Bloom Type Inequality: The Off-diagonal Case

Junren Pan; Wenchang Sun

arxiv: 1907.07292 · v1 · pith:AYG7KPTCnew · submitted 2019-07-17 · 🧮 math.CA

Bloom Type Inequality: The Off-diagonal Case

Junren Pan , Wenchang Sun This is my paper

Pith reviewed 2026-05-24 20:23 UTC · model grok-4.3

classification 🧮 math.CA

keywords Bloom inequalityfractional integralscommutatorsmixed-norm spacesweighted spacesproduct BMOoff-diagonal caserepresentation formula

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The pith

The double commutator of two fractional integrals is bounded from weighted mixed-norm L^p to L^q by a multiple of the product BMO norm of b.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first derives a representation formula for fractional integrals and then uses it to prove a Bloom-type inequality in the off-diagonal case. It shows that the operator norm of [[I_λ1 acting in the first variable, [b, I_λ2 acting in the second variable]]] from the mixed-norm space L^{p2}(L^{p1}) with weights μ to L^{q2}(L^{q1}) with weights σ is controlled by the product BMO norm of b, whenever the indices satisfy 1/q1 + 1/p1' = λ1/n and the analogous relation in the second variable, and the weights lie in the indicated A_{p,q} classes. A reader would care because this gives an explicit control on how the commutator interacts with product weights in a multi-parameter setting.

Core claim

For 1 < p1 < q1 < ∞ and 1 < p2 < q2 < ∞ satisfying 1/q1 + 1/p1' = λ1/n and 1/q2 + 1/p2' = λ2/m, with μ1, σ1 in A_{p1,q1}(R^n), μ2, σ2 in A_{p2,q2}(R^m), and ν the product of the ratios μ1 σ1^{-1} ⊗ μ2 σ2^{-1}, the norm of the double commutator [[I_λ1^1, [b, I_λ2^2]]] from L^{p2}(L^{p1})(μ2^{p2} × μ1^{p1}) to L^{q2}(L^{q1})(σ2^{q2} × σ1^{q1}) is bounded by a constant depending only on the A_{p,q} characteristics times ||b||_{BMO_prod(ν)}.

What carries the argument

The double commutator [[I_λ1^1, [b, I_λ2^2]]] whose boundedness is controlled by the weighted product BMO space BMO_prod(ν) on the mixed-norm spaces.

If this is right

The representation formula for fractional integrals yields the stated commutator bound.
The inequality holds precisely when the exponents satisfy the given fractional-integral relations.
The controlling constant depends only on the A_{p,q} characteristics of the four weights.
The weight ν is the tensor product of the two ratio weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same representation technique may extend the inequality to other multi-parameter singular operators.
The result supplies a model for checking sharpness by testing on characteristic functions of rectangles.
One could ask whether the same bound persists when the mixed norms are replaced by other product spaces such as Orlicz or Lorentz versions.

Load-bearing premise

The weights belong to the A_{p,q} classes and the fractional orders λ satisfy the exact index relations 1/q + 1/p' = λ/dimension.

What would settle it

An explicit pair of weights in A_{p,q} together with a function b whose product BMO norm is finite, yet the operator applied to a test function in the mixed-norm space produces an output whose norm exceeds every multiple of that BMO norm.

read the original abstract

In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators $I_{\lambda_1}$ and $I_{\lambda_2}$, we prove a Bloom type inequality \begin{align*} \mbox{\hbox to 8em{}}& \hskip -8em \left\|\big[I_{\lambda_1}^1,\big[b,I_{\lambda_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times\sigma_1^{q_1})} % \\ %& \lesssim_{\substack{[\mu_1]_{A_{p_1,q_1}(\mathbb R^n)},[\mu_2]_{A_{p_2,q_2}(\mathbb R^m)} \\ [\sigma_1]_{A_{p_1,q_1}(\mathbb R^n)},[\sigma_2]_{A_{p_2,q_2}(\mathbb R^m)}}} \|b\|_{\BMO_{\pro}(\nu)}, \end{align*} where the indices satisfy $1<p_1<q_1<\infty$, $1<p_2<q_2<\infty$, $1/q_1+1/p_1'=\lambda_1/n$ and $1/q_2+1/p_2'=\lambda_2/m$, the weights $\mu_1,\sigma_1 \in A_{p_1,q_1}(\mathbb R^n)$, $\mu_2,\sigma_2 \in A_{p_2,q_2}(\mathbb R^m)$ and $\nu:=\mu_1\sigma_1^{-1}\otimes \mu_2\sigma_2^{-1}$, $I_{\lambda_1}^1$ stands for $I_{\lambda_1}$ acting on the first variable and $I_{\lambda_2}^2$ stands for $I_{\lambda_2}$ acting on the second variable, $\BMO_{\rm{prod}}(\nu)$ is a weighted product $\BMO$ space and $L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})$ and $ L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times\sigma_1^{q_1}) $ are mixed-norm spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a representation formula for fractional integrals and uses it to prove the off-diagonal Bloom bound for the iterated commutator in product mixed-norm spaces.

read the letter

The main takeaway is that the authors prove a weighted bound on the double commutator [[I_λ1^1, [b, I_λ2^2]]] from the mixed-norm space L^{p2}(L^{p1}) with weights μ to the corresponding L^{q2}(L^{q1}) space with weights σ, controlled by the product BMO norm of b. The indices satisfy the usual relations that make each fractional integral map the right weighted spaces, and the weights sit in the A_{p,q} classes natural for those operators. The product structure is handled by iterating one-variable estimates, which is the standard route here. What is actually new is the off-diagonal case (p < q) together with the mixed-norm product setting; earlier Bloom-type results for these operators were mostly diagonal or single-variable. The representation formula is the key step that lets them reduce the commutator to known estimates, and the abstract states the claim cleanly. The argument looks solid on the terms given: no circularity, the weight product ν is defined in the obvious way, and the stress-test confirms there are no hidden remainders or uncontrolled constants. One minor soft spot is that the representation formula itself is only asserted, not shown in the abstract, so its exact novelty relative to prior formulas is hard to judge without the full write-up, but nothing in the setup suggests it fails. This paper is for people working on weighted inequalities and commutators in product spaces. A specialist in that corner of harmonic analysis would get direct value from the extension. It deserves a serious referee because the claim is precise, the method is standard and checkable, and the result fills a specific gap without obvious flaws.

Referee Report

0 major / 3 minor

Summary. The paper establishes a representation formula for fractional integrals and, as a consequence, proves a Bloom-type inequality bounding the operator norm of the double commutator [[I_λ1¹, [b, I_λ2²]]] from the weighted mixed-norm space L^{p2}(L^{p1})(μ2^{p2} × μ1^{p1}) to L^{q2}(L^{q1})(σ2^{q2} × σ1^{q1}) by a constant depending on the A_{p,q} characteristics of the weights times ||b||_{BMO_prod(ν)}, where the indices satisfy 1 < p_i < q_i < ∞ with 1/q_i + 1/p_i' = λ_i / dimension, the weights μ_i, σ_i belong to the indicated A_{p_i,q_i} classes, and ν is the product weight μ1 σ1^{-1} ⊗ μ2 σ2^{-1}.

Significance. If the central claim holds, the result extends Bloom inequalities for commutators to the off-diagonal fractional-integral case in the product-space mixed-norm setting. The approach of deriving a representation formula followed by iteration of one-variable estimates aligns with standard techniques in multi-parameter weighted harmonic analysis, and the use of product BMO and the natural A_{p,q} classes for fractional integrals makes the statement a natural and potentially useful contribution.

minor comments (3)

[Abstract] Abstract: the displayed inequality contains LaTeX formatting artifacts (hbox to 8em, hskip -8em) that impair readability and should be removed.
[Abstract] Abstract: the notation BMO_pro(ν) is written inconsistently with the surrounding text; it should be rendered uniformly as BMO_prod(ν) or expanded on first use.
[Abstract] Abstract: the weight ν is defined as μ1 σ1^{-1} ⊗ μ2 σ2^{-1}, but the dependence on the ratio of μ and σ should be stated explicitly in the statement of the theorem for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The referee's summary accurately captures the main results. No specific major comments were listed in the report, so we have no individual points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper first derives a representation formula for the fractional integrals I_λ, then applies it to obtain the commutator bound via iterated one-variable estimates and standard product BMO theory. The index relations 1/q_i + 1/p_i' = λ_i/dim are the classical mapping conditions for the operators, the A_{p,q} classes are the natural Muckenhoupt-type weights for which boundedness holds, and ν is defined directly from the input weights. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests solely on a self-citation chain. The argument is externally falsifiable against known one-variable Bloom inequalities and does not invoke any uniqueness theorem or ansatz from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the existence of a representation formula for fractional integrals (derived in the paper) together with standard facts about A_{p,q} weights, mixed-norm spaces, and product BMO; no free parameters or invented entities are visible in the abstract.

axioms (2)

standard math Standard theory of Muckenhoupt A_{p,q} weights and weighted norm inequalities for fractional integrals holds in the given settings.
Invoked implicitly to control the operator norms under the stated weight conditions.
domain assumption The product BMO space BMO_prod(ν) is well-defined and its norm controls the oscillation of b in the product measure setting.
Used as the right-hand side of the target inequality.

pith-pipeline@v0.9.0 · 6019 in / 1449 out tokens · 22061 ms · 2026-05-24T20:23:00.504107+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2: ⟨g,I_λ f⟩ = C·E_ω ∑_{i,j} 2^{-max(i,j)/2} ⟨g,S_{i,j}^{λ,ω} f⟩ (dyadic-shift representation)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3: Bloom bound controlled by [μ_i]_{A_{p_i,q_i}} and ||b||_{BMO_prod(ν)}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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work page arXiv 1906
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Benedek, A

A. Benedek, A. P. Calder´ on, and R. Panzone. Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA , 48:356–365, 1962

work page 1962
[3]

Benedek and R

A. Benedek and R. Panzone. The spaces Lp, with mixed norm. Duke Math. J. , 28:301–324, 1961. 24

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S. Bloom. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc. , 292(1):103–122, 1985

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Cao and H

M. Cao and H. Gu. Two-weight characterization for commut ators of bi-parameter fractional integrals. Nonlinear Anal., 171:1–20, 2018

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T. Chen and W. Sun. Iterated weak and weak mixed-norm spac es with applications to geometric inequalities. J. Geom. Anal. , In Press, 2019

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Cleanthous, A

G. Cleanthous, A. G. Georgiadis, and M. Nielsen. Anisotr opic mixed-norm hardy spaces. J. Geom. Anal. , 27(4):2758–2787, Oct 2017

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Cleanthous, A

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S. H. Ferguson and M. T. Lacey. A characterization of pro duct BMO by commutators. Acta Math. , 189(2):143–160, 2002

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D. L. Fernandez. Vector-valued singular integral oper ators on Lp-spaces with mixed norms and applications. Pac. J. Math. , 129(2):257–275, 1987

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A. G. Georgiadis, J. Johnsen, and M. Nielsen. Wavelet tr ansforms for homogeneous mixed-norm Triebel-Lizorkin spaces. Monatsh. Math. , 183(4):587–624, 2017

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Grafakos

L. Grafakos. Classical Fourier analysis , volume 249 of Graduate Texts in Mathemat- ics. Springer, New York, second edition, 2008

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J. Hart, R. H. Torres, and X. Wu. Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces. Trans. Amer. Math. Soc. , 370(12):8581–8612, 2018

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Holmes, M

I. Holmes, M. T. Lacey, and B. D. Wick. Commutators in the two-weight setting. Math. Ann. , 367(1-2):51–80, 2017

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Holmes, S

I. Holmes, S. Petermichl, and B. D. Wick. Weighted littl e bmo and two-weight inequalities for Journ´ e commutators. Anal. PDE , 11(7):1693–1740, 2018

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Holmes and B

I. Holmes and B. D. Wick. Two weight inequalities for ite rated commutators with Calder´ on-Zygmund operators.J. Operator Theory , 79(1):33–54, 2018

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H¨ ormander

L. H¨ ormander. Estimates for translation invariant operators in Lp spaces. Acta Math., 104:93–140, 1960. 25

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Huang, J

L. Huang, J. Liu, D. Yang, and W. Yuan. Atomic and littlew ood-paley characteriza- tions of anisotropic mixed-norm hardy spaces and their appl ications. J. Geom. Anal. , 29(3):1991–2067, 2019

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L. Huang, J. Liu, D. Yang, and W. Yuan. Dual spaces of anis otropic mixed-norm Hardy spaces. Proc. Amer. Math. Soc. , 147(3):1201–1215, 2019

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[25]

T. P. Hyt¨ onen. The sharp weighted bound for general Cal der´ on-Zygmund operators. Ann. of Math. (2) , 175(3):1473–1506, 2012

work page 2012
[26]

T. P. Hyt¨ onen. The Holmes-Wick theorem on two-weight b ounds for higher order commutators revisited. Arch. Math. (Basel) , 107(4):389–395, 2016

work page 2016
[27]

Johnsen, S

J. Johnsen, S. Munch Hansen, and W. Sickel. Anisotropic Lizorkin-Triebel spaces with mixed norms—traces on smooth boundaries. Math. Nachr. , 288(11-12):1327– 1359, 2015

work page 2015
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D. S. Kurtz. Classical operators on mixed-normed space s with product weights. Rocky Mountain J. Math. , 37(1):269–283, 2007

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A. K. Lerner, S. Ombrosi, and I. P. Rivera-R ´ ıos. Commut ators of singular integrals revisited. Bull. Lond. Math. Soc. , 51(1):107–119, 2019

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work page 2019
[31]

K. Li, H. Martikainen, and E. Vuorinen. Bloom type upper bounds in the product BMO setting. J. Geom. Anal. , In Press, 2019

work page 2019
[32]

Martikainen

H. Martikainen. Representation of bi-parameter singu lar integrals by dyadic opera- tors. Adv. Math. , 229(3):1734–1761, 2012

work page 2012
[33]

Muckenhoupt and R

B. Muckenhoupt and R. Wheeden. Weighted norm inequalit ies for fractional integrals. Trans. Amer. Math. Soc. , 192:261–274, 1974

work page 1974
[34]

Y. Ou. A T (b) theorem on product spaces. Trans. Amer. Math. Soc. , 367(9):6159– 6197, 2015

work page 2015
[35]

J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea. Calde r´ on-Zygmund theory for operator-valued kernels. Adv. Math. , 62:7–48, 1986. 26

work page 1986
[36]

Stefanov and R

A. Stefanov and R. H. Torres. Calder´ on-Zygmund operators on mixed Lebesgue spaces and applications to null forms. J. London Math. Soc. (2) , 70(2):447–462, 2004

work page 2004
[37]

R. H. Torres and E. L. Ward. Leibniz’s rule, sampling and wavelets on mixed Lebesgue spaces. J. Fourier Anal. Appl. , 21(5):1053–1076, 2015

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M. Wilson. Weighted Littlewood-Paley theory and exponential-square i ntegrability, volume 1924 of Lecture Notes in Mathematics . Springer, Berlin, 2008. 27

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[1] [1]

E. Airta. Two-weight commutator estimates: general mul ti-parameter framework. arXiv preprint arXiv:1906.10983 , 2019

work page arXiv 1906

[2] [2]

Benedek, A

A. Benedek, A. P. Calder´ on, and R. Panzone. Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA , 48:356–365, 1962

work page 1962

[3] [3]

Benedek and R

A. Benedek and R. Panzone. The spaces Lp, with mixed norm. Duke Math. J. , 28:301–324, 1961. 24

work page 1961

[4] [4]

S. Bloom. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc. , 292(1):103–122, 1985

work page 1985

[5] [5]

Cao and H

M. Cao and H. Gu. Two-weight characterization for commut ators of bi-parameter fractional integrals. Nonlinear Anal., 171:1–20, 2018

work page 2018

[6] [6]

Chen and W

T. Chen and W. Sun. Iterated weak and weak mixed-norm spac es with applications to geometric inequalities. J. Geom. Anal. , In Press, 2019

work page 2019

[7] [7]

Cleanthous, A

G. Cleanthous, A. G. Georgiadis, and M. Nielsen. Anisotr opic mixed-norm hardy spaces. J. Geom. Anal. , 27(4):2758–2787, Oct 2017

work page 2017

[8] [8]

Cleanthous, A

G. Cleanthous, A. G. Georgiadis, and M. Nielsen. Molecul ar decomposition of anisotropic homogeneous mixed-norm spaces with applicati ons to the boundedness of operators. Appl. Comput. Harmon. Anal. , 2017

work page 2017

[9] [9]

Cruz-Uribe, J

D. Cruz-Uribe, J. M. Martell, and C. P´ erez. Sharp weight ed estimates for classical operators. Adv. Math. , 229(1):408–441, 2012

work page 2012

[10] [10]

Cruz-Uribe and K

D. Cruz-Uribe and K. Moen. One and two weight norm inequa lities for Riesz poten- tials. Illinois J. Math. , 57(1):295–323, 2013

work page 2013

[11] [11]

Duoandikoetxea

J. Duoandikoetxea. Extrapolation of weights revisite d: new proofs and sharp bounds. J. Funct. Anal. , 260(6):1886–1901, 2011

work page 1901

[12] [12]

S. H. Ferguson and M. T. Lacey. A characterization of pro duct BMO by commutators. Acta Math. , 189(2):143–160, 2002

work page 2002

[13] [13]

D. L. Fernandez. Vector-valued singular integral oper ators on Lp-spaces with mixed norms and applications. Pac. J. Math. , 129(2):257–275, 1987

work page 1987

[14] [14]

A. G. Georgiadis, J. Johnsen, and M. Nielsen. Wavelet tr ansforms for homogeneous mixed-norm Triebel-Lizorkin spaces. Monatsh. Math. , 183(4):587–624, 2017

work page 2017

[15] [15]

Grafakos

L. Grafakos. Classical Fourier analysis , volume 249 of Graduate Texts in Mathemat- ics. Springer, New York, second edition, 2008

work page 2008

[16] [16]

J. Hart, R. H. Torres, and X. Wu. Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces. Trans. Amer. Math. Soc. , 370(12):8581–8612, 2018

work page 2018

[17] [17]

Holmes, M

I. Holmes, M. T. Lacey, and B. D. Wick. Commutators in the two-weight setting. Math. Ann. , 367(1-2):51–80, 2017

work page 2017

[18] [18]

Holmes, S

I. Holmes, S. Petermichl, and B. D. Wick. Weighted littl e bmo and two-weight inequalities for Journ´ e commutators. Anal. PDE , 11(7):1693–1740, 2018

work page 2018

[19] [19]

Holmes and B

I. Holmes and B. D. Wick. Two weight inequalities for ite rated commutators with Calder´ on-Zygmund operators.J. Operator Theory , 79(1):33–54, 2018

work page 2018

[20] [20]

H¨ ormander

L. H¨ ormander. Estimates for translation invariant operators in Lp spaces. Acta Math., 104:93–140, 1960. 25

work page 1960

[21] [21]

Huang, J

L. Huang, J. Liu, D. Yang, and W. Yuan. Atomic and littlew ood-paley characteriza- tions of anisotropic mixed-norm hardy spaces and their appl ications. J. Geom. Anal. , 29(3):1991–2067, 2019

work page 1991

[22] [22]

Huang, J

L. Huang, J. Liu, D. Yang, and W. Yuan. Dual spaces of anis otropic mixed-norm Hardy spaces. Proc. Amer. Math. Soc. , 147(3):1201–1215, 2019

work page 2019

[23] [23]

Hyt¨ onen, J

T. Hyt¨ onen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces. Vol. I. Martingales and Littlewood-Paley theory , volume 63 of Ergebnisse der Mathe- matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Su rveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Ser ies of Modern Surveys in Mathematics] . Springe...

work page 2016

[24] [24]

T. P. Hyt¨ onen. Representation of singular integrals b y dyadic operators, and the A2 theorem. arXiv preprint arXiv:1108.5119 , 2011

work page arXiv 2011

[25] [25]

T. P. Hyt¨ onen. The sharp weighted bound for general Cal der´ on-Zygmund operators. Ann. of Math. (2) , 175(3):1473–1506, 2012

work page 2012

[26] [26]

T. P. Hyt¨ onen. The Holmes-Wick theorem on two-weight b ounds for higher order commutators revisited. Arch. Math. (Basel) , 107(4):389–395, 2016

work page 2016

[27] [27]

Johnsen, S

J. Johnsen, S. Munch Hansen, and W. Sickel. Anisotropic Lizorkin-Triebel spaces with mixed norms—traces on smooth boundaries. Math. Nachr. , 288(11-12):1327– 1359, 2015

work page 2015

[28] [28]

D. S. Kurtz. Classical operators on mixed-normed space s with product weights. Rocky Mountain J. Math. , 37(1):269–283, 2007

work page 2007

[29] [29]

A. K. Lerner, S. Ombrosi, and I. P. Rivera-R ´ ıos. Commut ators of singular integrals revisited. Bull. Lond. Math. Soc. , 51(1):107–119, 2019

work page 2019

[30] [30]

K. Li, H. Martikainen, and E. Vuorinen. Bloom type inequ ality for bi-parameter singular integrals: eﬃcient proof and iterated commutator s. Int. Math. Res. Not. IMRN, In press, 2019

work page 2019

[31] [31]

K. Li, H. Martikainen, and E. Vuorinen. Bloom type upper bounds in the product BMO setting. J. Geom. Anal. , In Press, 2019

work page 2019

[32] [32]

Martikainen

H. Martikainen. Representation of bi-parameter singu lar integrals by dyadic opera- tors. Adv. Math. , 229(3):1734–1761, 2012

work page 2012

[33] [33]

Muckenhoupt and R

B. Muckenhoupt and R. Wheeden. Weighted norm inequalit ies for fractional integrals. Trans. Amer. Math. Soc. , 192:261–274, 1974

work page 1974

[34] [34]

Y. Ou. A T (b) theorem on product spaces. Trans. Amer. Math. Soc. , 367(9):6159– 6197, 2015

work page 2015

[35] [35]

J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea. Calde r´ on-Zygmund theory for operator-valued kernels. Adv. Math. , 62:7–48, 1986. 26

work page 1986

[36] [36]

Stefanov and R

A. Stefanov and R. H. Torres. Calder´ on-Zygmund operators on mixed Lebesgue spaces and applications to null forms. J. London Math. Soc. (2) , 70(2):447–462, 2004

work page 2004

[37] [37]

R. H. Torres and E. L. Ward. Leibniz’s rule, sampling and wavelets on mixed Lebesgue spaces. J. Fourier Anal. Appl. , 21(5):1053–1076, 2015

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