I} I ∈ I
ρ w , p , I ( u ) : = ( 1 ∣ I ∣ ∫ I ρ w , p , t ( u ) p d t ) 1 / p , ρ w , p , I ∗ ( u ) : = ( 1 ∣ I ∣ ∫ I ρ w , p , t ∗ ( u ) p ′ d t ) 1 / p ′ . \rho_{w,p,I}(u):=\left(\frac{1}{|I|}\int_I\rho_{w,p,t}(u)^p\,dt\right)^{1/p},\qquad
\rho^*_{w,p,I}(u):=\left(\frac{1}{|I|}\int_I\rho^*_{w,p,t}(u)^{p'}\,dt\right)^{1/p'}. ρ w , p , I ( u ) := ( ∣ I ∣ 1 ∫ I ρ w , p , t ( u ) p d t ) 1/ p , ρ w , p , I ∗ ( u ) := ( ∣ I ∣ 1 ∫ I ρ w , p , t ∗ ( u ) p ′ d t ) 1/ p ′ . Say w ∈ A p ( R ; M d ( C ) ) w\in A_p(\mathbb{R};M_d(\mathbb{C})) w ∈ A p ( R ; M d ( C )) C > 0 C>0 C > 0
ρ w , p , I ∗ ≤ C 1 / p ( ρ w , p , I ) ∗ \rho^*_{w,p,I}\le C^{1/p}(\rho_{w,p,I})^* ρ w , p , I ∗ ≤ C 1/ p ( ρ w , p , I ) ∗ for all I ∈ I I\in\mathcal{I} I ∈ I C C C [ w ] A p ( R ; M d ( C ) ) [w]_{A_p(\mathbb{R};M_d(\mathbb{C}))} [ w ] A p ( R ; M d ( C )) [30 ] for more details.
We will use the following quantitative version of Goldberg's maximal inequality given by Isralowitz and Moen in [12 ] :
Lemma 3.1. Let w : R → M d ( C ) w:\mathbb{R}\to M_d(\mathbb{C}) w : R → M d ( C ) M w M_w M w (1) . Then we have the following estimate:
∥ M w ∥ L p ( R ; ℓ 2 d ) → L p ( R ) ≤ C [ w ] A p ( R ; M d ( C ) ) p ′ / p . (29) \left\|M_w\right\|_{L^p(\mathbb{R};\ell^d_2)\to L^p(\mathbb{R})}\le C[w]_{A_p(\mathbb{R};M_d(\mathbb{C}))}^{p'/p}. \tag*{(29)} ∥ M w ∥ L p ( R ; ℓ 2 d ) → L p ( R ) ≤ C [ w ] A p ( R ; M d ( C )) p ′ / p . (29) Moreover, it is sharp with respect to the exponent p ′ / p p'/p p ′ / p A p A_p A p G p , d \mathcal{G}_{p,d} G p , d C C C (29) holds for every w ∈ A p ( R ; M d ( C ) ) w\in A_p(\mathbb{R};M_d(\mathbb{C})) w ∈ A p ( R ; M d ( C ))
Remark 3.2. Following the proof given in [12 ] , one may check that, for instance, G p , d ≲ p d 7 / 2 \mathcal{G}_{p,d}\lesssim_p d^{7/2} G p , d ≲ p d 7/2 7 / 2 7/2 7/2
The new matrix A p A_p A p N \mathcal{N} N
We give our new matrix A p A_p A p w : R → M d ( C ) w:\mathbb{R}\to M_d(\mathbb{C}) w : R → M d ( C ) t ∈ R t\in\mathbb{R} t ∈ R 1 < p < ∞ 1<p<\infty 1 < p < ∞ M d ( C ) M_d(\mathbb{C}) M d ( C )
ρ w , p , t ( A ) = ∥ w ( t ) 1 / 2 p A w ( t ) 1 / 2 p ∥ S p d , A ∈ M d ( C ) , (30) \rho_{w,p,t}(A)=\left\|w(t)^{1/2p}Aw(t)^{1/2p}\right\|_{S^d_p},\qquad A\in M_d(\mathbb{C}), \tag*{(30)} ρ w , p , t ( A ) = w ( t ) 1/2 p A w ( t ) 1/2 p S p d , A ∈ M d ( C ) , (30) and define its dual by
ρ w , p , t ∗ ( A ) : = ( ρ w , p , t ) ∗ ( A ) : = sup 0 ≠ B ∈ M d ( C ) ∣ tr ( B ∗ A ) ∣ ρ w , p , t ( B ) . \rho^*_{w,p,t}(A):=(\rho_{w,p,t})^*(A):=\sup_{0\ne B\in M_d(\mathbb{C})}\frac{|\operatorname{tr}(B^*A)|}{\rho_{w,p,t}(B)}. ρ w , p , t ∗ ( A ) := ( ρ w , p , t ) ∗ ( A ) := 0 = B ∈ M d ( C ) sup ρ w , p , t ( B ) ∣ tr ( B ∗ A ) ∣ . Hölder's inequality and the positivity of w w w
ρ w , p , t ( A ) p ≤ ∥ A ∥ S ∞ d p tr ( w ( t ) ) , \rho_{w,p,t}(A)^p\le\|A\|_{S^d_\infty}^p\operatorname{tr}(w(t)), ρ w , p , t ( A ) p ≤ ∥ A ∥ S ∞ d p tr ( w ( t )) , so ρ w , p , t ( A ) p \rho_{w,p,t}(A)^p ρ w , p , t ( A ) p
Lemma 3.3. For almost every t ∈ R t \in\mathbb{R} t ∈ R A ∈ M d ( C ) A \in M_d(\mathbb{C}) A ∈ M d ( C )
ρ w , p , t ∗ ( A ) = ∥ w ( t ) − 1 / 2 p A w ( t ) − 1 / 2 p ∥ S p ′ d (31) \rho^{*}_{w,p,t}(A)=\left\|w(t)^{-1/2p}Aw(t)^{-1/2p}\right\|_{S^{d}_{p'}} \tag*{(31)} ρ w , p , t ∗ ( A ) = w ( t ) − 1/2 p A w ( t ) − 1/2 p S p ′ d (31) Proof. Fix t t t w ( t ) > 0 w(t)>0 w ( t ) > 0 C = w ( t ) 1 / 2 p B w ( t ) 1 / 2 p C=w(t)^{1/2p}Bw(t)^{1/2p} C = w ( t ) 1/2 p B w ( t ) 1/2 p M d ( C ) M_d(\mathbb{C}) M d ( C )
tr ( B ∗ A ) = tr ( C ∗ w ( t ) − 1 / 2 p A w ( t ) − 1 / 2 p ) . \operatorname{tr}(B^{*}A)=\operatorname{tr}\left(C^{*}w(t)^{-1/2p}Aw(t)^{-1/2p}\right). tr ( B ∗ A ) = tr ( C ∗ w ( t ) − 1/2 p A w ( t ) − 1/2 p ) . Taking the supremum over C C C S p d S^{d}_{p} S p d S p ′ d S^{d}_{p'} S p ′ d
For I ∈ I I \in\mathcal{I} I ∈ I
ρ w , p , I ( A ) = ( 1 ∣ I ∣ ∫ I ρ w , p , t ( A ) p d t ) 1 / p , ρ w , p , I ∗ ( A ) = ( 1 ∣ I ∣ ∫ I ρ w , p , t ∗ ( A ) p ′ d t ) 1 / p ′ . (32) \rho_{w,p,I}(A)=\left(\frac{1}{|I|}\int_I\rho_{w,p,t}(A)^p\,dt\right)^{1/p},\qquad
\rho^{*}_{w,p,I}(A)=\left(\frac{1}{|I|}\int_I\rho^{*}_{w,p,t}(A)^{p'}\,dt\right)^{1/p'}. \tag*{(32)} ρ w , p , I ( A ) = ( ∣ I ∣ 1 ∫ I ρ w , p , t ( A ) p d t ) 1/ p , ρ w , p , I ∗ ( A ) = ( ∣ I ∣ 1 ∫ I ρ w , p , t ∗ ( A ) p ′ d t ) 1/ p ′ . (32) The second quantity is allowed to be infinite.
Lemma 3.4. For every I ∈ I I \in\mathcal{I} I ∈ I A ∈ M d ( C ) A \in M_d(\mathbb{C}) A ∈ M d ( C )
( ρ w , p , I ) ∗ ( A ) ≤ ρ w , p , I ∗ ( A ) . (33) (\rho_{w,p,I})^{*}(A)\leq\rho^{*}_{w,p,I}(A). \tag*{(33)} ( ρ w , p , I ) ∗ ( A ) ≤ ρ w , p , I ∗ ( A ) . (33) Proof. Pointwise duality gives ∣ tr ( B ∗ A ) ∣ ≤ ρ w , p , t ( B ) ρ w , p , t ∗ ( A ) |\operatorname{tr}(B^{*}A)|\leq\rho_{w,p,t}(B)\rho^{*}_{w,p,t}(A) ∣ tr ( B ∗ A ) ∣ ≤ ρ w , p , t ( B ) ρ w , p , t ∗ ( A ) ρ w , p , I ∗ ( A ) = ∞ \rho^{*}_{w,p,I}(A)=\infty ρ w , p , I ∗ ( A ) = ∞
∣ tr ( B ∗ A ) ∣ ≤ 1 ∣ I ∣ ∫ I ρ w , p , t ( B ) ρ w , p , t ∗ ( A ) d t ≤ ρ w , p , I ( B ) ρ w , p , I ∗ ( A ) . |\operatorname{tr}(B^{*}A)|\leq\frac{1}{|I|}\int_I\rho_{w,p,t}(B)\rho^{*}_{w,p,t}(A)\,dt\leq\rho_{w,p,I}(B)\rho^{*}_{w,p,I}(A). ∣ tr ( B ∗ A ) ∣ ≤ ∣ I ∣ 1 ∫ I ρ w , p , t ( B ) ρ w , p , t ∗ ( A ) d t ≤ ρ w , p , I ( B ) ρ w , p , I ∗ ( A ) . Taking the supremum over B ≠ 0 B\neq0 B = 0 (33) . □
Definition 3.5. We say that w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) C ≥ 1 C\geq1 C ≥ 1
ρ w , p , I ∗ ( A ) ≤ C 1 / p ( ρ w , p , I ) ∗ ( A ) (34) \rho^{*}_{w,p,I}(A)\leq C^{1/p}(\rho_{w,p,I})^{*}(A) \tag*{(34)} ρ w , p , I ∗ ( A ) ≤ C 1/ p ( ρ w , p , I ) ∗ ( A ) (34) for all I ∈ I I\in\mathcal{I} I ∈ I A ∈ M d ( C ) A\in M_d(\mathbb{C}) A ∈ M d ( C ) C C C (34) holds is denoted by [ w ] A p ( N ) [w]_{A_p(\mathcal{N})} [ w ] A p ( N )
Given w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) A ∈ M d ( C ) A\in M_d(\mathbb{C}) A ∈ M d ( C ) ρ w , p , t ∗ ( A ) p ′ \rho^{*}_{w,p,t}(A)^{p'} ρ w , p , t ∗ ( A ) p ′ w − p ′ / p w^{-p'/p} w − p ′ / p
Proposition 3.6. Given 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) w − p ′ / p ∈ A p ′ ( N ) w^{-p'/p}\in A_{p'}(\mathcal{N}) w − p ′ / p ∈ A p ′ ( N )
[ w − p ′ / p ] A p ′ ( N ) = [ w ] A p ( N ) . (35) [w^{-p'/p}]_{A_{p'}(\mathcal{N})}=[w]_{A_p(\mathcal{N})}. \tag*{(35)} [ w − p ′ / p ] A p ′ ( N ) = [ w ] A p ( N ) . (35) Proof. By Lemma 3.3, the pointwise norm associated with w − p ′ / p w^{-p'/p} w − p ′ / p p ′ p' p ′ ρ w , p , t ∗ \rho^{*}_{w,p,t} ρ w , p , t ∗ ρ w , p , t \rho_{w,p,t} ρ w , p , t
ρ w , p , I ∗ ≤ [ w ] A p ( N ) 1 / p ( ρ w , p , I ) ∗ . \rho^{*}_{w,p,I}\leq[w]_{A_p(\mathcal{N})}^{1/p}(\rho_{w,p,I})^{*}. ρ w , p , I ∗ ≤ [ w ] A p ( N ) 1/ p ( ρ w , p , I ) ∗ . Dualizing this inequality between finite-dimensional normed spaces gives
ρ w , p , I ≤ [ w ] A p ( N ) 1 / p ( ρ w , p , I ∗ ) ∗ , \rho_{w,p,I}\leq[w]_{A_p(\mathcal{N})}^{1/p}(\rho^{*}_{w,p,I})^{*}, ρ w , p , I ≤ [ w ] A p ( N ) 1/ p ( ρ w , p , I ∗ ) ∗ , which is the A p ′ A_{p'} A p ′ w − p ′ / p w^{-p'/p} w − p ′ / p w − p ′ / p w^{-p'/p} w − p ′ / p [ w ] A p ( N ) = [ w − p ′ / p ] A p ′ ( N ) 1 / p ′ [w]_{A_p(\mathcal{N})}=[w^{-p'/p}]_{A_{p'}(\mathcal{N})}^{1/p'} [ w ] A p ( N ) = [ w − p ′ / p ] A p ′ ( N ) 1/ p ′ (35) . □
In this section, the key idea is that one may relate w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) W ∈ A p ( R ; M d 2 ( C ) ) \mathbb{W}\in A_p(\mathbb{R};M_{d^2}(\mathbb{C})) W ∈ A p ( R ; M d 2 ( C ))
W ( t ) : A ⟼ w 1 / 2 ( t ) A w 1 / 2 ( t ) , A ∈ S 2 d ( C ) . (36) \mathbb{W}(t):A\longmapsto w^{1/2}(t)Aw^{1/2}(t),\qquad A\in S_2^d(\mathbb{C}). \tag*{(36)} W ( t ) : A ⟼ w 1/2 ( t ) A w 1/2 ( t ) , A ∈ S 2 d ( C ) . (36) Then by functional calculus, we have
W ± 1 / p ( t ) ( A ) = w ± 1 / 2 p ( t ) A w ± 1 / 2 p ( t ) , A ∈ S 2 d ( C ) . (37) \mathbb{W}^{\pm1/p}(t)(A)=w^{\pm1/2p}(t)Aw^{\pm1/2p}(t),\qquad A\in S_2^d(\mathbb{C}). \tag*{(37)} W ± 1/ p ( t ) ( A ) = w ± 1/2 p ( t ) A w ± 1/2 p ( t ) , A ∈ S 2 d ( C ) . (37) Based on this observation, we can compare our A p ( N ) A_p(\mathcal{N}) A p ( N ) A p ( R ; M d 2 ( C ) ) A_p(\mathbb{R};M_{d^2}(\mathbb{C})) A p ( R ; M d 2 ( C ))
Lemma 3.7. Given a matrix weight w : R → M d ( C ) w:\mathbb{R}\to M_d(\mathbb{C}) w : R → M d ( C ) W \mathsf{W} W (36) . Then we have
d − ∣ 1 − p 2 ∣ [ W ] A p ( R ; M d 2 ( C ) ) ≤ [ w ] A p ( N ) ≤ d ∣ 1 − p 2 ∣ [ W ] A p ( R ; M d 2 ( C ) ) . (38) d^{-\left|1-\frac{p}{2}\right|}[\mathsf{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))}\leq[w]_{A_p(\mathcal{N})}\leq d^{\left|1-\frac{p}{2}\right|}[\mathsf{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))}. \tag*{(38)} d − ∣ 1 − 2 p ∣ [ W ] A p ( R ; M d 2 ( C )) ≤ [ w ] A p ( N ) ≤ d ∣ 1 − 2 p ∣ [ W ] A p ( R ; M d 2 ( C )) . (38) Proof. Define
κ p ( d ) = d max { 0 , 1 p − 1 2 } , η p ( d ) = d min { 0 , 1 p − 1 2 } . \kappa_p(d)=d^{\max\left\{0,\frac{1}{p}-\frac{1}{2}\right\}},\qquad\eta_p(d)=d^{\min\left\{0,\frac{1}{p}-\frac{1}{2}\right\}}. κ p ( d ) = d m a x { 0 , p 1 − 2 1 } , η p ( d ) = d m i n { 0 , p 1 − 2 1 } . On the one hand, we have
ρ w , p , t ( A ) = ∥ w ( t ) 1 / 2 p A w ( t ) 1 / 2 p ∥ S p d ≤ κ p ( d ) ∥ W 1 / p ( t ) A ∥ ℓ 2 d = κ p ( d ) ρ W , p , t ( A ) , \rho_{w,p,t}(A)=\left\|w(t)^{1/2p}Aw(t)^{1/2p}\right\|_{\mathcal{S}_p^d}\leq\kappa_p(d)\left\|\mathsf{W}^{1/p}(t)A\right\|_{\ell_2^d}=\kappa_p(d)\rho_{\mathsf{W},p,t}(A), ρ w , p , t ( A ) = w ( t ) 1/2 p A w ( t ) 1/2 p S p d ≤ κ p ( d ) W 1/ p ( t ) A ℓ 2 d = κ p ( d ) ρ W , p , t ( A ) , ρ w , p , t ( A ) ≥ η p ( d ) ∥ W 1 / p ( t ) A ∥ ℓ 2 d = η p ( d ) ρ W , p , t ( A ) , \rho_{w,p,t}(A)\geq\eta_p(d)\left\|\mathsf{W}^{1/p}(t)A\right\|_{\ell_2^d}=\eta_p(d)\rho_{\mathsf{W},p,t}(A), ρ w , p , t ( A ) ≥ η p ( d ) W 1/ p ( t ) A ℓ 2 d = η p ( d ) ρ W , p , t ( A ) , thus for any I ∈ I I\in\mathcal{I} I ∈ I
η p ( d ) ρ W , p , I ( A ) ≤ ρ w , p , I ( A ) = ( 1 ∣ I ∣ ∫ I ρ w , p , t ( A ) p d t ) 1 / p ≤ κ p ( d ) ρ W , p , I ( A ) . \eta_p(d)\rho_{\mathsf{W},p,I}(A)\leq\rho_{w,p,I}(A)=\left(\frac{1}{|I|}\int_I\rho_{w,p,t}(A)^p\,dt\right)^{1/p}\leq\kappa_p(d)\rho_{\mathsf{W},p,I}(A). η p ( d ) ρ W , p , I ( A ) ≤ ρ w , p , I ( A ) = ( ∣ I ∣ 1 ∫ I ρ w , p , t ( A ) p d t ) 1/ p ≤ κ p ( d ) ρ W , p , I ( A ) . Taking dual norms gives
κ p − 1 ( d ) ( ρ w , p , I ) ∗ ( A ) ≤ ( ρ w , p , I ) ∗ ( A ) ≤ η p − 1 ( d ) ( ρ W , p , I ) ∗ ( A ) . \kappa_p^{-1}(d)(\rho_{w,p,I})^*(A)\leq(\rho_{w,p,I})^*(A)\leq\eta_p^{-1}(d)(\rho_{\mathsf{W},p,I})^*(A). κ p − 1 ( d ) ( ρ w , p , I ) ∗ ( A ) ≤ ( ρ w , p , I ) ∗ ( A ) ≤ η p − 1 ( d ) ( ρ W , p , I ) ∗ ( A ) . On the other hand, η p ( d ) ρ W , p , t ( A ) ≤ ρ w , p , t ( A ) ≤ κ p ( d ) ρ W , p , t ( A ) \eta_p(d)\rho_{\mathsf{W},p,t}(A)\leq\rho_{w,p,t}(A)\leq\kappa_p(d)\rho_{\mathsf{W},p,t}(A) η p ( d ) ρ W , p , t ( A ) ≤ ρ w , p , t ( A ) ≤ κ p ( d ) ρ W , p , t ( A )
κ p − 1 ( d ) ρ W , p , t ∗ ( A ) ≤ ρ w , p , t ∗ ( A ) ≤ η p − 1 ( d ) ρ W , p , t ∗ ( A ) , \kappa_p^{-1}(d)\rho_{\mathsf{W},p,t}^*(A)\leq\rho_{w,p,t}^*(A)\leq\eta_p^{-1}(d)\rho_{\mathsf{W},p,t}^*(A), κ p − 1 ( d ) ρ W , p , t ∗ ( A ) ≤ ρ w , p , t ∗ ( A ) ≤ η p − 1 ( d ) ρ W , p , t ∗ ( A ) , thus for any I ∈ I I\in\mathcal{I} I ∈ I
κ p − 1 ( d ) ρ W , p , I ∗ ( A ) ≤ ρ w , p , I ∗ ( A ) ≤ η p − 1 ( d ) ρ W , p , I ∗ ( A ) . \kappa_p^{-1}(d)\rho_{\mathsf{W},p,I}^*(A)\leq\rho_{w,p,I}^*(A)\leq\eta_p^{-1}(d)\rho_{\mathsf{W},p,I}^*(A). κ p − 1 ( d ) ρ W , p , I ∗ ( A ) ≤ ρ w , p , I ∗ ( A ) ≤ η p − 1 ( d ) ρ W , p , I ∗ ( A ) . Combining the argument above, we have
[ w ] A p ( N ) 1 / p = sup I ∈ I sup A ≠ 0 ρ w , p , I ∗ ( A ) ( ρ w , p , I ) ∗ ( A ) ≤ η p − 1 ( d ) κ p ( d ) [ W ] A p ( R ; M d 2 ( C ) ) 1 / p = d ∣ 1 p − 1 2 ∣ [ W ] A p ( R ; M d 2 ( C ) ) 1 / p . [w]_{A_p(\mathcal{N})}^{1/p}=\sup_{I\in\mathcal{I}}\sup_{A\ne0}\frac{\rho_{w,p,I}^*(A)}{(\rho_{w,p,I})^*(A)}\leq\eta_p^{-1}(d)\kappa_p(d)[\mathsf{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))}^{1/p}=d^{\left|\frac{1}{p}-\frac{1}{2}\right|}[\mathsf{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))}^{1/p}. [ w ] A p ( N ) 1/ p = I ∈ I sup A = 0 sup ( ρ w , p , I ) ∗ ( A ) ρ w , p , I ∗ ( A ) ≤ η p − 1 ( d ) κ p ( d ) [ W ] A p ( R ; M d 2 ( C )) 1/ p = d ∣ p 1 − 2 1 ∣ [ W ] A p ( R ; M d 2 ( C )) 1/ p . Similarly, we have [ w ] A p ( N ) ≥ d − ∣ 1 − p 2 ∣ [ W ] A p ( R ; M d 2 ( C ) ) [w]_{A_p(\mathcal{N})}\geq d^{-\left|1-\frac{p}{2}\right|}[\mathsf{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))} [ w ] A p ( N ) ≥ d − ∣ 1 − 2 p ∣ [ W ] A p ( R ; M d 2 ( C ))
Averaging operators
Given I ∈ I I\in\mathcal{I} I ∈ I
A I : N ⟶ N A_I:\mathcal{N}\longrightarrow\mathcal{N} A I : N ⟶ N f ⟼ ⟨ f ⟩ I ⋅ χ I (39) f\longmapsto\langle f\rangle_I\cdot\chi_I \tag*{(39)} f ⟼ ⟨ f ⟩ I ⋅ χ I (39) where ⟨ f ⟩ I : = 1 ∣ I ∣ ∫ I f ( t ) d t \langle f\rangle_I:=\frac{1}{|I|}\int_I f(t)\,dt ⟨ f ⟩ I := ∣ I ∣ 1 ∫ I f ( t ) d t f f f I I I
Lemma 3.8. Let w w w I ∈ I I\in\mathcal{I} I ∈ I A I A_I A I
∥ A I ∥ L p ( N , w ) → L p ( N , w ) = sup A ≠ 0 ρ w , p , I ∗ ( A ) ( ρ w , p , I ) ∗ ( A ) . (40) \|A_I\|_{L_p(\mathcal{N},w)\to L_p(\mathcal{N},w)}=\sup_{A\ne0}\frac{\rho_{w,p,I}^*(A)}{(\rho_{w,p,I})^*(A)}. \tag*{(40)} ∥ A I ∥ L p ( N , w ) → L p ( N , w ) = A = 0 sup ( ρ w , p , I ) ∗ ( A ) ρ w , p , I ∗ ( A ) . (40) Consequently,
sup I ∈ I ∥ A I ∥ L p ( N , w ) → L p ( N , w ) = [ w ] A p ( N ) 1 / p . (41) \sup_{I\in\mathcal{I}}\|A_I\|_{L_p(\mathcal{N},w)\to L_p(\mathcal{N},w)}=[w]_{A_p(\mathcal{N})}^{1/p}. \tag*{(41)} I ∈ I sup ∥ A I ∥ L p ( N , w ) → L p ( N , w ) = [ w ] A p ( N ) 1/ p . (41) If w ∉ A p ( N ) w\notin A_p(\mathcal{N}) w ∈ / A p ( N ) [ w ] A p ( N ) [w]_{A_p(\mathcal{N})} [ w ] A p ( N )
Proof. We first consider an operator
P I : L p ( N ) ⟶ ( M d ( C ) , ρ w , p , I ) P_I:L_p(\mathcal{N})\longrightarrow(M_d(\mathbb{C}),\rho_{w,p,I}) P I : L p ( N ) ⟶ ( M d ( C ) , ρ w , p , I ) f ⟼ ⟨ w − 1 / 2 p f w − 1 / 2 p ⟩ I f\longmapsto\left\langle w^{-1/2p}fw^{-1/2p}\right\rangle_I f ⟼ ⟨ w − 1/2 p f w − 1/2 p ⟩ I then
ρ w , p , I ( P I f ) = ( 1 ∣ I ∣ ∫ I ∥ w 1 / 2 p ( t ) ⟨ w − 1 / 2 p f w − 1 / 2 p ⟩ I w 1 / 2 p ( t ) ∥ S d p p d t ) 1 / p = ∣ I ∣ − 1 / p ∥ A I ( w − 1 / 2 p f w − 1 / 2 p ) ∥ L p ( N , w ) . \begin{aligned}
\rho_{w,p,I}(P_I f)=\left(\frac{1}{|I|}\int_I\left\|w^{1/2p}(t)\langle w^{-1/2p}fw^{-1/2p}\rangle_Iw^{1/2p}(t)\right\|_{S_d^p}^{p}\,dt\right)^{1/p} \\
&=|I|^{-1/p}\left\|A_I\left(w^{-1/2p}fw^{-1/2p}\right)\right\|_{L_p(\mathcal{N},w)}.
\end{aligned} ρ w , p , I ( P I f ) = ( ∣ I ∣ 1 ∫ I w 1/2 p ( t ) ⟨ w − 1/2 p f w − 1/2 p ⟩ I w 1/2 p ( t ) S d p p d t ) 1/ p = ∣ I ∣ − 1/ p A I ( w − 1/2 p f w − 1/2 p ) L p ( N , w ) . Thus
∥ P I ∥ L p ( N ) → ( M d ( C ) , ρ w , p , I ) = sup 0 ≠ f ∈ L p ( N ) ρ w , p , I ( P I f ) ∥ f ∥ L p ( N ) = ∣ I ∣ − 1 / p ⋅ sup 0 ≠ g ∈ L p ( N , w ) ∥ A I g ∥ L p ( N , w ) ∥ g ∥ L p ( N , w ) . \begin{aligned}
\left\|P_I\right\|_{L_p(\mathcal{N})\to(M_d(\mathbb{C}),\rho_{w,p,I})}
=\sup_{0\ne f\in L_p(\mathcal{N})}\frac{\rho_{w,p,I}(P_I f)}{\|f\|_{L_p(\mathcal{N})}} \\
&=|I|^{-1/p}\cdot\sup_{0\ne g\in L_p(\mathcal{N},w)}\frac{\|A_Ig\|_{L_p(\mathcal{N},w)}}{\|g\|_{L_p(\mathcal{N},w)}}.
\end{aligned} ∥ P I ∥ L p ( N ) → ( M d ( C ) , ρ w , p , I ) = 0 = f ∈ L p ( N ) sup ∥ f ∥ L p ( N ) ρ w , p , I ( P I f ) = ∣ I ∣ − 1/ p ⋅ 0 = g ∈ L p ( N , w ) sup ∥ g ∥ L p ( N , w ) ∥ A I g ∥ L p ( N , w ) . Thus we only need to calculate the operator norm of P I P_I P I P I P_I P I
P I ∗ : ( M d ( C ) , ( ρ w , p , I ) ∗ ) ⟶ L p ′ ( N ) P_I^*:(M_d(\mathbb{C}),(\rho_{w,p,I})^*)\longrightarrow L_{p'}(\mathcal{N}) P I ∗ : ( M d ( C ) , ( ρ w , p , I ) ∗ ) ⟶ L p ′ ( N ) A ⟼ w − 1 / 2 p A w − 1 / 2 p ⋅ χ I ∣ I ∣ . A\longmapsto w^{-1/2p}Aw^{-1/2p}\cdot\frac{\chi_I}{|I|}. A ⟼ w − 1/2 p A w − 1/2 p ⋅ ∣ I ∣ χ I . Then
∥ A I ∥ L p ( N , w ) → L p ( N , w ) = ∣ I ∣ 1 / p ⋅ sup A ≠ 0 ∥ P I ∗ A ∥ L p ′ ( N ) ( ρ w , p , I ) ∗ ( A ) = sup A ≠ 0 ρ w , p , I ∗ ( A ) ( ρ w , p , I ) ∗ ( A ) . \|A_I\|_{L_p(\mathcal{N},w)\to L_p(\mathcal{N},w)}
=|I|^{1/p}\cdot\sup_{A\ne0}\frac{\|P_I^*A\|_{L_{p'}(\mathcal{N})}}{(\rho_{w,p,I})^*(A)}
=\sup_{A\ne0}\frac{\rho_{w,p,I}^*(A)}{(\rho_{w,p,I})^*(A)}. ∥ A I ∥ L p ( N , w ) → L p ( N , w ) = ∣ I ∣ 1/ p ⋅ A = 0 sup ( ρ w , p , I ) ∗ ( A ) ∥ P I ∗ A ∥ L p ′ ( N ) = A = 0 sup ( ρ w , p , I ) ∗ ( A ) ρ w , p , I ∗ ( A ) . Taking the supremum over I I I (41) . □
The noncommutative weighted Hardy-Littlewood maximal inequality
Our proof of the only if part of Theorem 1.1 needs a comparison between the noncommutative maximal norm and the vector-valued maximal norm.
Lemma 3.9. Let F \mathcal{F} F ( h α ) α ∈ F ⊂ L p ( N ) (h_\alpha)_{\alpha\in\mathcal{F}}\subset L_p(\mathcal{N}) ( h α ) α ∈ F ⊂ L p ( N )
∥ max α ∈ F ∥ h α ( ⋅ ) ∥ S d p ∥ L p ( R ) ≤ ∥ sup α ∈ F + h α ∥ L p ( N ) ≤ d 1 / p ∥ max α ∈ F ∥ h α ( ⋅ ) ∥ S d p ∥ L p ( R ) . (42) \left\|\max_{\alpha\in\mathcal{F}}\|h_\alpha(\cdot)\|_{S_d^p}\right\|_{L_p(\mathbb{R})}
\le\left\|\sup_{\alpha\in\mathcal{F}}{}^+h_\alpha\right\|_{L_p(\mathcal{N})}
\le d^{1/p}\left\|\max_{\alpha\in\mathcal{F}}\|h_\alpha(\cdot)\|_{S_d^p}\right\|_{L_p(\mathbb{R})}. \tag*{(42)} α ∈ F max ∥ h α ( ⋅ ) ∥ S d p L p ( R ) ≤ α ∈ F sup + h α L p ( N ) ≤ d 1/ p α ∈ F max ∥ h α ( ⋅ ) ∥ S d p L p ( R ) . (42) Proof. Set h ~ ( t ) = max α ∥ h α ( t ) ∥ S d p \widetilde h(t)=\max_\alpha\|h_\alpha(t)\|_{S_d^p} h ( t ) = max α ∥ h α ( t ) ∥ S d p h α = a y α b h_\alpha=ay_\alpha b h α = a y α b
∥ h ~ ∥ L p ( R ) ≤ ∥ a ∥ 2 p sup α ∥ y α ∥ ∞ ∥ b ∥ 2 p . \|\widetilde h\|_{L_p(\mathbb{R})}\le\|a\|_{2p}\sup_\alpha\|y_\alpha\|_\infty\|b\|_{2p}. ∥ h ∥ L p ( R ) ≤ ∥ a ∥ 2 p α sup ∥ y α ∥ ∞ ∥ b ∥ 2 p . Taking the infimum proves the first inequality. Conversely, on the set E : = { x ∈ R : h ~ ( x ) > 0 } E:=\{x\in\mathbb{R}:\widetilde h(x)>0\} E := { x ∈ R : h ( x ) > 0 }
h α = ( h ~ 1 / 2 ⋅ 1 d ) ( h ~ − 1 h α ) ( h ~ 1 / 2 ⋅ 1 d ) , ∥ h ~ − 1 h α ∥ S d ∞ ≤ 1 , h_\alpha=(\widetilde h^{1/2}\cdot1_d)(\widetilde h^{-1}h_\alpha)(\widetilde h^{1/2}\cdot1_d),\qquad\|\widetilde h^{-1}h_\alpha\|_{S_d^\infty}\le1, h α = ( h 1/2 ⋅ 1 d ) ( h − 1 h α ) ( h 1/2 ⋅ 1 d ) , ∥ h − 1 h α ∥ S d ∞ ≤ 1 , and set the middle factor equal to zero on E c E^c E c ∥ h ~ 1 / 2 ⋅ 1 d ∥ 2 p 2 = d 1 / p ∥ h ~ ∥ L p ( R ) \|\widetilde h^{1/2}\cdot1_d\|_{2p}^2=d^{1/p}\|\widetilde h\|_{L_p(\mathbb{R})} ∥ h 1/2 ⋅ 1 d ∥ 2 p 2 = d 1/ p ∥ h ∥ L p ( R )
Remark 3.10. The preceding lemma shows that the noncommutative maximal norm dominates the scalar L p L_p L p p p p d 1 / p d^{1/p} d 1/ p h j = 1 ( 0 , 1 ) e j j h_j=1_{(0,1)}e_{jj} h j = 1 ( 0 , 1 ) e j j 1 ≤ j ≤ d 1\le j\le d 1 ≤ j ≤ d ( e i j ) (e_{ij}) ( e ij )
∥ max 1 ≤ j ≤ d ∥ h j ( ⋅ ) ∥ S d p ∥ L p ( R ) = 1 , ∥ sup 1 ≤ j ≤ d + h j ∥ L p ( N ) = d 1 / p . \left\|\max_{1\le j\le d}\|h_j(\cdot)\|_{S_d^p}\right\|_{L_p(\mathbb{R})}=1,\qquad
\left\|\sup_{1\le j\le d}{}^+h_j\right\|_{L_p(\mathcal{N})}=d^{1/p}. 1 ≤ j ≤ d max ∥ h j ( ⋅ ) ∥ S d p L p ( R ) = 1 , 1 ≤ j ≤ d sup + h j L p ( N ) = d 1/ p . Thus the reverse estimate cannot hold with a dimension-independent constant. For 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ B ( ℓ 2 ) B(\ell_2) B ( ℓ 2 ) L p L_p L p 1 1 1
We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. The only if part. Let w ∈ A p ( N ) w \in A_p(\mathcal{N}) w ∈ A p ( N )
∥ sup I + A I f ∥ L p ( N , w ) ≤ d 1 p + ( 1 + p ′ ) ∣ 1 p − 1 2 ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ f ∥ L p ( N , w ) , f ∈ S N c . \left\|\sup_{I}^{+} A_I f\right\|_{L_p(\mathcal{N},w)} \le d^{\frac{1}{p}+(1+p')\left|\frac{1}{p}-\frac{1}{2}\right|} G_{p,d^2}[w]_{A_p(\mathcal{N})}^{p'/p}\|f\|_{L_p(\mathcal{N},w)}, \qquad f \in S_{\mathcal{N}}^{c}. I sup + A I f L p ( N , w ) ≤ d p 1 + ( 1 + p ′ ) ∣ p 1 − 2 1 ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ f ∥ L p ( N , w ) , f ∈ S N c . Given f ∈ S N c f \in S_{\mathcal{N}}^{c} f ∈ S N c F = w 1 / 2 p f w 1 / 2 p F=w^{1/2p}fw^{1/2p} F = w 1/2 p f w 1/2 p
M W F ( t ) = sup I ∋ t 1 ∣ I ∣ ∥ ∫ I W ( t ) 1 / p W ( s ) − 1 / p F ( s ) d s ∥ S p d , M_{\mathbb{W}}F(t)=\sup_{I\ni t}\frac{1}{|I|}\left\|\int_I\mathbb{W}(t)^{1/p}\mathbb{W}(s)^{-1/p}F(s)\,\mathrm{d}s\right\|_{S_p^d}, M W F ( t ) = I ∋ t sup ∣ I ∣ 1 ∫ I W ( t ) 1/ p W ( s ) − 1/ p F ( s ) d s S p d , where W \mathbb{W} W w w w (36) . Lemma 3.1 and Lemma 3.7 give
∥ M W F ∥ L p ( R ) ≤ G p , d 2 [ W ] A p ( R ; M d 2 ( C ) ) p ′ / p ∥ F ∥ L p ( R ; S p d ) ≤ d max { 0 , 1 p − 1 2 } + ∣ 1 2 − 1 p ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ F ∥ L p ( R ; S p d ) . (43) \begin{aligned}
\|M_{\mathbb{W}}F\|_{L_p(\mathbb{R})}\le G_{p,d^2}[\mathbb{W}]_{A_p(\mathbb{R};M_{d^2}(\mathbb{C}))}^{p'/p}\|F\|_{L_p(\mathbb{R};S_p^d)} \\
&\le d^{\max\left\{0,\frac{1}{p}-\frac{1}{2}\right\}+\left| \frac{1}{2}-\frac{1}{p}\right|}G_{p,d^2}[w]_{A_p(\mathcal{N})}^{p'/p}\|F\|_{L_p(\mathbb{R};S_p^d)}. \tag*{(43)}
\end{aligned} ∥ M W F ∥ L p ( R ) ≤ G p , d 2 [ W ] A p ( R ; M d 2 ( C )) p ′ / p ∥ F ∥ L p ( R ; S p d ) ≤ d m a x { 0 , p 1 − 2 1 } + ∣ 2 1 − p 1 ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ F ∥ L p ( R ; S p d ) . (43) By Minkowski’s inequality and (37) ,
sup I ∋ t ∥ w ( t ) 1 / 2 p A I f ( t ) w ( t ) 1 / 2 p ∥ S p d = sup I ∋ t 1 ∣ I ∣ ∥ ∫ I w ( t ) 1 / 2 p f ( s ) w ( t ) 1 / 2 p d s ∥ S p d ≤ d max { 0 , 1 p − 1 2 } sup I ∋ t 1 ∣ I ∣ ∫ I ∥ W 1 / p ( t ) W − 1 / p ( s ) F ( s ) ∥ S p d d s = d max { 0 , 1 p − 1 2 } M W F ( t ) . \begin{aligned}
\sup_{I\ni t}\left\|w(t)^{1/2p}A_I f(t)w(t)^{1/2p}\right\|_{S_p^d}
=\sup_{I\ni t}\frac{1}{|I|}\left\|\int_Iw(t)^{1/2p}f(s)w(t)^{1/2p}\,\mathrm{d}s\right\|_{S_p^d} \\
&\le d^{\max\left\{0,\frac{1}{p}-\frac{1}{2}\right\}}\sup_{I\ni t}\frac{1}{|I|}\int_I\left\|\mathbb{W}^{1/p}(t)\mathbb{W}^{-1/p}(s)F(s)\right\|_{S_p^d}\,\mathrm{d}s \\
&=d^{\max\left\{0,\frac{1}{p}-\frac{1}{2}\right\}}M_{\mathbb{W}}F(t).
\end{aligned} I ∋ t sup w ( t ) 1/2 p A I f ( t ) w ( t ) 1/2 p S p d = I ∋ t sup ∣ I ∣ 1 ∫ I w ( t ) 1/2 p f ( s ) w ( t ) 1/2 p d s S p d ≤ d m a x { 0 , p 1 − 2 1 } I ∋ t sup ∣ I ∣ 1 ∫ I W 1/ p ( t ) W − 1/ p ( s ) F ( s ) S p d d s = d m a x { 0 , p 1 − 2 1 } M W F ( t ) . Thus, for every finite F ⊂ I \mathcal{F}\subset\mathcal{I} F ⊂ I (43) , and equivalence of Schatten norms yield
∥ sup I ∈ F + A I f ∥ L p ( N , w ) ≤ d 1 / p ∥ max I ∈ F ∥ w 1 / 2 p ( A I f ) w 1 / 2 p ∥ S p d ∥ L p ( R ) ≤ d 1 p + ( 1 + p ′ ) ∣ 1 p − 1 2 ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ f ∥ L p ( N , w ) . \begin{aligned}
\left\|\sup_{I\in\mathcal{F}}^{+}A_I f\right\|_{L_p(\mathcal{N},w)}
\le d^{1/p}\left\|\max_{I\in\mathcal{F}}\left\|w^{1/2p}(A_I f)w^{1/2p}\right\|_{S_p^d}\right\|_{L_p(\mathbb{R})} \\
&\le d^{\frac{1}{p}+(1+p')\left|\frac{1}{p}-\frac{1}{2}\right|}G_{p,d^2}[w]_{A_p(\mathcal{N})}^{p'/p}\|f\|_{L_p(\mathcal{N},w)}.
\end{aligned} I ∈ F sup + A I f L p ( N , w ) ≤ d 1/ p I ∈ F max w 1/2 p ( A I f ) w 1/2 p S p d L p ( R ) ≤ d p 1 + ( 1 + p ′ ) ∣ p 1 − 2 1 ∣ G p , d 2 [ w ] A p ( N ) p ′ / p ∥ f ∥ L p ( N , w ) . The finite-subfamily formula (9) gives the full interval family, and density of S N c S_{\mathcal{N}}^{c} S N c
The if part. If the maximal operator has finite norm C p , d ( w ) C_{p,d}(w) C p , d ( w ) A I A_I A I C p , d ( w ) C_{p,d}(w) C p , d ( w )
[ w ] A p ( N ) 1 / p = sup I ∈ I ∥ A I ∥ L p ( N , w ) → L p ( N , w ) ≤ C p , d ( w ) , [w]_{A_p(\mathcal{N})}^{1/p}=\sup_{I\in\mathcal{I}}\|A_I\|_{L_p(\mathcal{N},w)\to L_p(\mathcal{N},w)}\le C_{p,d}(w), [ w ] A p ( N ) 1/ p = I ∈ I sup ∥ A I ∥ L p ( N , w ) → L p ( N , w ) ≤ C p , d ( w ) , which proves the lower bound and completes the proof. □
A noncommutative analogue of Rubio de Francia's abstract extrapolation theory
In this section, we establish a noncommutative version of Rubio de Francia’s abstract extrapolation theory. For technical reasons, we can only work on trace-finite von Neumann algebras at the moment of writing. Let ( M , τ ) (\mathcal{M},\tau) ( M , τ ) ( E β ) β ∈ B (E_\beta)_{\beta\in B} ( E β ) β ∈ B M \mathcal{M} M ( E β ) β ∈ B (E_\beta)_{\beta\in B} ( E β ) β ∈ B
The weight classes and factorization
Definition 4.1. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ W p ( ( E β ) β ) w\in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β ) w ∈ L + 0 ( M ) w\in L_+^0(\mathcal{M}) w ∈ L + 0 ( M )
x ⟼ ( E β ( x ) ) β ∈ B , x ∈ M ∩ L p ( M , w ) , x\longmapsto(E_\beta(x))_{\beta\in B},\qquad x\in\mathcal{M}\cap L_p(\mathcal{M},w), x ⟼ ( E β ( x ) ) β ∈ B , x ∈ M ∩ L p ( M , w ) , extends boundedly from L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) L p ( M , w ; ℓ ∞ ( B ) ) L_p(\mathcal{M},w;\ell_\infty(B)) L p ( M , w ; ℓ ∞ ( B )) [ w ] W p ( ( E β ) β ) [w]_{W_p((E_\beta)_\beta)} [ w ] W p (( E β ) β ) [ w ] W p [w]_{W_p} [ w ] W p
∥ sup β ∈ B + E β ( x ) ∥ L p ( M , w ) ≤ [ w ] W p ∥ x ∥ L p ( M , w ) , x ∈ L p ( M , w ) . (44) \left\|\sup_{\beta\in B}^{+}E_\beta(x)\right\|_{L_p(\mathcal{M},w)}
\le[w]_{W_p}\|x\|_{L_p(\mathcal{M},w)},\qquad x\in L_p(\mathcal{M},w). \tag*{(44)} β ∈ B sup + E β ( x ) L p ( M , w ) ≤ [ w ] W p ∥ x ∥ L p ( M , w ) , x ∈ L p ( M , w ) . (44) The extension is unique because M ∩ L p ( M , w ) ⊂ L p ( M , w ) \mathcal{M}\cap L_p(\mathcal{M},w)\subset L_p(\mathcal{M},w) M ∩ L p ( M , w ) ⊂ L p ( M , w )
Assumption 4.2. For every 1 < p < ∞ 1 < p < \infty 1 < p < ∞ w ∈ W p ( ( E β ) β ) w \in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β ) w − p ′ / p ∈ W p ′ ( ( E β ) β ) w^{-p'/p} \in W_{p'}((E_\beta)_\beta) w − p ′ / p ∈ W p ′ (( E β ) β ) γ p : [ 0 , ∞ ) → [ 0 , ∞ ) \gamma_p : [0,\infty) \to[0,\infty) γ p : [ 0 , ∞ ) → [ 0 , ∞ )
[ w − p ′ / p ] W p ′ ≤ γ p ( [ w ] W p ) . (45) [w^{-p'/p}]_{W_{p'}} \le\gamma_p([w]_{W_p}). \tag*{(45)} [ w − p ′ / p ] W p ′ ≤ γ p ([ w ] W p ) . (45) Applying the same assumption at p ′ p' p ′ w − p ′ / p w^{-p'/p} w − p ′ / p
For 1 < p , r < ∞ 1 < p,r < \infty 1 < p , r < ∞
Θ p , r ( t ) = { γ r ′ ( 4 1 − p ′ / r ′ max { 1 , γ p ( t ) } ) , 1 < r < p , 4 1 − p / r max { 1 , t } , 1 < p ≤ r . (46) \Theta_{p,r}(t) =
\begin{cases}
\gamma_{r'}\left(4^{1-p'/r'}\max\{1,\gamma_p(t)\}\right), & 1 < r < p,\\
4^{1-p/r}\max\{1,t\}, & 1 < p \le r.
\end{cases} \tag*{(46)} Θ p , r ( t ) = { γ r ′ ( 4 1 − p ′ / r ′ max { 1 , γ p ( t )} ) , 4 1 − p / r max { 1 , t } , 1 < r < p , 1 < p ≤ r . (46) The following equivalence statement will play an instrumental role in showing a factorization result.
Lemma 4.3. Let 1 < p < ∞ 1 < p < \infty 1 < p < ∞ w ∈ L 0 ( M ) w \in L_0(\mathcal{M}) w ∈ L 0 ( M ) ∣ w ∣ 2 p ∈ W p ( ( E β ) β ) |w|^{2p} \in W_p((E_\beta)_\beta) ∣ w ∣ 2 p ∈ W p (( E β ) β )
∥ sup β ∈ B + w E β ( w − 1 x ( w ∗ ) − 1 ) w ∗ ∥ p ≤ C ∥ x ∥ p , x ∈ L p ( M ) . (47) \left\|\sup_{\beta\in\mathcal{B}}^{+} wE_\beta\left(w^{-1}x(w^*)^{-1}\right)w^*\right\|_p \le C\|x\|_p,\qquad x\in L_p(\mathcal{M}). \tag*{(47)} β ∈ B sup + w E β ( w − 1 x ( w ∗ ) − 1 ) w ∗ p ≤ C ∥ x ∥ p , x ∈ L p ( M ) . (47) The least constant in (47) equals [ ∣ w ∣ 2 p ] W p ( ( E β ) β ) [|w|^{2p}]_{W_p((E_\beta)_\beta)} [ ∣ w ∣ 2 p ] W p (( E β ) β )
Proof. Firstly, we claim that the map
ι : L p ( M , ∣ w ∣ 2 p ) ⟶ L p ( M ) \iota:L_p(\mathcal{M},|w|^{2p})\longrightarrow L_p(\mathcal{M}) ι : L p ( M , ∣ w ∣ 2 p ) ⟶ L p ( M ) x ⟼ w x w ∗ x\longmapsto wxw^* x ⟼ w x w ∗ is an onto isometry. Indeed, on the one hand,
∥ x ∥ L p ( M , ∣ w ∣ 2 p ) = ∥ ∣ w ∣ ⋅ x ⋅ ∣ w ∣ ∥ p = ∥ w x w ∗ ∥ p = ∥ ι ( x ) ∥ p , \|x\|_{L_p(\mathcal{M},|w|^{2p})}=\||w|\cdot x\cdot|w|\|_p=\|wxw^*\|_p=\|\iota(x)\|_p, ∥ x ∥ L p ( M , ∣ w ∣ 2 p ) = ∥∣ w ∣ ⋅ x ⋅ ∣ w ∣ ∥ p = ∥ w x w ∗ ∥ p = ∥ ι ( x ) ∥ p , which shows that ι \iota ι x ∈ L p ( M ) x\in L_p(\mathcal{M}) x ∈ L p ( M ) ι ( w − 1 x ( w ∗ ) − 1 ) = x \iota(w^{-1}x(w^*)^{-1})=x ι ( w − 1 x ( w ∗ ) − 1 ) = x (47) is equivalent to
∥ sup β ∈ B + E β ( y ) ∥ L p ( M , ∣ w ∣ 2 p ) = ∥ sup β ∈ B + w E β ( y ) w ∗ ∥ p ≤ C ∥ y ∥ L p ( M , ∣ w ∣ 2 p ) , y ∈ L p ( M , ∣ w ∣ 2 p ) , \left\|\sup_{\beta\in\mathcal{B}}^{+}E_\beta(y)\right\|_{L_p(\mathcal{M},|w|^{2p})}
=
\left\|\sup_{\beta\in\mathcal{B}}^{+}wE_\beta(y)w^*\right\|_p
\le C\|y\|_{L_p(\mathcal{M},|w|^{2p})},\qquad y\in L_p(\mathcal{M},|w|^{2p}), β ∈ B sup + E β ( y ) L p ( M , ∣ w ∣ 2 p ) = β ∈ B sup + w E β ( y ) w ∗ p ≤ C ∥ y ∥ L p ( M , ∣ w ∣ 2 p ) , y ∈ L p ( M , ∣ w ∣ 2 p ) , which gives the estimate as desired. □
Theorem 4.4. Suppose that Assumption 4.2 holds. Let 1 < p , r < ∞ 1 < p,r < \infty 1 < p , r < ∞ p ≠ r p\ne r p = r s > 1 s>1 s > 1 1 / s = ∣ 1 / p − 1 / r ∣ 1/s=|1/p-1/r| 1/ s = ∣1/ p − 1/ r ∣ w ∈ W p ( ( E β ) β ) w\in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β ) u ∈ L s + ( M ) u\in L_s^+(\mathcal{M}) u ∈ L s + ( M ) v ∈ L s + ( M ) v\in L_s^+(\mathcal{M}) v ∈ L s + ( M )
u ≤ v , ∥ v ∥ s ≤ 2 ∥ u ∥ s , (48) u\le v,\qquad\|v\|_s\le2\|u\|_s, \tag*{(48)} u ≤ v , ∥ v ∥ s ≤ 2∥ u ∥ s , (48) and
{ ∣ v 1 / 2 p w 1 / 2 p ∣ 2 r ∈ W r ( ( E β ) β ) , 1 < r < p , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ∈ W r ( ( E β ) β ) , 1 < p < r . (49) \begin{cases}
|v^{1/2p}w^{1/2p}|^{2r}\in W_r((E_\beta)_\beta), & 1<r<p,\\
|v^{-1/2}w^{1/2p}|^{2r}\in W_r((E_\beta)_\beta), & 1<p<r. \tag*{(49)}
\end{cases} { ∣ v 1/2 p w 1/2 p ∣ 2 r ∈ W r (( E β ) β ) , ∣ v − 1/2 w 1/2 p ∣ 2 r ∈ W r (( E β ) β ) , 1 < r < p , 1 < p < r . (49) Quantitatively,
[ ∣ v 1 / 2 p w 1 / 2 p ∣ 2 r ] W r ≤ γ r ′ ( 4 1 − p ′ / r ′ max { 1 , γ p ( [ w ] W p ) } ) , 1 < r < p , (50) [|v^{1/2p}w^{1/2p}|^{2r}]_{W_r}
\le\gamma_{r'}\left(4^{1-p'/r'}\max\{1,\gamma_p([w]_{W_p})\}\right),\qquad1<r<p, \tag*{(50)} [ ∣ v 1/2 p w 1/2 p ∣ 2 r ] W r ≤ γ r ′ ( 4 1 − p ′ / r ′ max { 1 , γ p ([ w ] W p )} ) , 1 < r < p , (50) [ ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ] W r ≤ 4 1 − p / r max { 1 , [ w ] W p } , 1 < p < r . [|v^{-1/2}w^{1/2p}|^{2r}]_{W_r}
\le4^{1-p/r}\max\{1,[w]_{W_p}\},\qquad1<p<r. [ ∣ v − 1/2 w 1/2 p ∣ 2 r ] W r ≤ 4 1 − p / r max { 1 , [ w ] W p } , 1 < p < r . Proof. Firstly, if v v v w w w ∣ v 1 / 2 p w 1 / 2 p ∣ 2 r |v^{1/2p}w^{1/2p}|^{2r} ∣ v 1/2 p w 1/2 p ∣ 2 r ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r |v^{-1/2}w^{1/2p}|^{2r} ∣ v − 1/2 w 1/2 p ∣ 2 r
For the proof, denote by S β S_\beta S β L p ( M ) L_p(\mathcal{M}) L p ( M )
S β ( x ) = w 1 / 2 p E β ( w − 1 / 2 p x w − 1 / 2 p ) w 1 / 2 p . (51) S_\beta(x)=w^{1/2p}E_\beta(w^{-1/2p}xw^{-1/2p})w^{1/2p}. \tag*{(51)} S β ( x ) = w 1/2 p E β ( w − 1/2 p x w − 1/2 p ) w 1/2 p . (51) Definition 4.1 gives
∥ sup β ∈ B + S β ( x ) ∥ p ≤ C ∥ x ∥ p , C = max { 1 , [ w ] W p } . (52) \left\|\sup_{\beta\in\mathcal{B}}^{+}S_\beta(x)\right\|_p\le C\|x\|_p,\qquad C=\max\{1,[w]_{W_p}\}. \tag*{(52)} β ∈ B sup + S β ( x ) p ≤ C ∥ x ∥ p , C = max { 1 , [ w ] W p } . (52) We first establish an endpoint construction. Let h ∈ L p + ( M ) h \in L_p^+(\mathcal{M}) h ∈ L p + ( M ) h 0 = h h_0=h h 0 = h (10) allows us to choose inductively h k + 1 ∈ L p + ( M ) h_{k+1}\in L_p^+(\mathcal{M}) h k + 1 ∈ L p + ( M )
S β ( h k ) ≤ h k + 1 ( β ∈ B ) , ∥ h k + 1 ∥ p ≤ 2 C ∥ h k ∥ p . S_\beta(h_k)\le h_{k+1}\quad(\beta\in\mathcal{B}),\qquad\lVert h_{k+1}\rVert_p\le2C\lVert h_k\rVert_p. S β ( h k ) ≤ h k + 1 ( β ∈ B ) , ∥ h k + 1 ∥ p ≤ 2 C ∥ h k ∥ p . Consequently,
v 0 = ∑ k = 0 ∞ ( 4 C ) − k h k (53) v_0=\sum_{k=0}^{\infty}(4C)^{-k}h_k \tag*{(53)} v 0 = k = 0 ∑ ∞ ( 4 C ) − k h k (53) converges in L p + ( M ) L_p^+(\mathcal{M}) L p + ( M )
h ≤ v 0 , ∥ v 0 ∥ p ≤ 2 ∥ h ∥ p , S β ( v 0 ) ≤ 4 C v 0 . (54) h\le v_0,\qquad\lVert v_0\rVert_p\le2\lVert h\rVert_p,\qquad S_\beta(v_0)\le4Cv_0. \tag*{(54)} h ≤ v 0 , ∥ v 0 ∥ p ≤ 2 ∥ h ∥ p , S β ( v 0 ) ≤ 4 C v 0 . (54) The last inequality follows by applying the positive map S β S_\beta S β
For x ∈ M x\in\mathcal{M} x ∈ M
x ⟼ v 0 − 1 / 2 S β ( v 0 1 / 2 x v 0 1 / 2 ) v 0 − 1 / 2 . x\longmapsto v_0^{-1/2}S_\beta(v_0^{1/2}xv_0^{1/2})v_0^{-1/2}. x ⟼ v 0 − 1/2 S β ( v 0 1/2 x v 0 1/2 ) v 0 − 1/2 . If 0 ≤ x ≤ 1 0\le x\le1 0 ≤ x ≤ 1 (54) bounds the image by 4 C 1 4C1 4 C 1 M \mathcal{M} M 4 C 4C 4 C L ∞ ( M ; ℓ ∞ ( B ) ) = ℓ ∞ ( B ; M ) L_\infty(\mathcal{M};\ell_\infty(\mathcal{B}))=\ell_\infty(\mathcal{B};\mathcal{M}) L ∞ ( M ; ℓ ∞ ( B )) = ℓ ∞ ( B ; M ) x ∈ M x\in\mathcal{M} x ∈ M
∥ sup β ∈ B + v 0 − 1 / 2 S β ( v 0 1 / 2 x v 0 1 / 2 ) v 0 − 1 / 2 ∥ ∞ ≤ 4 C ∥ x ∥ ∞ . (55) \left\lVert\sup_{\beta\in\mathcal{B}}^{+}v_0^{-1/2}S_\beta(v_0^{1/2}xv_0^{1/2})v_0^{-1/2}\right\rVert_\infty\le4C\lVert x\rVert_\infty. \tag*{(55)} β ∈ B sup + v 0 − 1/2 S β ( v 0 1/2 x v 0 1/2 ) v 0 − 1/2 ∞ ≤ 4 C ∥ x ∥ ∞ . (55) Suppose now that p < r p<r p < r 1 / s = 1 / p − 1 / r 1/s=1/p-1/r 1/ s = 1/ p − 1/ r h = u s / p h=u^{s/p} h = u s / p u u u ∥ u s / p ∥ p = ∥ u ∥ s s / p \lVert u^{s/p}\rVert_p=\lVert u\rVert_s^{s/p} ∥ u s / p ∥ p = ∥ u ∥ s s / p
θ = p s = 1 − p r , v = v 0 θ . \theta=\frac{p}{s}=1-\frac{p}{r},\qquad v=v_0^\theta. θ = s p = 1 − r p , v = v 0 θ . The Löwner–Heinz inequality for positive operators [15 ] , applied with 0 < θ < 1 0<\theta<1 0 < θ < 1 u s / p ≤ v 0 u^{s/p}\le v_0 u s / p ≤ v 0
u ≤ v , ∥ v ∥ s = ∥ v 0 ∥ p θ ≤ 2 θ ∥ u ∥ s ≤ 2 ∥ u ∥ s . (56) u\le v,\qquad\lVert v\rVert_s=\lVert v_0\rVert_p^\theta\le2^\theta\lVert u\rVert_s\le2\lVert u\rVert_s. \tag*{(56)} u ≤ v , ∥ v ∥ s = ∥ v 0 ∥ p θ ≤ 2 θ ∥ u ∥ s ≤ 2 ∥ u ∥ s . (56) We include the truncation needed for complex interpolation. Let e k = 1 [ 1 / k , k ] ( v 0 ) e_k=1_{[1/k,k]}(v_0) e k = 1 [ 1/ k , k ] ( v 0 ) F ⊂ B \mathcal{F}\subset\mathcal{B} F ⊂ B m , n ≥ 1 m,n\ge1 m , n ≥ 1
T m , n ( z ) x = ( e m v 0 − z / 2 S β ( v 0 z / 2 e n x e n v 0 z / 2 ) v 0 − z / 2 e m ) β ∈ F , 0 ≤ ℜ z ≤ 1. (57) \mathcal{T}_{m,n}(z)x=\left(e_m v_0^{-z/2}S_\beta(v_0^{z/2}e_nxe_nv_0^{z/2})v_0^{-z/2}e_m\right)_{\beta\in\mathcal{F}},\qquad0\le\Re z\le1. \tag*{(57)} T m , n ( z ) x = ( e m v 0 − z /2 S β ( v 0 z /2 e n x e n v 0 z /2 ) v 0 − z /2 e m ) β ∈ F , 0 ≤ ℜ z ≤ 1. (57) The spectral projections make sure that T m , n \mathcal{T}_{m,n} T m , n M \mathcal{M} M L p ( M ; ℓ ∞ ( F ) ) L_p(\mathcal{M};\ell_\infty(\mathcal{F})) L p ( M ; ℓ ∞ ( F ))
By (52) and (55) , we have the following estimates on the lines ℜ z = 0 , 1 \Re z=0,1 ℜ z = 0 , 1
∥ T m , n ( i t ) x ∥ L p ( M ; ℓ ∞ ) ≤ ∥ ( S β ( v 0 i t / 2 e n x e n v 0 i t / 2 ) ) β ∈ F ∥ L p ( M ; ℓ ∞ ) ≤ C ∥ v 0 i t / 2 e n x e n v 0 i t / 2 ∥ p ≤ C ∥ x ∥ p . (58) \begin{aligned}
\lVert\mathcal{T}_{m,n}(it)x\rVert_{L_p(\mathcal{M};\ell_\infty)}
&\le\left\lVert\left(S_\beta(v_0^{it/2}e_nxe_nv_0^{it/2})\right)_{\beta\in\mathcal{F}}\right\rVert_{L_p(\mathcal{M};\ell_\infty)}\\
&\le C\lVert v_0^{it/2}e_nxe_nv_0^{it/2}\rVert_p\le C\lVert x\rVert_p. \tag*{(58)}
\end{aligned} ∥ T m , n ( i t ) x ∥ L p ( M ; ℓ ∞ ) ≤ ( S β ( v 0 i t /2 e n x e n v 0 i t /2 ) ) β ∈ F L p ( M ; ℓ ∞ ) ≤ C ∥ v 0 i t /2 e n x e n v 0 i t /2 ∥ p ≤ C ∥ x ∥ p . (58) ∥ T m , n ( 1 + i t ) x ∥ L ∞ ( M ; ℓ ∞ ) ≤ 4 C ∥ v 0 i t / 2 e n x e n v 0 i t / 2 ∥ ∞ = 4 C ∥ e n x e n ∥ ∞ ≤ 4 C ∥ x ∥ ∞ . (59) \lVert\mathcal{T}_{m,n}(1+it)x\rVert_{L_\infty(\mathcal{M};\ell_\infty)}
\le4C\lVert v_0^{it/2}e_nxe_nv_0^{it/2}\rVert_\infty
=4C\lVert e_nxe_n\rVert_\infty\le4C\lVert x\rVert_\infty. \tag*{(59)} ∥ T m , n ( 1 + i t ) x ∥ L ∞ ( M ; ℓ ∞ ) ≤ 4 C ∥ v 0 i t /2 e n x e n v 0 i t /2 ∥ ∞ = 4 C ∥ e n x e n ∥ ∞ ≤ 4 C ∥ x ∥ ∞ . (59) Stein’s interpolation [27 ] , together with the interpolation identity in Lemma 2.2, gives at θ = 1 − p / r \theta=1-p/r θ = 1 − p / r
∥ T m , n ( θ ) x ∥ L r ( M ; ℓ ∞ ) ≤ 4 θ C ∥ x ∥ r . (60) \lVert\mathcal{T}_{m,n}(\theta)x\rVert_{L_r(\mathcal{M};\ell_\infty)}\le4^\theta C\lVert x\rVert_r. \tag*{(60)} ∥ T m , n ( θ ) x ∥ L r ( M ; ℓ ∞ ) ≤ 4 θ C ∥ x ∥ r . (60) Letting m → ∞ m\to\infty m → ∞ (60) gives that
∥ sup β ∈ F + v − 1 / 2 S β ( v 1 / 2 e n x e n v 1 / 2 ) v − 1 / 2 ∥ r ≤ 4 θ C ∥ x ∥ r , x ∈ M . \left\lVert\sup_{\beta\in\mathcal{F}}^{+}v^{-1/2}S_\beta(v^{1/2}e_nxe_nv^{1/2})v^{-1/2}\right\rVert_r\le4^\theta C\lVert x\rVert_r,\qquad x\in\mathcal{M}. β ∈ F sup + v − 1/2 S β ( v 1/2 e n x e n v 1/2 ) v − 1/2 r ≤ 4 θ C ∥ x ∥ r , x ∈ M . For x ∈ C ( v 0 ) x \in\mathcal{C}(v_0) x ∈ C ( v 0 ) n ∈ N + n \in\mathbb{N}_+ n ∈ N + e n x e n = x e_nxe_n=x e n x e n = x
∥ sup β ∈ F + v − 1 / 2 S β ( v 1 / 2 x v 1 / 2 ) v − 1 / 2 ∥ r ≤ 4 θ C ∥ x ∥ r , x ∈ C ( v 0 ) . \left\|\sup_{\beta\in\mathcal{F}}^{+}v^{-1/2}S_\beta(v^{1/2}xv^{1/2})v^{-1/2}\right\|_r \le4^{\theta}\mathcal{C}\|x\|_r,\qquad x\in\mathcal{C}(v_0). β ∈ F sup + v − 1/2 S β ( v 1/2 x v 1/2 ) v − 1/2 r ≤ 4 θ C ∥ x ∥ r , x ∈ C ( v 0 ) . The density of C ( v 0 ) ⊂ L r ( M ) \mathcal{C}(v_0)\subset L_r(\mathcal{M}) C ( v 0 ) ⊂ L r ( M ) Lemma 2.6 and the finite-subfamily property give that
∥ sup β ∈ B + v − 1 / 2 S β ( v 1 / 2 x v 1 / 2 ) v − 1 / 2 ∥ r ≤ 4 1 − p / r C ∥ x ∥ r , x ∈ L r ( M ) . \left\|\sup_{\beta\in\mathcal{B}}^{+}v^{-1/2}S_\beta(v^{1/2}xv^{1/2})v^{-1/2}\right\|_r \le4^{1-p/r}\mathcal{C}\|x\|_r,\qquad x\in L_r(\mathcal{M}). β ∈ B sup + v − 1/2 S β ( v 1/2 x v 1/2 ) v − 1/2 r ≤ 4 1 − p / r C ∥ x ∥ r , x ∈ L r ( M ) . Then Lemma 4.3 , applied to w = v − 1 / 2 w 1 / 2 p \mathbf{w}=v^{-1/2}w^{1/2p} w = v − 1/2 w 1/2 p p < r p<r p < r
Finally, suppose that r < p r<p r < p Assumption 4.2 gives
w − p ′ / p ∈ W p ′ ( ( E β ) β ) , [ w − p ′ / p ] W p ′ ≤ γ p ( [ w ] W p ) . w^{-p'/p}\in W_{p'}((E_\beta)_\beta),\qquad[w^{-p'/p}]_{W_{p'}}\le\gamma_p([w]_{W_p}). w − p ′ / p ∈ W p ′ (( E β ) β ) , [ w − p ′ / p ] W p ′ ≤ γ p ([ w ] W p ) . Here p ′ < r ′ p'<r' p ′ < r ′ 1 / s = 1 / p ′ − 1 / r ′ 1/s=1/p'-1/r' 1/ s = 1/ p ′ − 1/ r ′ v v v
∣ v − 1 / 2 w − 1 / 2 p ∣ 2 r ′ ∈ W r ′ ( ( E β ) β ) |v^{-1/2}w^{-1/2p}|^{2r'}\in W_{r'}((E_\beta)_\beta) ∣ v − 1/2 w − 1/2 p ∣ 2 r ′ ∈ W r ′ (( E β ) β ) with characteristic at most 4 1 − p ′ / r ′ max { 1 , γ p ′ ( [ w ] W p ′ ) } 4^{1-p'/r'}\max\{1,\gamma_{p'}([w]_{W_{p'}})\} 4 1 − p ′ / r ′ max { 1 , γ p ′ ([ w ] W p ′ )} Assumption 4.2 at r ′ r' r ′
( ∣ v − 1 / 2 w − 1 / 2 p ∣ 2 r ′ ) − r / r ′ = ( w − 1 / 2 p v − 1 w − 1 / 2 p ) − r = ( w 1 / 2 p v w 1 / 2 p ) r = ∣ v 1 / 2 w 1 / 2 p ∣ 2 r . (61) \begin{aligned}
\left(|v^{-1/2}w^{-1/2p}|^{2r'}\right)^{-r/r'}=(w^{-1/2p}v^{-1}w^{-1/2p})^{-r} \\
&=(w^{1/2p}vw^{1/2p})^r=|v^{1/2}w^{1/2p}|^{2r}. \tag*{(61)}
\end{aligned} ( ∣ v − 1/2 w − 1/2 p ∣ 2 r ′ ) − r / r ′ = ( w − 1/2 p v − 1 w − 1/2 p ) − r = ( w 1/2 p v w 1/2 p ) r = ∣ v 1/2 w 1/2 p ∣ 2 r . (61) This proves the case of r < p r<p r < p (50) . □
A noncommutative Rubio de Francia extrapolation theorem
Now we exploit the factorization for the weights to show the extrapolation result for the operators.
Theorem 4.5. Suppose that Assumption 4.2 holds. Let ( T α ) α ∈ A (T_\alpha)_{\alpha\in A} ( T α ) α ∈ A M \mathcal{M} M L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) 1 < r < ∞ 1<r<\infty 1 < r < ∞ Φ : [ 0 , ∞ ) → [ 0 , ∞ ) \Phi:[0,\infty)\to[0,\infty) Φ : [ 0 , ∞ ) → [ 0 , ∞ )
∥ sup α ∈ A + T α ( x ) ∥ L r ( M , w ) ≤ Φ ( [ w ] W r ) ∥ x ∥ L r ( M , w ) (62) \left\|\sup_{\alpha\in A}^{+}T_\alpha(x)\right\|_{L_r(\mathcal{M},w)}\le\Phi([w]_{W_r})\|x\|_{L_r(\mathcal{M},w)} \tag*{(62)} α ∈ A sup + T α ( x ) L r ( M , w ) ≤ Φ ([ w ] W r ) ∥ x ∥ L r ( M , w ) (62) for every x ∈ M ∩ L r ( M , w ) x\in\mathcal{M}\cap L_r(\mathcal{M},w) x ∈ M ∩ L r ( M , w ) w ∈ W r ( ( E β ) β ) w\in W_r((E_\beta)_\beta) w ∈ W r (( E β ) β ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ W p ( ( E β ) β ) w\in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β ) x ↦ ( T α x ) α x\mapsto(T_\alpha x)_\alpha x ↦ ( T α x ) α L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) L p ( M , w ; ℓ ∞ ( A ) ) L_p(\mathcal{M},w;\ell_\infty(A)) L p ( M , w ; ℓ ∞ ( A ))
∥ sup α ∈ A + T α ( x ) ∥ L p ( M , w ) ≤ 8 Φ ( Θ p , r [ w ] W p ) ∥ x ∥ L p ( M , w ) . (63) \left\|\sup_{\alpha\in A}^{+}T_\alpha(x)\right\|_{L_p(\mathcal{M},w)}
\le8\Phi(\Theta_{p,r}[w]_{W_p})\|x\|_{L_p(\mathcal{M},w)}. \tag*{(63)} α ∈ A sup + T α ( x ) L p ( M , w ) ≤ 8Φ ( Θ p , r [ w ] W p ) ∥ x ∥ L p ( M , w ) . (63) As mentioned in § 1 , in Theorem 4.5 , letting T α ≡ T T_\alpha\equiv T T α ≡ T ∥ T ( x ) ∥ L p ( M , w ) \|T(x)\|_{L_p(\mathcal{M},w)} ∥ T ( x ) ∥ L p ( M , w )
Corollary 4.6. Suppose that Assumption 4.2 holds. Let T T T M \mathcal{M} M L 0 ( M ) L_0(\mathcal{M}) L 0 ( M ) 1 < r < ∞ 1<r<\infty 1 < r < ∞ Φ \Phi Φ
∥ T ( x ) ∥ L r ( M , w ) ≤ Φ ( [ w ] W r ) ∥ x ∥ L r ( M , w ) \|T(x)\|_{L_r(\mathcal{M},w)}\le\Phi([w]_{W_r})\|x\|_{L_r(\mathcal{M},w)} ∥ T ( x ) ∥ L r ( M , w ) ≤ Φ ([ w ] W r ) ∥ x ∥ L r ( M , w ) for every w ∈ W r ( ( E β ) β ) w\in W_r((E_\beta)_\beta) w ∈ W r (( E β ) β ) x ∈ M ∩ L r ( M , w ) x\in\mathcal{M}\cap L_r(\mathcal{M},w) x ∈ M ∩ L r ( M , w ) T T T L p ( M , w ) L_p(\mathcal{M},w) L p ( M , w ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ W p ( ( E β ) β ) w\in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β )
∥ T ( x ) ∥ L p ( M , w ) ≤ 8 Φ ( Θ p , r [ w ] W p ) ∥ x ∥ L p ( M , w ) . \|T(x)\|_{L_p(\mathcal{M},w)}\le8\Phi(\Theta_{p,r}[w]_{W_p})\|x\|_{L_p(\mathcal{M},w)}. ∥ T ( x ) ∥ L p ( M , w ) ≤ 8Φ ( Θ p , r [ w ] W p ) ∥ x ∥ L p ( M , w ) . With the factorization for weights—Theorem 4.4 , the Hölder type testing estimate established in Lemma 2.7 yields Theorem 4.5 .
Proof of Theorem 4.5 . Fix w ∈ W p ( ( E β ) β ) w\in W_p((E_\beta)_\beta) w ∈ W p (( E β ) β ) x ∈ M ∩ L p ( M , w ) x\in\mathcal{M}\cap L_p(\mathcal{M},w) x ∈ M ∩ L p ( M , w ) r < p r<p r < p 1 / r = 1 / p + 1 / s 1/r=1/p+1/s 1/ r = 1/ p + 1/ s u ∈ L s ( M ) u\in L_s(\mathcal{M}) u ∈ L s ( M ) Theorem 4.4 gives v ≥ u v\ge u v ≥ u ∥ v ∥ s ≤ 2 ∥ u ∥ s \|v\|_s\le2\|u\|_s ∥ v ∥ s ≤ 2∥ u ∥ s ∣ v 1 / 2 w 1 / 2 p ∣ 2 r ∈ W r ( ( E β ) β ) |v^{1/2}w^{1/2p}|^{2r}\in W_r((E_\beta)_\beta) ∣ v 1/2 w 1/2 p ∣ 2 r ∈ W r (( E β ) β ) v 1 / 2 w 1 / 2 p v^{1/2}w^{1/2p} v 1/2 w 1/2 p
∥ x ∥ L r ( M , ∣ v 1 / 2 w 1 / 2 p ∣ 2 r ) = ∥ v 1 / 2 w 1 / 2 p x w 1 / 2 p v 1 / 2 ∥ r ≤ ∥ v ∥ s ∥ x ∥ L p ( M , w ) < ∞ . \|x\|_{L_r(\mathcal{M},|v^{1/2}w^{1/2p}|^{2r})}
=\|v^{1/2}w^{1/2p}xw^{1/2p}v^{1/2}\|_r
\le\|v\|_s\|x\|_{L_p(\mathcal{M},w)}<\infty. ∥ x ∥ L r ( M , ∣ v 1/2 w 1/2 p ∣ 2 r ) = ∥ v 1/2 w 1/2 p x w 1/2 p v 1/2 ∥ r ≤ ∥ v ∥ s ∥ x ∥ L p ( M , w ) < ∞. That is, x ∈ M ∩ L r ( M , ∣ v 1 / 2 w 1 1 / 2 p ∣ 2 r ) x \in\mathcal{M} \cap L_r(\mathcal{M}, |v^{1/2}w_1^{1/2p}|^{2r}) x ∈ M ∩ L r ( M , ∣ v 1/2 w 1 1/2 p ∣ 2 r ) (62) , followed by the monotonicity property—Lemma 2.8 , yields
∥ sup α ∈ A + u 1 / 2 w 1 1 / 2 p T α ( x ) w 1 1 / 2 p u 1 1 / 2 ∥ r ≤ ∥ sup α ∈ A + v 1 / 2 w 1 1 / 2 p T α ( x ) w 1 1 / 2 p v 1 1 / 2 ∥ r = ∥ sup α ∈ A + T α ( x ) ∥ L r ( M , ∣ v 1 / 2 w 1 1 / 2 p ∣ 2 r ) ≤ Φ ( [ ∣ v 1 / 2 w 1 1 / 2 p ∣ 2 r ] W r ) ⋅ ∥ x ∥ L r ( M , ∣ v 1 / 2 w 1 1 / 2 p ∣ 2 r ) ≤ 2 Φ ( Θ p , r ( [ w ] W p ) ) ⋅ ∥ u ∥ s ⋅ ∥ x ∥ L p ( M , w ) . \begin{aligned}
\left\|\sup_{\alpha\in\mathcal{A}}^{+}u^{1/2}w_1^{1/2p}T_\alpha(x)w_1^{1/2p}u_1^{1/2}\right\|_r
&\leq\left\|\sup_{\alpha\in\mathcal{A}}^{+}v^{1/2}w_1^{1/2p}T_\alpha(x)w_1^{1/2p}v_1^{1/2}\right\|_r \\
&= \left\|\sup_{\alpha\in\mathcal{A}}^{+}T_\alpha(x)\right\|_{L_r(\mathcal{M},|v^{1/2}w_1^{1/2p}|^{2r})} \\
&\leq\Phi([|v^{1/2}w_1^{1/2p}|^{2r}]_{W_r})\cdot\|x\|_{L_r(\mathcal{M},|v^{1/2}w_1^{1/2p}|^{2r})} \\
&\leq2\Phi(\Theta_{p,r}([w]_{W_p}))\cdot\|u\|_s\cdot\|x\|_{L_p(\mathcal{M},w)}.
\end{aligned} α ∈ A sup + u 1/2 w 1 1/2 p T α ( x ) w 1 1/2 p u 1 1/2 r ≤ α ∈ A sup + v 1/2 w 1 1/2 p T α ( x ) w 1 1/2 p v 1 1/2 r = α ∈ A sup + T α ( x ) L r ( M , ∣ v 1/2 w 1 1/2 p ∣ 2 r ) ≤ Φ ([ ∣ v 1/2 w 1 1/2 p ∣ 2 r ] W r ) ⋅ ∥ x ∥ L r ( M , ∣ v 1/2 w 1 1/2 p ∣ 2 r ) ≤ 2Φ ( Θ p , r ([ w ] W p )) ⋅ ∥ u ∥ s ⋅ ∥ x ∥ L p ( M , w ) . Divide by ∥ u ∥ s \|u\|_s ∥ u ∥ s u u u Lemma 2.7 (ii). This gives (63) , with the factor 8 = 4 ⋅ 2 8=4\cdot2 8 = 4 ⋅ 2
Suppose next that p < r p<r p < r 1 / p = 1 / r + 1 / s 1/p=1/r+1/s 1/ p = 1/ r + 1/ s ε > 0 \varepsilon>0 ε > 0 Lemma 2.7 (i), applied to w 1 / 2 p x w 1 / 2 p w^{1/2p}xw^{1/2p} w 1/2 p x w 1/2 p u ∈ L s ( M ) u\in L_s(\mathcal{M}) u ∈ L s ( M )
∥ u − 1 / 2 w 1 / 2 p x w 1 / 2 p u − 1 / 2 ∥ r ∥ u ∥ s ≤ 2 ∥ x ∥ L p ( M , w ) + ε . (64) \left\|u^{-1/2}w^{1/2p}xw^{1/2p}u^{-1/2}\right\|_r\|u\|_s\leq2\|x\|_{L_p(\mathcal{M},w)}+\varepsilon. \tag*{(64)} u − 1/2 w 1/2 p x w 1/2 p u − 1/2 r ∥ u ∥ s ≤ 2∥ x ∥ L p ( M , w ) + ε . (64) Theorem 4.4 gives v ≥ u v\geq u v ≥ u ∥ v ∥ s ≤ 2 ∥ u ∥ s \|v\|_s\leq2\|u\|_s ∥ v ∥ s ≤ 2∥ u ∥ s ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ∈ W r ( ( E β ) β ) |v^{-1/2}w^{1/2p}|^{2r}\in W_r((E_\beta)_\beta) ∣ v − 1/2 w 1/2 p ∣ 2 r ∈ W r (( E β ) β )
∥ x ∥ L r ( M , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ) = ∥ v − 1 / 2 w 1 / 2 p x w 1 / 2 p v − 1 / 2 ∥ r ≤ ∥ u − 1 / 2 w 1 / 2 p x w 1 / 2 p u − 1 / 2 ∥ r ≤ 1 ∥ u ∥ s ⋅ ( 2 ∥ x ∥ L p ( M , w ) + ε ) < ∞ \begin{aligned}
\|x\|_{L_r(\mathcal{M},|v^{-1/2}w^{1/2p}|^{2r})}
&=\|v^{-1/2}w^{1/2p}xw^{1/2p}v^{-1/2}\|_r \\
&\leq\|u^{-1/2}w^{1/2p}xw^{1/2p}u^{-1/2}\|_r \\
&\leq\frac{1}{\|u\|_s}\cdot(2\|x\|_{L_p(\mathcal{M},w)}+\varepsilon)<\infty
\end{aligned} ∥ x ∥ L r ( M , ∣ v − 1/2 w 1/2 p ∣ 2 r ) = ∥ v − 1/2 w 1/2 p x w 1/2 p v − 1/2 ∥ r ≤ ∥ u − 1/2 w 1/2 p x w 1/2 p u − 1/2 ∥ r ≤ ∥ u ∥ s 1 ⋅ ( 2∥ x ∥ L p ( M , w ) + ε ) < ∞ gives that x ∈ M ∩ L r ( M , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ) x\in\mathcal{M}\cap L_r(\mathcal{M},|v^{-1/2}w^{1/2p}|^{2r}) x ∈ M ∩ L r ( M , ∣ v − 1/2 w 1/2 p ∣ 2 r ) Lemma 2.7 (ii) and the assumption (62) give
∥ sup α ∈ A + T α ( x ) ∥ L p ( M , w ) ≤ ∥ v ∥ s ∥ sup α ∈ A + v − 1 / 2 w 1 / 2 p T α ( x ) w 1 / 2 p v − 1 / 2 ∥ r ≤ 2 ∥ u ∥ s ⋅ ∥ sup α ∈ A + T α ( x ) ∥ L r ( M , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ) ≤ 2 Φ ( [ ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ] W r ) ∥ u ∥ s ⋅ ∥ x ∥ L r ( M , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ) ≤ 2 Φ ( Θ p , r ( [ w ] W p ) ) ⋅ ( 2 ∥ x ∥ L p ( M , w ) + ε ) . \begin{aligned}
\left\|\sup_{\alpha\in\mathcal{A}}^{+}T_\alpha(x)\right\|_{L_p(\mathcal{M},w)}
&\leq\|v\|_s\left\|\sup_{\alpha\in\mathcal{A}}^{+}v^{-1/2}w^{1/2p}T_\alpha(x)w^{1/2p}v^{-1/2}\right\|_r \\
&\leq2\|u\|_s\cdot\left\|\sup_{\alpha\in\mathcal{A}}^{+}T_\alpha(x)\right\|_{L_r(\mathcal{M},|v^{-1/2}w^{1/2p}|^{2r})} \\
&\leq2\Phi([|v^{-1/2}w^{1/2p}|^{2r}]_{W_r})\|u\|_s\cdot\|x\|_{L_r(\mathcal{M},|v^{-1/2}w^{1/2p}|^{2r})} \\
&\leq2\Phi(\Theta_{p,r}([w]_{W_p}))\cdot(2\|x\|_{L_p(\mathcal{M},w)}+\varepsilon).
\end{aligned} α ∈ A sup + T α ( x ) L p ( M , w ) ≤ ∥ v ∥ s α ∈ A sup + v − 1/2 w 1/2 p T α ( x ) w 1/2 p v − 1/2 r ≤ 2∥ u ∥ s ⋅ α ∈ A sup + T α ( x ) L r ( M , ∣ v − 1/2 w 1/2 p ∣ 2 r ) ≤ 2Φ ([ ∣ v − 1/2 w 1/2 p ∣ 2 r ] W r ) ∥ u ∥ s ⋅ ∥ x ∥ L r ( M , ∣ v − 1/2 w 1/2 p ∣ 2 r ) ≤ 2Φ ( Θ p , r ([ w ] W p )) ⋅ ( 2∥ x ∥ L p ( M , w ) + ε ) . Letting ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 4 4 4 8 8 8
Matrix-weighted extrapolation and applications
In this section, we prove Theorem 1.2 by applying the argument of § 4 directly to interval averages on N \mathcal{N} N
The factorization structure on N \mathcal{N} N
Recall that N = L ∞ ( R ) ⊗ ‾ M d ( C ) \mathcal{N}=L_\infty(\mathbb{R})\overline{\otimes}M_d(\mathbb{C}) N = L ∞ ( R ) ⊗ M d ( C ) ( A I ) I ∈ I (A_I)_{I\in\mathcal{I}} ( A I ) I ∈ I § 4 , for a matrix weight w w w w ∈ W p ( ( A I ) I ∈ I ) w\in W_p((A_I)_{I\in\mathcal{I}}) w ∈ W p (( A I ) I ∈ I )
f ⟼ ( A I f ) I ∈ I , f ∈ S N c , f\longmapsto(A_If)_{I\in\mathcal{I}},\qquad f\in\mathcal{S}_{\mathcal{N}}^c, f ⟼ ( A I f ) I ∈ I , f ∈ S N c , extends boundedly from L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p ( N , w ; ℓ ∞ ) L_p(\mathcal{N},w;\ell_\infty) L p ( N , w ; ℓ ∞ ) [ w ] W p ( ( A I ) I ∈ I ) [w]_{W_p((A_I)_{I\in\mathcal{I}})} [ w ] W p (( A I ) I ∈ I )
[ w ] A p ( N ) 1 / p ≤ [ w ] W p ( ( A I ) I ∈ I ) ≤ W p , d [ w ] A p ( N ) 1 / p ′ , (65) [w]_{A_p(\mathcal{N})}^{1/p}\leq[w]_{W_p((A_I)_{I\in\mathcal{I}})}\leq\mathcal{W}_{p,d}[w]_{A_p(\mathcal{N})}^{1/p'}, \tag*{(65)} [ w ] A p ( N ) 1/ p ≤ [ w ] W p (( A I ) I ∈ I ) ≤ W p , d [ w ] A p ( N ) 1/ p ′ , (65) with W p , d ≤ d 1 p + ( 1 + p ′ ) ∣ 1 p − 1 2 ∣ G p , d 2 \mathcal{W}_{p,d}\leq d^{\frac{1}{p}+(1+p')\left|\frac{1}{p}-\frac{1}{2}\right|}\mathcal{G}_{p,d}^2 W p , d ≤ d p 1 + ( 1 + p ′ ) ∣ p 1 − 2 1 ∣ G p , d 2
Lemma 5.1. The class W p ( ( A I ) I ∈ I ) W_p((A_I)_{I\in\mathcal{I}}) W p (( A I ) I ∈ I ) 1 < p < ∞ 1<p<\infty 1 < p < ∞ Assumption 4.2 . More precisely, one may take
γ p ( t ) = γ p , d ( t ) = W p ′ , d t p . (66) \gamma_p(t)=\gamma_{p,d}(t)=\mathcal{W}_{p',d}t^{p}. \tag*{(66)} γ p ( t ) = γ p , d ( t ) = W p ′ , d t p . (66) Proof. Let w ∈ W p ( ( A I ) I ) w \in\mathcal{W}_p((A_I)_I) w ∈ W p (( A I ) I ) w w w w ∈ A p ( N ) w \in A_p(\mathcal{N}) w ∈ A p ( N ) w − p ′ / p ∈ A p ′ ( N ) w^{-p'/p} \in A_{p'}(\mathcal{N}) w − p ′ / p ∈ A p ′ ( N ) (35) . Applying the upper comparison in Theorem 1.1 at p ′ p' p ′ p p p
[ w − p ′ / p ] W p ′ ( ( A I ) I ) ≤ W p ′ , d [ w − p ′ / p ] A p ′ ( N ) p ′ / p = W p ′ , d [ w ] A p ( N ) ≤ W p ′ , d [ w ] W p ( ( A I ) I ) p ′ / p . [w^{-p'/p}]_{\mathcal{W}_{p'}((A_I)_I)}
\leq\mathcal{W}_{p',d}[w^{-p'/p}]_{A_{p'}(\mathcal{N})}^{p'/p}
= \mathcal{W}_{p',d}[w]_{A_p(\mathcal{N})}
\leq\mathcal{W}_{p',d}[w]_{\mathcal{W}_p((A_I)_I)}^{p'/p}. [ w − p ′ / p ] W p ′ (( A I ) I ) ≤ W p ′ , d [ w − p ′ / p ] A p ′ ( N ) p ′ / p = W p ′ , d [ w ] A p ( N ) ≤ W p ′ , d [ w ] W p (( A I ) I ) p ′ / p . Geometric duality also ensures local integrability of the inverse power. The same argument at p ′ p' p ′
Remark 5.2. The realization of Assumption 4.2 on N \mathcal{N} N
γ p , d ( t ) = W p ′ , d t p , \gamma_{p,d}(t)=\mathcal{W}_{p',d}t^p, γ p , d ( t ) = W p ′ , d t p , which depends on the matrix dimension d d d d d d γ p , d \gamma_{p,d} γ p , d d d d
The factorization theorem now transfers to N \mathcal{N} N ∣ v 1 / 2 w 1 / 2 p ∣ 2 r |v^{1/2}w^{1/2p}|^{2r} ∣ v 1/2 w 1/2 p ∣ 2 r r < p r<p r < p ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r |v^{-1/2}w^{1/2p}|^{2r} ∣ v − 1/2 w 1/2 p ∣ 2 r p < r p<r p < r (50) and (65) .
Theorem 5.3. Let 1 < p , r < ∞ 1<p,r<\infty 1 < p , r < ∞ p ≠ r p\ne r p = r s > 1 s>1 s > 1 1 / s = ∣ 1 / p − 1 / r ∣ 1/s=|1/p-1/r| 1/ s = ∣1/ p − 1/ r ∣ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) u ∈ L s + ( N ) u\in L_s^+(\mathcal{N}) u ∈ L s + ( N ) v ∈ L s + ( N ) v\in L_s^+(\mathcal{N}) v ∈ L s + ( N )
(5.3) u ≤ v u\leq v u ≤ v ∥ v ∥ s ≤ 2 ∥ u ∥ s \|v\|_s\leq2\|u\|_s ∥ v ∥ s ≤ 2∥ u ∥ s
and
(5.4)
{ ∣ v 1 / 2 w 1 / 2 p ∣ 2 r ∈ A r ( N ) , 1 < r < p , ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ∈ A r ( N ) , 1 < p < r . \begin{cases}
|v^{1/2}w^{1/2p}|^{2r}\in A_r(\mathcal{N}), & 1<r<p,\\
|v^{-1/2}w^{1/2p}|^{2r}\in A_r(\mathcal{N}), & 1<p<r.
\end{cases} { ∣ v 1/2 w 1/2 p ∣ 2 r ∈ A r ( N ) , ∣ v − 1/2 w 1/2 p ∣ 2 r ∈ A r ( N ) , 1 < r < p , 1 < p < r . Quantitatively,
(5.5)
∣ v 1 / 2 w 1 / 2 p ∣ A r ( N ) 2 r ≤ ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) , 1 < r < p , |v^{1/2}w^{1/2p}|_{A_r(\mathcal{N})}^{2r}
\leq(4\mathcal{W}_{p',d})^{(p-r)/(p-1)}[w]_{A_p(\mathcal{N})},
\qquad1<r<p, ∣ v 1/2 w 1/2 p ∣ A r ( N ) 2 r ≤ ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) , 1 < r < p , ∣ v − 1 / 2 w 1 / 2 p ∣ A r ( N ) 2 r ≤ ( 4 W p , d ) ( r − 1 ) / ( p − 1 ) [ w ] A p ( N ) , 1 < p < r , |v^{-1/2}w^{1/2p}|_{A_r(\mathcal{N})}^{2r}
\leq(4\mathcal{W}_{p,d})^{(r-1)/(p-1)}[w]_{A_p(\mathcal{N})},
\qquad1<p<r, ∣ v − 1/2 w 1/2 p ∣ A r ( N ) 2 r ≤ ( 4 W p , d ) ( r − 1 ) / ( p − 1 ) [ w ] A p ( N ) , 1 < p < r , where W p , d \mathcal{W}_{p,d} W p , d (65) .
Proof. On the one hand, suppose first that p < r p<r p < r ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r |v^{-1/2}w^{1/2p}|^{2r} ∣ v − 1/2 w 1/2 p ∣ 2 r J ∈ I J\in\mathcal{I} J ∈ I
E n : = ( − n , n ) ∩ { t : ∥ ( v − 1 / 2 w 1 / 2 p ) ( t ) ∥ S d ∞ ≤ n , ∥ ( w − 1 / 2 p v 1 / 2 ) ( t ) ∥ S d ∞ ≤ n } , E_n:=(-n,n)\cap\left\{t:\|(v^{-1/2}w^{1/2p})(t)\|_{S_d^\infty}\leq n,\ \|(w^{-1/2p}v^{1/2})(t)\|_{S_d^\infty}\leq n\right\}, E n := ( − n , n ) ∩ { t : ∥ ( v − 1/2 w 1/2 p ) ( t ) ∥ S d ∞ ≤ n , ∥ ( w − 1/2 p v 1/2 ) ( t ) ∥ S d ∞ ≤ n } , and
f n : = χ J ∩ E n v − 1 / 2 w 1 / p v − 1 / 2 . f_n:=\chi_{J\cap E_n}v^{-1/2}w^{1/p}v^{-1/2}. f n := χ J ∩ E n v − 1/2 w 1/ p v − 1/2 . Then f n f_n f n f n ∈ S N c ⊂ L r ( N ) f_n\in S_{\mathcal{N}}^c\subset L_r(\mathcal{N}) f n ∈ S N c ⊂ L r ( N )
∥ sup I ∈ I + v − 1 / 2 w 1 / 2 p A I ( w − 1 / 2 p v 1 / 2 f v 1 / 2 w − 1 / 2 p ) w 1 / 2 p v − 1 / 2 ∥ r ≲ ∥ f ∥ r , f ∈ S N c . \left\|\sup_{I\in\mathcal{I}}^+v^{-1/2}w^{1/2p}A_I(w^{-1/2p}v^{1/2}fv^{1/2}w^{-1/2p})w^{1/2p}v^{-1/2}\right\|_r
\lesssim\|f\|_r,\qquad f\in S_{\mathcal{N}}^c. I ∈ I sup + v − 1/2 w 1/2 p A I ( w − 1/2 p v 1/2 f v 1/2 w − 1/2 p ) w 1/2 p v − 1/2 r ≲ ∥ f ∥ r , f ∈ S N c . In particular, we have
∥ v − 1 / 2 w 1 / 2 p A J ( w − 1 / 2 p v 1 / 2 f n v 1 / 2 w − 1 / 2 p ) w 1 / 2 p v − 1 / 2 ∥ r ≲ ∥ f n ∥ r < ∞ . (67) \left\|v^{-1/2}w^{1/2p}A_J(w^{-1/2p}v^{1/2}f_nv^{1/2}w^{-1/2p})w^{1/2p}v^{-1/2}\right\|_r
\lesssim\|f_n\|_r<\infty. \tag*{(67)} v − 1/2 w 1/2 p A J ( w − 1/2 p v 1/2 f n v 1/2 w − 1/2 p ) w 1/2 p v − 1/2 r ≲ ∥ f n ∥ r < ∞. (67) The left hand side of (67) is exactly
∣ J ∩ E n ∣ ∣ J ∣ ( ∫ J tr ( ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ( t ) ) d t ) 1 / r \frac{|J \cap E_n|}{|J|}\left(\int_J \operatorname{tr}(|v^{-1/2}w^{1/2p}|^{2r}(t))\,dt\right)^{1/r} ∣ J ∣ ∣ J ∩ E n ∣ ( ∫ J tr ( ∣ v − 1/2 w 1/2 p ∣ 2 r ( t )) d t ) 1/ r Taking n n n ∣ J ∩ E n ∣ > 0 |J \cap E_n|>0 ∣ J ∩ E n ∣ > 0 r < p r<p r < p
On the other hand, we improve the quantitative estimate. For p < r p<r p < r Theorem 4.4 to construct the factorization v v v I ∈ I I\in\mathcal{I} I ∈ I F I = { I } \mathcal{F}_I=\{I\} F I = { I } Lemma 3.8 gives the estimate on ℜ z = 0 \Re z=0 ℜ z = 0
∥ T m , n ( i t ) f ∥ L p ( N ; ℓ ∞ ( F I ) ) = ∥ e m v 0 − i t / 2 w 1 / 2 p A I ( w − 1 / 2 p v 0 i t / 2 e n f e n v 0 i t / 2 w − 1 / 2 p ) w 1 / 2 p v 0 − i t / 2 e m ∥ p ≤ [ w ] A p ( N ) 1 / p ∥ f ∥ p . \|\mathcal{T}_{m,n}(it)f\|_{L^p(\mathcal{N};\ell^\infty(\mathcal{F}_I))}
=\|e_m v_0^{-it/2}w^{1/2p}A_I(w^{-1/2p}v_0^{it/2}e_nf e_nv_0^{it/2}w^{-1/2p})w^{1/2p}v_0^{-it/2}e_m\|_p
\le[w]_{A_p(\mathcal{N})}^{1/p}\|f\|_p. ∥ T m , n ( i t ) f ∥ L p ( N ; ℓ ∞ ( F I )) = ∥ e m v 0 − i t /2 w 1/2 p A I ( w − 1/2 p v 0 i t /2 e n f e n v 0 i t /2 w − 1/2 p ) w 1/2 p v 0 − i t /2 e m ∥ p ≤ [ w ] A p ( N ) 1/ p ∥ f ∥ p . Similar to the proof of (59) , we have the estimate on ℜ z = 1 \Re z=1 ℜ z = 1
∥ T m , n ( 1 + i t ) f ∥ L ∞ ( N ; ℓ ∞ ( F I ) ) ≤ 4 [ w ] W p ( ( A I ) I ) ∥ f ∥ ∞ ≤ 4 W p , d [ w ] A p ( N ) 1 / ( p − 1 ) ∥ f ∥ ∞ , \|\mathcal{T}_{m,n}(1+it)f\|_{L^\infty(\mathcal{N};\ell^\infty(\mathcal{F}_I))}
\le4[w]W_p((A_I)_I)\|f\|_\infty
\le4W_{p,d}[w]_{A_p(\mathcal{N})}^{1/(p-1)}\|f\|_\infty, ∥ T m , n ( 1 + i t ) f ∥ L ∞ ( N ; ℓ ∞ ( F I )) ≤ 4 [ w ] W p (( A I ) I ) ∥ f ∥ ∞ ≤ 4 W p , d [ w ] A p ( N ) 1/ ( p − 1 ) ∥ f ∥ ∞ , where the last inequality comes from (65) . Interpolation at θ = 1 − p / r \theta=1-p/r θ = 1 − p / r
∥ T m , n ( θ ) f ∥ r ≤ [ w ] A p ( N ) 1 / r ( 4 W p , d [ w ] A p ( N ) 1 / ( p − 1 ) ) 1 − p / r ∥ f ∥ r . \|\mathcal{T}_{m,n}(\theta)f\|_r
\le[w]_{A_p(\mathcal{N})}^{1/r}\left(4W_{p,d}[w]_{A_p(\mathcal{N})}^{1/(p-1)}\right)^{1-p/r}\|f\|_r. ∥ T m , n ( θ ) f ∥ r ≤ [ w ] A p ( N ) 1/ r ( 4 W p , d [ w ] A p ( N ) 1/ ( p − 1 ) ) 1 − p / r ∥ f ∥ r . Passing to the cutoff limits as in the proof of Theorem 4.4 , and using Lemma 4.3 and Lemma 3.8 again, gives
[ ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ] A r ( N ) ≤ [ w ] A p ( N ) ( 4 W p , d [ w ] A p ( N ) 1 / ( p − 1 ) ) r − p = ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) . [|v^{-1/2}w^{1/2p}|^{2r}]_{A_r(\mathcal{N})}
\le[w]_{A_p(\mathcal{N})}\left(4W_{p,d}[w]_{A_p(\mathcal{N})}^{1/(p-1)}\right)^{r-p}
=(4W_{p,d})^{r-p}[w]_{A_p(\mathcal{N})}^{(r-1)/(p-1)}. [ ∣ v − 1/2 w 1/2 p ∣ 2 r ] A r ( N ) ≤ [ w ] A p ( N ) ( 4 W p , d [ w ] A p ( N ) 1/ ( p − 1 ) ) r − p = ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) . If r < p r<p r < p
[ ∣ v 1 / 2 w 1 / 2 p ∣ 2 r ] A r ( N ) = [ ∣ v − 1 / 2 w − 1 / 2 p ∣ 2 r ′ ] A r ′ ( N ) r / r ′ ≤ ( ( 4 W p ′ , d ) r ′ − p ′ [ w − p ′ / p ] A p ′ ( N ) ( r ′ − 1 ) / ( p ′ − 1 ) ) r / r ′ = ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) . \begin{aligned}
[|v^{1/2}w^{1/2p}|^{2r}]_{A_r(\mathcal{N})}
=[|v^{-1/2}w^{-1/2p}|^{2r'}]_{A_{r'}(\mathcal{N})}^{r/r'} \\
&\le\left((4W_{p',d})^{r'-p'}[w^{-p'/p}]_{A_{p'}(\mathcal{N})}^{(r'-1)/(p'-1)}\right)^{r/r'} \\
&=(4W_{p',d})^{(p-r)/(p-1)}[w]_{A_p(\mathcal{N})}.
\end{aligned} [ ∣ v 1/2 w 1/2 p ∣ 2 r ] A r ( N ) = [ ∣ v − 1/2 w − 1/2 p ∣ 2 r ′ ] A r ′ ( N ) r / r ′ ≤ ( ( 4 W p ′ , d ) r ′ − p ′ [ w − p ′ / p ] A p ′ ( N ) ( r ′ − 1 ) / ( p ′ − 1 ) ) r / r ′ = ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) . where the first and last equalities use (35) . This proves (5.5). □
Remark 5.4. The quantitative estimate (5.5) is better than the direct combination of (50) and (65) . Indeed, (50) and (65) give
[ ∣ v 1 / 2 w 1 / 2 p ∣ 2 r ] A r ( N ) ≤ 4 r ( r ′ − p ′ ) W r , d W p ′ , d r r ′ W p , d r r ′ [ w ] A p ( N ) r r ′ / ( p − 1 ) , 1 < r < p [|v^{1/2}w^{1/2p}|^{2r}]_{A_r(\mathcal{N})}
\le4^{r(r'-p')}W_{r,d}W_{p',d}^{rr'}W_{p,d}^{rr'}[w]_{A_p(\mathcal{N})}^{rr'/(p-1)},
\qquad1<r<p [ ∣ v 1/2 w 1/2 p ∣ 2 r ] A r ( N ) ≤ 4 r ( r ′ − p ′ ) W r , d W p ′ , d r r ′ W p , d r r ′ [ w ] A p ( N ) r r ′ / ( p − 1 ) , 1 < r < p [ ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ] A r ( N ) ≤ 4 r − p W r , d W p , d r [ w ] A p ( N ) r / ( p − 1 ) , 1 < p < r . [|v^{-1/2}w^{1/2p}|^{2r}]_{A_r(\mathcal{N})}
\le4^{r-p}W_{r,d}W_{p,d}^{r}[w]_{A_p(\mathcal{N})}^{r/(p-1)},
\qquad1<p<r. [ ∣ v − 1/2 w 1/2 p ∣ 2 r ] A r ( N ) ≤ 4 r − p W r , d W p , d r [ w ] A p ( N ) r / ( p − 1 ) , 1 < p < r . And it is easily verified that p r r ′ / ( p − 1 ) ≥ 1 prr'/(p-1)\ge1 p r r ′ / ( p − 1 ) ≥ 1 1 < r < p 1<r<p 1 < r < p r / ( p − 1 ) ≥ ( r − 1 ) / ( p − 1 ) r/(p-1)\ge(r-1)/(p-1) r / ( p − 1 ) ≥ ( r − 1 ) / ( p − 1 ) 1 < p < r 1<p<r 1 < p < r
Matrix-weighted extrapolation
Proof of Theorem 1.2. Let w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) f ∈ S N c f\in\mathcal{S}_{\mathcal{N}}^c f ∈ S N c Theorem 4.5 , using Theorem 5.3 for factorization and Lemma 2.10 for the Hölder type testing estimate. The factorized weights belong to A r ( N ) A_r(\mathcal{N}) A r ( N ) f f f L r L_r L r (5) applies.
Suppose first that p < r p<r p < r Theorem 5.3 , the weight produced by factorization satisfies
[ ∣ v − 1 / 2 w 1 / 2 p ∣ 2 r ] A r ( N ) ≤ ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) . [|v^{-1/2}w^{1/2p}|^{2r}]_{A_r(\mathcal{N})}
\le(4W_{p,d})^{r-p}[w]_{A_p(\mathcal{N})}^{(r-1)/(p-1)}. [ ∣ v − 1/2 w 1/2 p ∣ 2 r ] A r ( N ) ≤ ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) . Applying (5) with this weight and completing the Hölder type testing argument using Lemma 2.10 (i), as in the proof of Theorem 4.5 , gives
∥ sup α ∈ A + T α ( f ) ∥ L p ( N , w ) ≤ 4 Φ ( ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) ) ∥ f ∥ L p ( N , w ) . \left\|\sup_{\alpha\in\mathcal{A}}^{+}T_\alpha(f)\right\|_{L_p(\mathcal{N},w)}
\le4\Phi\left((4W_{p,d})^{r-p}[w]_{A_p(\mathcal{N})}^{(r-1)/(p-1)}\right)\|f\|_{L_p(\mathcal{N},w)}. α ∈ A sup + T α ( f ) L p ( N , w ) ≤ 4Φ ( ( 4 W p , d ) r − p [ w ] A p ( N ) ( r − 1 ) / ( p − 1 ) ) ∥ f ∥ L p ( N , w ) . Suppose now that r < p r < p r < p
[ v 1 / 2 w 1 / 2 p ] 2 r [ w ] A r ( N ) ≤ ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) . \left[v^{1/2}w^{1/2p}\right]_{2r} [w]_{A_r(\mathcal{N})} \leq(4\mathcal{W}_{p',d})^{(p-r)/(p-1)}[w]_{A_p(\mathcal{N})}. [ v 1/2 w 1/2 p ] 2 r [ w ] A r ( N ) ≤ ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) . Applying (5) with this weight, using Lemma 2.10(ii) as in the proof of Theorem 4.5, yields
∥ sup α ∈ A + T α ( f ) ∥ L p ( N , w ) ≤ 8 Φ ( ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) . \left\|\sup_{\alpha\in\mathcal{A}}^{+}T_\alpha(f)\right\|_{L_p(\mathcal{N},w)}
\leq8\Phi\left((4\mathcal{W}_{p',d})^{(p-r)/(p-1)}[w]_{A_p(\mathcal{N})}\right)\|f\|_{L_p(\mathcal{N},w)}. α ∈ A sup + T α ( f ) L p ( N , w ) ≤ 8Φ ( ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) [ w ] A p ( N ) ) ∥ f ∥ L p ( N , w ) . Finally, S N c \mathcal{S}_{\mathcal{N}}^c S N c L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) f ↦ ( T α ( f ) ) α ∈ A f\mapsto(T_\alpha(f))_{\alpha\in\mathcal{A}} f ↦ ( T α ( f ) ) α ∈ A L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w )
Thus we complete the proof by choosing the increasing function Φ p , r , d \Phi_{p,r,d} Φ p , r , d
Φ p , r , d ( t ) = { 4 Φ ( ( 4 W p ′ , d ) r − p t ( r − 1 ) / ( p − 1 ) ) , 1 < p ≤ r , 8 Φ ( ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) t ) , 1 < r < p . (68) \Phi_{p,r,d}(t)=
\begin{cases}
4\Phi\left((4\mathcal{W}_{p',d})^{r-p}t^{(r-1)/(p-1)}\right), & 1<p\leq r,\\
8\Phi\left((4\mathcal{W}_{p',d})^{(p-r)/(p-1)}t\right), & 1<r<p.
\end{cases} \tag*{(68)} Φ p , r , d ( t ) = { 4Φ ( ( 4 W p ′ , d ) r − p t ( r − 1 ) / ( p − 1 ) ) , 8Φ ( ( 4 W p ′ , d ) ( p − r ) / ( p − 1 ) t ) , 1 < p ≤ r , 1 < r < p . (68) Remark 5.5. There is an alternative finite-corner proof. One may work on
N m = L ∞ ( − m , m ) ⊗ ‾ M d ( C ) , \mathcal{N}_m=L_\infty(-m,m)\mathbin{\overline{\otimes}}M_d(\mathbb{C}), N m = L ∞ ( − m , m ) ⊗ M d ( C ) , extend a local matrix weight to the line by translation and reflection, apply the trace-finite theorem with constants independent of m m m A p A_p A p
Remark 5.6. The dependence on d d d W p , d \mathcal{W}_{p,d} W p , d N \mathcal{N} N
Applications
We apply Theorem 1.2 to the sharp truncations of Calderón–Zygmund operators acting entrywise on matrix-valued functions. The starting estimate at exponent 2 follows from the classical Cotlar inequality and existing matrix-weighted estimates.
Let T T T L 2 ( R ) L_2(\mathbb{R}) L 2 ( R ) K : { ( t , s ) ∈ R 2 : t ≠ s } → C K:\{(t,s)\in\mathbb{R}^2:t\ne s\}\to\mathbb{C} K : {( t , s ) ∈ R 2 : t = s } → C
∣ K ( t , s ) ∣ ≤ C K ∣ t − s ∣ , |K(t,s)|\leq\frac{C_K}{|t-s|}, ∣ K ( t , s ) ∣ ≤ ∣ t − s ∣ C K , ∣ K ( t , s ) − K ( t ′ , s ) ∣ + ∣ K ( s , t ) − K ( s , t ′ ) ∣ ≤ C K ∣ t − t ′ ∣ γ ∣ t − s ∣ 1 + γ , ∣ t − t ′ ∣ ≤ ∣ t − s ∣ 2 . (69) |K(t,s)-K(t',s)|+|K(s,t)-K(s,t')|
\leq C_K\frac{|t-t'|^\gamma}{|t-s|^{1+\gamma}},
\qquad|t-t'|\leq\frac{|t-s|}{2}. \tag*{(69)} ∣ K ( t , s ) − K ( t ′ , s ) ∣ + ∣ K ( s , t ) − K ( s , t ′ ) ∣ ≤ C K ∣ t − s ∣ 1 + γ ∣ t − t ′ ∣ γ , ∣ t − t ′ ∣ ≤ 2 ∣ t − s ∣ . (69) where C K > 0 C_K>0 C K > 0 0 < γ < 1 0<\gamma<1 0 < γ < 1
T f ( t ) = ∫ R K ( t , s ) f ( s ) d s , t ∉ supp f . Tf(t)=\int_{\mathbb{R}}K(t,s)f(s)\,ds,\qquad t\notin\operatorname{supp}f. T f ( t ) = ∫ R K ( t , s ) f ( s ) d s , t ∈ / supp f . For ε > 0 \varepsilon>0 ε > 0
T ε f ( t ) = ∫ ∣ t − s ∣ > ε K ( t , s ) f ( s ) d s , f ∈ S N c . (70) T_\varepsilon f(t)=\int_{|t-s|>\varepsilon}K(t,s)f(s)\,ds,\qquad f\in\mathcal{S}_{\mathcal{N}}^c. \tag*{(70)} T ε f ( t ) = ∫ ∣ t − s ∣ > ε K ( t , s ) f ( s ) d s , f ∈ S N c . (70) Recall the operator weight on S 2 d \mathcal{S}_2^d S 2 d § 3 :
W ( t ) ( A ) = w ( t ) 1 / 2 A w ( t ) 1 / 2 , W ( t ) 1 / 2 ( A ) = w ( t ) 1 / 4 A w ( t ) 1 / 4 . \mathsf{W}(t)(A)=w(t)^{1/2}Aw(t)^{1/2},\qquad
\mathsf{W}(t)^{1/2}(A)=w(t)^{1/4}Aw(t)^{1/4}. W ( t ) ( A ) = w ( t ) 1/2 A w ( t ) 1/2 , W ( t ) 1/2 ( A ) = w ( t ) 1/4 A w ( t ) 1/4 . At exponent 2, the identification S 2 d ≃ ℓ 2 d 2 \mathcal{S}_2^d\simeq\ell_2^{d^2} S 2 d ≃ ℓ 2 d 2
∥ W 1 / 2 f ∥ L 2 ( R ; S 2 d ) = ∥ f ∥ L 2 ( N , w ) (71) \|\mathsf{W}^{1/2}f\|_{L_2(\mathbb{R};\mathcal{S}_2^d)}
=\|f\|_{L_2(\mathcal{N},w)} \tag*{(71)} ∥ W 1/2 f ∥ L 2 ( R ; S 2 d ) = ∥ f ∥ L 2 ( N , w ) (71) and
[ W ] A 2 ( R ; M d 2 ( C ) ) = [ w ] A 2 ( N ) (72) [\mathsf{W}]_{A_2(\mathbb{R};M_{d^2}(\mathbb{C}))}
=[w]_{A_2(\mathcal{N})} \tag*{(72)} [ W ] A 2 ( R ; M d 2 ( C )) = [ w ] A 2 ( N ) (72) The Christ–Goldberg maximal operator at exponent 2 is
M W g ( t ) = sup I ∋ t 1 ∣ I ∣ ∫ I ∥ W ( t ) 1 / 2 W ( s ) − 1 / 2 g ( s ) ∥ S 2 d d s . M_{\mathsf{W}}g(t)=\sup_{I\ni t}\frac{1}{|I|}\int_I
\|\mathsf{W}(t)^{1/2}\mathsf{W}(s)^{-1/2}g(s)\|_{\mathcal{S}_2^d}\,ds. M W g ( t ) = I ∋ t sup ∣ I ∣ 1 ∫ I ∥ W ( t ) 1/2 W ( s ) − 1/2 g ( s ) ∥ S 2 d d s . Lemma 5.7. Let w ∈ A 2 ( N ) w \in A_2(\mathcal{N}) w ∈ A 2 ( N )
∥ sup ε > 0 + T ε f ∥ L 2 ( N , w ) ≤ C d , γ C T [ w ] A 2 ( N ) 5 / 2 ∥ f ∥ L 2 ( N , w ) , f ∈ S N c . (73) \left\|\sup_{\varepsilon>0}^{+}T_\varepsilon f\right\|_{L_2(\mathcal{N},w)} \leq C_{d,\gamma}C_T[w]_{A_2(\mathcal{N})}^{5/2}\|f\|_{L_2(\mathcal{N},w)}, \qquad f\in\mathcal{S}_{\mathcal{N}}^c. \tag*{(73)} ε > 0 sup + T ε f L 2 ( N , w ) ≤ C d , γ C T [ w ] A 2 ( N ) 5/2 ∥ f ∥ L 2 ( N , w ) , f ∈ S N c . (73) Here C T = ∥ T ∥ L 2 ( R ) → L 2 ( R ) + C K C_T=\|T\|_{L_2(\mathbb{R})\to L_2(\mathbb{R})}+C_K C T = ∥ T ∥ L 2 ( R ) → L 2 ( R ) + C K C d , γ C_{d,\gamma} C d , γ d d d γ \gamma γ
Proof. Firstly, we show that
∥ sup ε > 0 + T ε f ∥ L 2 ( N , w ) ≤ C d , γ ′ [ w ] A 2 ( N ) ( ∥ T f ∥ L 2 ( N , w ) + C T ∥ f ∥ L 2 ( N , w ) ) . (74) \left\|\sup_{\varepsilon>0}^{+}T_\varepsilon f\right\|_{L_2(\mathcal{N},w)} \leq C'_{d,\gamma}[w]_{A_2(\mathcal{N})}\left(\|Tf\|_{L_2(\mathcal{N},w)}+C_T\|f\|_{L_2(\mathcal{N},w)}\right). \tag*{(74)} ε > 0 sup + T ε f L 2 ( N , w ) ≤ C d , γ ′ [ w ] A 2 ( N ) ( ∥ T f ∥ L 2 ( N , w ) + C T ∥ f ∥ L 2 ( N , w ) ) . (74) The classical Cotlar inequality [17 (1) and Section 3.3, Theorem 5] , applied to T / C T T/C_T T / C T
sup ε > 0 ∣ T ε f ( t ) ∣ ≤ C γ M ( ∣ T f ∣ ) ( t ) + C γ C T M ( ∣ f ∣ ) ( t ) (75) \sup_{\varepsilon>0}|T_\varepsilon f(t)|\leq C_\gamma M(|Tf|)(t)+C_\gamma C_T M(|f|)(t) \tag*{(75)} ε > 0 sup ∣ T ε f ( t ) ∣ ≤ C γ M ( ∣ T f ∣ ) ( t ) + C γ C T M ( ∣ f ∣ ) ( t ) (75) for scalar inputs, where M M M g ∈ S N c g\in\mathcal{S}_{\mathcal{N}}^c g ∈ S N c A ∈ M d ( C ) A\in M_d(\mathbb{C}) A ∈ M d ( C ) (75) to the scalar-valued function t ↦ tr ( A ∗ g ( t ) ) t\mapsto\operatorname{tr}(A^*g(t)) t ↦ tr ( A ∗ g ( t ))
sup ε > 0 ∣ tr ( A ∗ T ε g ( t ) ) ∣ ≤ C γ M ( ∣ tr ( A ∗ ( T g ) ) ∣ ) ( t ) + C γ C T M ( ∣ tr ( A ∗ g ) ∣ ) ( t ) ≤ C γ ∥ A ∥ S 2 d M ( ∥ T g ∥ S 2 d ) ( t ) + C γ C T ∥ A ∥ S 2 d M ( ∥ g ∥ S 2 d ) ( t ) . \begin{aligned}
\sup_{\varepsilon>0}|\operatorname{tr}(A^*T_\varepsilon g(t))|\leq C_\gamma M(|\operatorname{tr}(A^*(Tg))|)(t)+C_\gamma C_T M(|\operatorname{tr}(A^*g)|)(t) \\
&\leq C_\gamma\|A\|_{S_2^d}M(\|Tg\|_{S_2^d})(t)+C_\gamma C_T\|A\|_{S_2^d}M(\|g\|_{S_2^d})(t).
\end{aligned} ε > 0 sup ∣ tr ( A ∗ T ε g ( t )) ∣ ≤ C γ M ( ∣ tr ( A ∗ ( T g )) ∣ ) ( t ) + C γ C T M ( ∣ tr ( A ∗ g ) ∣ ) ( t ) ≤ C γ ∥ A ∥ S 2 d M ( ∥ T g ∥ S 2 d ) ( t ) + C γ C T ∥ A ∥ S 2 d M ( ∥ g ∥ S 2 d ) ( t ) . Taking the supremum over ∥ A ∥ S 2 d ≤ 1 \|A\|_{S_2^d}\leq1 ∥ A ∥ S 2 d ≤ 1
sup ε > 0 ∥ T ε g ( t ) ∥ S 2 d ≤ C γ M ( ∥ T g ∥ S 2 d ) ( t ) + C γ C T M ( ∥ g ∥ S 2 d ) ( t ) . \sup_{\varepsilon>0}\|T_\varepsilon g(t)\|_{S_2^d}\leq C_\gamma M(\|Tg\|_{S_2^d})(t)+C_\gamma C_T M(\|g\|_{S_2^d})(t). ε > 0 sup ∥ T ε g ( t ) ∥ S 2 d ≤ C γ M ( ∥ T g ∥ S 2 d ) ( t ) + C γ C T M ( ∥ g ∥ S 2 d ) ( t ) . Fix t ∈ R t\in\mathbb{R} t ∈ R g : u ↦ w ( t ) 1 / 4 f ( u ) w ( t ) 1 / 4 g:u\mapsto w(t)^{1/4}f(u)w(t)^{1/4} g : u ↦ w ( t ) 1/4 f ( u ) w ( t ) 1/4
sup ε > 0 ∥ W 1 / 2 ( t ) T ε f ( t ) ∥ S 2 d ≤ C γ M W ( W 1 / 2 T f ) ( t ) + C γ C T M W ( W 1 / 2 f ) ( t ) . (76) \sup_{\varepsilon>0}\|W^{1/2}(t)T_\varepsilon f(t)\|_{S_2^d}\leq C_\gamma M_W(W^{1/2}Tf)(t)+C_\gamma C_T M_W(W^{1/2}f)(t). \tag*{(76)} ε > 0 sup ∥ W 1/2 ( t ) T ε f ( t ) ∥ S 2 d ≤ C γ M W ( W 1/2 T f ) ( t ) + C γ C T M W ( W 1/2 f ) ( t ) . (76) Lemma 3.1 gives Goldberg’s quantitative maximal inequality
∥ M W g ∥ L 2 ( R ) ≤ G 2 , d 2 [ W ] A 2 ∥ g ∥ L 2 ( R ; S 2 d ) . (77) \|M_Wg\|_{L_2(\mathbb{R})}\leq G_{2,d^2}[W]_{A_2}\|g\|_{L_2(\mathbb{R};S_2^d)}. \tag*{(77)} ∥ M W g ∥ L 2 ( R ) ≤ G 2 , d 2 [ W ] A 2 ∥ g ∥ L 2 ( R ; S 2 d ) . (77) Moreover, Lemma 3.9 gives
∥ sup ε > 0 + T ε f ∥ L 2 ( N , w ) ≤ d ∥ sup ε > 0 ∥ W 1 / 2 T ε f ∥ S 2 d ∥ L 2 ( R ) . (78) \left\|\sup_{\varepsilon>0}^{+}T_\varepsilon f\right\|_{L_2(\mathcal{N},w)}
\leq\sqrt{d}\left\|\sup_{\varepsilon>0}\|W^{1/2}T_\varepsilon f\|_{S_2^d}\right\|_{L_2(\mathbb{R})}. \tag*{(78)} ε > 0 sup + T ε f L 2 ( N , w ) ≤ d ε > 0 sup ∥ W 1/2 T ε f ∥ S 2 d L 2 ( R ) . (78) Thus combining (71) , (72) , and (76) –(78) proves (74) with constant C d , γ ′ = d ⋅ G 2 , d 2 C γ C'_{d,\gamma}=\sqrt{d}\cdot G_{2,d^2}C_\gamma C d , γ ′ = d ⋅ G 2 , d 2 C γ
Finally, the matrix A 2 A_2 A 2 [20 Theorem 1.1 and Remark 1.3] , applied to W W W S 2 d S_2^d S 2 d
∥ T f ∥ L 2 ( N , w ) ≤ C d , γ ′ ′ C T [ w ] A 2 ( N ) 3 / 2 ∥ f ∥ L 2 ( N , w ) . (79) \|Tf\|_{L_2(\mathcal{N},w)}\leq C''_{d,\gamma}C_T[w]_{A_2(\mathcal{N})}^{3/2}\|f\|_{L_2(\mathcal{N},w)}. \tag*{(79)} ∥ T f ∥ L 2 ( N , w ) ≤ C d , γ ′′ C T [ w ] A 2 ( N ) 3/2 ∥ f ∥ L 2 ( N , w ) . (79) Substituting (79) into (74) , and using [ w ] A 2 ( N ) ≥ 1 [w]_{A_2(\mathcal{N})}\geq1 [ w ] A 2 ( N ) ≥ 1
∥ sup ε > 0 + T ε f ∥ L 2 ( N , w ) ≤ C d , γ ′ C T ( C d , γ ′ ′ [ w ] A 2 ( N ) 5 / 2 + [ w ] A 2 ( N ) ) ∥ f ∥ L 2 ( N , w ) ≤ C d , γ C T [ w ] A 2 ( N ) 5 / 2 ∥ f ∥ L 2 ( N , w ) , \begin{aligned}
\left\|\sup_{\varepsilon>0}^{+}T_\varepsilon f\right\|_{L_2(\mathcal{N},w)}
\leq C'_{d,\gamma}C_T\left(C''_{d,\gamma}[w]_{A_2(\mathcal{N})}^{5/2}+[w]_{A_2(\mathcal{N})}\right)\|f\|_{L_2(\mathcal{N},w)} \\
&\leq C_{d,\gamma}C_T[w]_{A_2(\mathcal{N})}^{5/2}\|f\|_{L_2(\mathcal{N},w)},
\end{aligned} ε > 0 sup + T ε f L 2 ( N , w ) ≤ C d , γ ′ C T ( C d , γ ′′ [ w ] A 2 ( N ) 5/2 + [ w ] A 2 ( N ) ) ∥ f ∥ L 2 ( N , w ) ≤ C d , γ C T [ w ] A 2 ( N ) 5/2 ∥ f ∥ L 2 ( N , w ) , where C d , γ = C d , γ ′ ( C d , γ ′ ′ + 1 ) C_{d,\gamma}=C'_{d,\gamma}(C''_{d,\gamma}+1) C d , γ = C d , γ ′ ( C d , γ ′′ + 1 ) (73) . □
Theorem 5.8. Let T T T 1 < p < ∞ 1<p<\infty 1 < p < ∞ w ∈ A p ( N ) w\in A_p(\mathcal{N}) w ∈ A p ( N ) f ↦ ( T ε f ) ε > 0 f\mapsto(T_\varepsilon f)_{\varepsilon>0} f ↦ ( T ε f ) ε > 0 S N c \mathcal{S}_{\mathcal{N}}^c S N c L p ( N , w ) L_p(\mathcal{N},w) L p ( N , w ) L p ( N , w ; ℓ ∞ ) L_p(\mathcal{N},w;\ell_\infty) L p ( N , w ; ℓ ∞ )
∥ sup ε > 0 + T ε f ∥ L p ( N , w ) ≤ C p , d , γ C T [ w ] A p ( N ) 5 2 max { 1 , 1 p − 1 } ∥ f ∥ L p ( N , w ) . (80) \left\|\sup_{\varepsilon>0}^{+}T_\varepsilon f\right\|_{L_p(\mathcal{N},w)}
\leq C_{p,d,\gamma}C_T[w]_{A_p(\mathcal{N})}^{\frac{5}{2}\max\left\{1,\frac{1}{p-1}\right\}}\|f\|_{L_p(\mathcal{N},w)}. \tag*{(80)} ε > 0 sup + T ε f L p ( N , w ) ≤ C p , d , γ C T [ w ] A p ( N ) 2 5 m a x { 1 , p − 1 1 } ∥ f ∥ L p ( N , w ) . (80) Proof. Lemma 5.7 verifies the hypothesis of Theorem 1.2 for the linear family ( T ε ) ε > 0 (T_\varepsilon)_{\varepsilon>0} ( T ε ) ε > 0 r = 2 r=2 r = 2 Φ ( t ) = C d , γ C T t 5 / 2 \Phi(t)=C_{d,\gamma}C_Tt^{5/2} Φ ( t ) = C d , γ C T t 5/2 (68) gives
Φ p , 2 , d ( t ) = 4 C d , γ C T ( 4 W p , d ) 5 2 ( 2 − p ) t 5 2 ( p − 1 ) , 1 < p < 2 , \Phi_{p,2,d}(t)=4C_{d,\gamma}C_T(4W_{p,d})^{\frac{5}{2}(2-p)}t^{\frac{5}{2(p-1)}}, \qquad1<p<2, Φ p , 2 , d ( t ) = 4 C d , γ C T ( 4 W p , d ) 2 5 ( 2 − p ) t 2 ( p − 1 ) 5 , 1 < p < 2 , Φ p , 2 , d ( t ) = 8 C d , γ C T ( 4 W p , d ′ ) 5 ( p − 2 ) 2 ( p − 1 ) t 5 / 2 , 2 < p < ∞ . \Phi_{p,2,d}(t)=8C_{d,\gamma}C_T(4W'_{p,d})^{\frac{5(p-2)}{2(p-1)}}t^{5/2}, \qquad2<p<\infty. Φ p , 2 , d ( t ) = 8 C d , γ C T ( 4 W p , d ′ ) 2 ( p − 1 ) 5 ( p − 2 ) t 5/2 , 2 < p < ∞. Together with (73) at p = 2 p=2 p = 2 (80) . The density of S N c S^c_{\mathcal N} S N c L p ( N , w ) L_p(\mathcal N,w) L p ( N , w )
Remark 5.9. The proof of Lemma 5.7 reduces the weighted L 2 L_2 L 2 S 2 d S^d_2 S 2 d d d d (73) . This raises the natural question whether the bound is sharp, which in turn motivates the question whether Lemma 5.7 admits a direct proof without reductions.
Open problems
We conclude with several questions about extensions of the symmetric weighted realization and the limitations of the present method.
Dimension dependence
In Lemma 5.1, the symmetry of the weight classes is obtained via
γ p ( t ) = γ p , d ( t ) = W p ′ , d t p \gamma_p(t)=\gamma_{p,d}(t)=\mathcal W_{p',d}t^p γ p ( t ) = γ p , d ( t ) = W p ′ , d t p with W p , d ≤ d 1 p + ( 1 + p ′ ) ∣ 1 p − 1 2 ∣ G p , d 2 \mathcal W_{p,d}\le d^{\frac{1}{p}+(1+p')\left|\frac{1}{p}-\frac{1}{2}\right|}\mathcal G_{p,d}^2 W p , d ≤ d p 1 + ( 1 + p ′ ) ∣ p 1 − 2 1 ∣ G p , d 2 d d d
What is the sharp order of γ p , d \gamma_{p,d} γ p , d d d d d d d
A related question is the sharp order of the bound having appeared in (80) with respect to d d d d d d
Another related question is the existence of another generalization of classical A p A_p A p [29 , 30 ] such that the resulting bounds of the weighted Hardy–Littlewood maximal operator, the weighted Hilbert transform or Calderón–Zygmund operators are independent of d d d
Other questions
In the abstract framework of § 4 , we work with trace-finite von Neumann algebras because our use of Assumption 4.2 involves w w w w − 1 w^{-1} w − 1 w ∈ L 0 ( M ) w\in L_0(\mathcal M) w ∈ L 0 ( M ) L 0 ( M ) L_0(\mathcal M) L 0 ( M )
For the particular semifinite algebra N = L ∞ ( R ) ⊗ ‾ M d ( C ) \mathcal N=L_\infty(\mathbb R)\mathbin{\overline\otimes}M_d(\mathbb C) N = L ∞ ( R ) ⊗ M d ( C ) R \mathbb R R
For a general semifinite von Neumann algebra, however, a positive full-support operator w ∈ L 0 ( M ) w\in L_0(\mathcal M) w ∈ L 0 ( M ) w − 1 ∈ L 0 ( M ) w^{-1}\in L_0(\mathcal M) w − 1 ∈ L 0 ( M )
In § 4 , the factorization proof uses the positivity of the symmetric weighted realization x ↦ w 1 / 2 p x w 1 / 2 p x\mapsto w^{1/2p}xw^{1/2p} x ↦ w 1/2 p x w 1/2 p
x ⟼ w θ / p x w ( 1 − θ ) / p , 0 ≤ θ ≤ 1 , θ ≠ 1 2 ? x\longmapsto w^{\theta/p}xw^{(1-\theta)/p},\qquad0\le\theta\le1,\quad\theta\ne\frac{1}{2}? x ⟼ w θ / p x w ( 1 − θ ) / p , 0 ≤ θ ≤ 1 , θ = 2 1 ? A positive answer would provide an alternative proof of the extrapolation for Treil–Volberg’s matrix-weighted theory. (3) In the classical setting, Rubio de Francia’s extrapolation theorem applies to sublinear operators [25 ] . This includes maximal operators and square functions, two central objects in harmonic analysis. In the noncommutative setting, however, these objects do not admit the same pointwise sublinear formulation. Theorem 1.2 has established extrapolation for maximal estimates. It is therefore natural to ask whether an analogous result holds for square-function estimates. A basic complication is that in the noncommutative case, we may have to deal with the column and row square-function inequalities simultaneously:
∥ ( ∑ n T n ( x ) ∗ T n ( x ) ) 1 / 2 ∥ L p ( M , w ) ≤ C ∥ x ∥ L p ( M , w ) \left\|\left(\sum_n T_n(x)^*T_n(x)\right)^{1/2}\right\|_{L_p(\mathcal{M},w)} \leq C\|x\|_{L_p(\mathcal{M},w)} ( n ∑ T n ( x ) ∗ T n ( x ) ) 1/2 L p ( M , w ) ≤ C ∥ x ∥ L p ( M , w ) ∥ ( ∑ n T n ( x ) T n ( x ) ∗ ) 1 / 2 ∥ L p ( M , w ) ≤ C ∥ x ∥ L p ( M , w ) . \left\|\left(\sum_n T_n(x)T_n(x)^*\right)^{1/2}\right\|_{L_p(\mathcal{M},w)} \leq C\|x\|_{L_p(\mathcal{M},w)}. ( n ∑ T n ( x ) T n ( x ) ∗ ) 1/2 L p ( M , w ) ≤ C ∥ x ∥ L p ( M , w ) . Acknowledgements. The authors would like to thank Marius Junge, Javier Parcet and Éric Ricard for some discussions or suggestions in establishing the abstract factorization and extrapolation theorem in § 4 . This work was partially supported by the National Natural Science Foundation of China (grant Nos. 12325105, W2441002, and W2611005).
AI statement. During the preparation of this manuscript, the authors used GPT only to assist with English-language editing, the organization and presentation of the manuscript, LaTeX formatting, and preliminary checks of mathematical consistency. All mathematical ideas, results, arguments, and proofs were developed and independently verified by the authors. The authors carefully reviewed and edited all AI-assisted suggestions and take full responsibility for the accuracy, originality, and content of the manuscript.
References
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
Email address: gxhong@hit.edu.cn
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
Email address: p.yin@stu.hit.edu.cn
