arxiv: 2605.09372 · v1 · submitted 2026-05-10 · 🧮 math.PR · math.FA

Recognition: 2 theorem links

· Lean Theorem

Sharp weighted norm estimates for martingale square functions

Lian Wu, Wei Chen, Xingyan Quan, Yong Jiao

Pith reviewed 2026-05-12 03:50 UTC · model grok-4.3

classification 🧮 math.PR math.FA

keywords martingale square functionsmatrix weightsweighted norm estimatessparse dominationmatrix A_p conditionsharp constantsmartingale theory

0 comments

The pith

Matrix-weighted martingale square functions obey sharp L_p bounds controlled by the A_p characteristic of the weight.

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

The paper defines a martingale square function S_W built from a matrix weight W and proves that its L_p norm is controlled by a power of the matrix A_p characteristic of W. The argument proceeds by sparse domination, which produces an explicit dependence on that characteristic. For 1 < p ≤ 2 the power is shown to be optimal; the same method, with adjustments, yields the optimal power in the scalar case for every 1 < p < ∞. This supplies the first sharp weighted bounds for these square functions in the matrix setting and resolves an open sharpness question from classical martingale theory.

Core claim

We introduce the matrix-weighted martingale square function S_W and show that ||S_W f||_p ≲ [W]_{A_p}^α ||f||_p, where the exponent α is sharp for 1 < p ≤ 2. Sparse domination reduces the estimate to a sparse maximal operator whose weighted bound is read off directly from the matrix A_p condition, yielding the explicit constant dependence. In the scalar case the same technique is modified to recover the optimal exponent for all 1 < p < ∞.

What carries the argument

Sparse domination of the matrix-weighted square function S_W, which converts the weighted estimate into a bound for a sparse maximal operator while retaining the precise matrix A_p data.

If this is right

The L_p operator norm of S_W is bounded by an explicit function of the matrix A_p characteristic alone.
The exponent of the characteristic cannot be lowered for 1 < p ≤ 2 without violating the inequality for some weights.
In the scalar setting the optimal exponent is recovered for the full range 1 < p < ∞.
The same sparse-domination argument applies verbatim to other martingale operators once they admit a suitable domination.

Where Pith is reading between the lines

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

The matrix-weighted square function may serve as a model for studying quadratic variations of matrix-valued stochastic integrals.
Sharpness in the matrix case suggests that analogous results could hold for vector-valued or operator-valued martingale transforms.
Numerical checks with random matrix weights could verify whether the predicted exponents are attained asymptotically.

Load-bearing premise

The matrix A_p condition is strong enough to control the L_p norm of S_W and that sparse domination does not lose the sharpness of the exponent.

What would settle it

Exhibit a matrix weight W in A_p with small characteristic such that for some p ≤ 2 the ratio ||S_W f||_p / ||f||_p exceeds every multiple of [W]_{A_p}^α for the claimed α, for a suitable choice of f.

read the original abstract

This paper is devoted to the study of quantitative weighted norm estimates for martingale square functions in both scalar-weighted and matrix-weighted settings. In particular, we introduce the martingale square functions $S_W$ via matrix weights $W$, and then use the matrix $A_p$ condition, introduced in our previous work \cite{ChenQuanJiaoWu}, to characterize the $L_p$ estimate for $S_W$. Our proof mainly relies on the idea of sparse dominations, which leads to the explicit information on the characteristic of the matrix weight involved. For the range $1<p\leq 2$, our result is sharp in terms of the characteristic of the matrix weight. With some modification on the arguments, we can further improve the result in scalar settings by obtaining the optimal exponent of the characteristic of the weight involved for all indices $1<p<\infty$, addressing a fundamental problem from the classical martingale theory.

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

Chen et al. define a matrix-weighted martingale square function S_W, extract explicit A_p dependence via sparse domination, claim sharpness for p≤2, and fix the scalar case to optimal exponents for all p, but the matrix sharpness needs explicit lower-bound verification.

read the letter

This paper defines a matrix-weighted version of the martingale square function called S_W. It then applies sparse domination to bound the L_p norm of S_W in terms of the matrix A_p characteristic from the authors' previous work. The authors state that the bound is sharp for 1 < p ≤ 2 in the matrix setting. They also modify the argument to get the best possible exponent on the weight characteristic in the ordinary scalar case for all 1 < p < ∞, which they say solves a basic open question from classical martingale theory.

Referee Report

2 major / 2 minor

Summary. The paper introduces a matrix-weighted martingale square function S_W and characterizes its L_p boundedness via the matrix A_p condition from the authors' prior work. It employs sparse domination to derive explicit dependence on the weight characteristic, claiming sharpness of the resulting exponent for 1 < p ≤ 2 in the matrix setting and optimal exponents for all 1 < p < ∞ in the scalar setting.

Significance. If the sharpness claims hold, the work would advance quantitative martingale theory by supplying explicit, optimal constants in weighted square-function inequalities, particularly extending to the less-developed matrix-weighted case. The sparse-domination approach is a methodological strength for obtaining concrete exponents rather than abstract boundedness.

major comments (2)

[Abstract] Abstract and the sharpness paragraph: the claim that the result is sharp for 1<p≤2 in terms of the matrix A_p characteristic is not supported by an explicit lower-bound construction (a concrete matrix weight W and test function f) showing that the operator norm grows at least as [W]_{A_p}^α with the same α obtained from the upper bound via sparse domination.
[Proof of the main estimate (around the sparse domination step)] The matrix A_p condition (defined in the cited prior work) is used to characterize boundedness of the newly introduced S_W, but the manuscript does not verify that sparse domination preserves the exact exponent without an extra factor that would appear only in the matrix setting and not in the scalar case.

minor comments (2)

[Introduction] The definition of the matrix-weighted square function S_W should be stated explicitly before the main theorem rather than introduced inline.
[References] Reference formatting for the self-citation ChenQuanJiaoWu is inconsistent with standard arXiv style; full bibliographic details should be supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below and outline the revisions we will make to strengthen the presentation of sharpness and the details of the sparse domination argument.

read point-by-point responses

Referee: [Abstract] Abstract and the sharpness paragraph: the claim that the result is sharp for 1<p≤2 in terms of the matrix A_p characteristic is not supported by an explicit lower-bound construction (a concrete matrix weight W and test function f) showing that the operator norm grows at least as [W]_{A_p}^α with the same α obtained from the upper bound via sparse domination.

Authors: We agree that the current version does not contain an explicit lower-bound example for the matrix-weighted case. Although the scalar case is included as a special instance of the matrix setting (by taking diagonal weights with equal entries), and the scalar sharpness is classical, we will add a concrete construction in the revised manuscript. Specifically, we will exhibit a diagonal matrix weight W whose matrix A_p characteristic reduces to the scalar A_p characteristic, together with a suitable test function f, showing that the operator norm is at least c [W]_{A_p}^{1/(p-1)} for 1 < p ≤ 2. This example will be placed in a new subsection following the main theorem. revision: yes
Referee: [Proof of the main estimate (around the sparse domination step)] The matrix A_p condition (defined in the cited prior work) is used to characterize boundedness of the newly introduced S_W, but the manuscript does not verify that sparse domination preserves the exact exponent without an extra factor that would appear only in the matrix setting and not in the scalar case.

Authors: In the sparse domination argument (Section 3), the constants are tracked explicitly using only the matrix A_p characteristic raised to the power appearing in the scalar case; no additional matrix-dependent factors (such as dimension-dependent logs) enter because the domination is applied entrywise after reducing to the positive-definite case via the definition in our prior work. Nevertheless, we acknowledge that this preservation is not stated as a separate remark. We will insert a clarifying paragraph immediately after the sparse domination lemma that compares the matrix and scalar exponents side-by-side and confirms the absence of extra factors. This will be a minor but explicit addition. revision: yes

Circularity Check

1 steps flagged

Self-cited matrix A_p condition underpins characterization of S_W boundedness and sharpness

specific steps

self citation load bearing [Abstract]
"we introduce the martingale square functions $S_W$ via matrix weights $W$, and then use the matrix $A_p$ condition, introduced in our previous work [ChenQuanJiaoWu], to characterize the $L_p$ estimate for $S_W$. Our proof mainly relies on the idea of sparse dominations, which leads to the explicit information on the characteristic of the matrix weight involved. For the range 1<p≤2, our result is sharp in terms of the characteristic of the matrix weight."

The characterization that the matrix A_p condition (defined in the authors' own prior paper) governs the L_p boundedness of the newly introduced S_W, together with the sharpness assertion, is justified solely by the self-citation; the current manuscript does not re-derive or independently verify the condition's necessity/sufficiency for S_W but treats it as the characterizing tool.

full rationale

The paper introduces S_W and immediately invokes the matrix A_p condition from the authors' prior work to characterize its L_p estimates, then applies sparse domination to extract explicit dependence on the weight characteristic. This creates moderate self-citation load-bearing for the central claim (pattern 3), as the sufficiency of the condition for this new operator and the asserted sharpness for 1<p≤2 rest on the imported definition rather than an independent derivation within the manuscript. Sparse domination supplies new proof machinery and explicit constants, preventing full reduction to the citation; the scalar-case improvement is presented as a modification rather than a new foundational result. No self-definitional equations, fitted predictions, or ansatz smuggling are exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard martingale theory and the authors' prior definition of matrix A_p weights. No numerical free parameters are fitted. The new entity is the matrix-weighted square function.

axioms (2)

standard math Standard martingale filtration and increment properties hold in the weighted L_p spaces.
Invoked implicitly when defining square functions and L_p norms.
domain assumption The matrix A_p condition from the authors' prior work is the correct characterizing condition for the weighted estimate.
Explicitly used to characterize the L_p estimate for S_W.

invented entities (1)

Matrix-weighted martingale square function S_W no independent evidence
purpose: Extend classical square functions to matrix weights to obtain L_p estimates under matrix A_p conditions.
Newly introduced in this paper to handle the matrix-weighted setting.

pith-pipeline@v0.9.0 · 5453 in / 1468 out tokens · 96287 ms · 2026-05-12T03:50:19.971834+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem B. … ∥S_W f∥_p ≲_{p,d} [W]_{A_p}^{max{1/2 + 1/(p(p-1)), 1/(p-1)}} ∥f∥_p … For 1<p≤2 the exponent … equals 1/(p-1) which is sharp.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Our proof mainly relies on the idea of sparse dominations … principal sets … γ(n,m) … τ_P …

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

[1]

Bañuelos and A

R. Bañuelos and A. Os¸ ekowski,WeightedL2 inequalities for square functions, Trans. Amer. Math. Soc.370(2018), no. 4, 2391–2422. MR3748572

work page 2018
[2]

,A weighted maximal inequality for differentially subordinate martingales, Proc. Amer. Math. Soc.146(2018), no. 5, 2263–2275. MR3767376

work page 2018
[3]

Bickel, S

K. Bickel, S. Petermichl, and B. D. Wick,Bounds for the Hilbert transform with matrixA2 weights, J. Funct. Anal.270(2016), no. 5, 1719–1743. MR3452715

work page 2016
[4]

Bonami and D

A. Bonami and D. Lépingle,Fonction maximale et variation quadratique des martingales en présence d’un poids, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), 1979, pp. 294–306. MR544802

work page 1977
[5]

Bownik and D

M. Bownik and D. Cruz-Uribe,Extrapolation and factorization of matrix weights, arXiv preprint arXiv:2210.09443 (2022)

work page arXiv 2022
[6]

S. M. Buckley,Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc.340(1993), no. 1, 253–272. MR1124164

work page 1993
[7]

D. L. Burkholder,Martingale transforms, Ann. Math. Statist.37(1966), 1494–1504. MR208647

work page 1966
[8]

Chen and Y

W. Chen and Y. Jiao,Weighted estimates for the bilinear maximal operator on filtered measure spaces, J. Geom. Anal.31(2021), no. 5, 5309–5335. MR4244905

work page 2021
[9]

W. Chen, Y. Jiao, X. Quan, and L. Wu,The sharp maxtrix-weighted Doob inequalities, Trans. Amer. Math. Soc. (to appear)

work page
[10]

W. Chen, C. Zhu, Y. Zuo, and Y. Jiao,Two-weighted estimates for positive operators and Doob maximal operators on filtered measure spaces, J. Math. Soc. Japan72(2020), no. 3, 795–817. MR4125846

work page 2020
[11]

Christ and M

M. Christ and M. Goldberg,VectorA2 weights and a Hardy-Littlewood maximal function, Trans. Amer. Math. Soc.353(2001), no. 5, 1995–2002. MR1813604

work page 2001
[12]

Cruz-Uribe, J

D. Cruz-Uribe, J. Martell, and C. Pérez,Sharp weighted estimates for classical operators, Adv. Math.229(2012), no. 1, 408–441. MR2854179

work page 2012
[13]

Domelevo, P

K. Domelevo, P. Ivanisvili, S. Petermichl, S. Treil, and A. Volberg,On the failure of lower square function estimates in the non-homogeneous weighted setting, Math. Ann.374(2019), no. 3-4, 1923–1952. MR3985127

work page 2019
[14]

Domelevo and S

K. Domelevo and S. Petermichl,Differential subordination under change of law, Ann. Probab.47 (2019), no. 2, 896–925. MR3916937

work page 2019
[15]

Domelevo, S

K. Domelevo, S. Petermichl, S. Treil, and A. Volberg,The matrixA2 conjecture fails, i.e.3/2>1, arXiv preprint arXiv:2402.06961 (2024)

work page arXiv 2024
[16]

Domelevo, S

K. Domelevo, S. Petermichl, and K. A. Škreb,Continuous sparse domination and dimension- less weighted estimates for the Bakry-Riesz vector, J. Reine Angew. Math.824(2025), 137–166. MR4926944

work page 2025
[17]

Goldberg,MatrixA p weights via maximal functions, Pacific J

M. Goldberg,MatrixA p weights via maximal functions, Pacific J. Math.211(2003), no. 2, 201–

work page 2003
[18]

Hukovic,Singular integral operators in weighted spaces and Bellman functions, ProQuest LLC, Ann Arbor, MI, 1998

S. Hukovic,Singular integral operators in weighted spaces and Bellman functions, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Brown University. MR2697378

work page 1998
[19]

Hukovic, S

S. Hukovic, S. Treil, and A. Volberg,The Bellman functions and sharp weighted inequalities for square functions, Complex analysis, operators, and related topics, 2000, pp. 97–113. MR1771755

work page 2000
[20]

Hytönen,The sharp weighted bound for general Calderón-Zygmund operators, Ann

T. Hytönen,The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2)175(2012), no. 3, 1473–1506. MR2912709

work page 2012
[21]

Hytönen, S

T. Hytönen, S. Petermichl,and A. Volberg,The sharp square function estimate with matrix weight, Discrete Anal. (2019), Paper No. 2, 8. MR3936542

work page 2019
[22]

Isralowitz,Sharp matrix weighted strong type inequalities for the dyadic square function, Po- tential Anal.53(2020), no

J. Isralowitz,Sharp matrix weighted strong type inequalities for the dyadic square function, Po- tential Anal.53(2020), no. 4, 1529–1540. MR4159390

work page 2020
[23]

Izumisawa and N

M. Izumisawa and N. Kazamaki,Weighted norm inequalities for martingales, Tohoku Math. J. (2)29(1977), no. 1, 115–124. MR436313 SHARP WEIGHTED ESTIMATES FOR SQUARE FUNCTIONS 25

work page 1977
[24]

Jawerth,Weighted inequalities for maximal operators: linearization, localization and factoriza- tion, Amer

B. Jawerth,Weighted inequalities for maximal operators: linearization, localization and factoriza- tion, Amer. J. Math.108(1986), no. 2, 361–414. MR833361

work page 1986
[25]

W. B. Johnson and G. Schechtman,Martingale inequalities in rearrangement invariant function spaces, Israel J. Math.64(1988), no. 3, 267–275. MR995572

work page 1988
[26]

M. T. Lacey,An elementary proof of theA2 bound, Israel J. Math.217(2017), no. 1, 181–195. MR3625108

work page 2017
[27]

M. T. Lacey, K. Moen, C. Pérez, and R. H. Torres,Sharp weighted bounds for fractional integral operators, J. Funct. Anal.259(2010), no. 5, 1073–1097. MR2652182

work page 2010
[28]

A. K. Lerner,On some weighted norm inequalities for Littlewood-Paley operators, Illinois J. Math. 52(2008), no. 2, 653–666. MR2524658

work page 2008
[29]

P. D. Liu,Martingale and the geometry of Banach spaces, Beijing: Science Precess, 2007

work page 2007
[30]

Muckenhoupt,Weighted norm inequalities for the Hardy maximal function, Trans

B. Muckenhoupt,Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc.165(1972), 207–226. MR293384

work page 1972
[31]

F. L. Nazarov and S. Treil,The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz8(1996), no. 5, 32–162. MR1428988

work page 1996
[32]

Os¸ ekowski,Weak-type inequality for the martingale square function and a related characteri- zation of Hilbert spaces, Probab

A. Os¸ ekowski,Weak-type inequality for the martingale square function and a related characteri- zation of Hilbert spaces, Probab. Math. Statist.31(2011), no. 2, 227–238. MR2853676

work page 2011
[33]

Petermichl and S

S. Petermichl and S. Pott,An estimate for weighted Hilbert transform via square functions, Trans. Amer. Math. Soc.354(2002), no. 4, 1699–1703. MR1873024

work page 2002
[34]

Pisier,Martingales with values in uniformly convex spaces, Israel J

G. Pisier,Martingales with values in uniformly convex spaces, Israel J. Math.20(1975), no. 3-4, 326–350. MR394135

work page 1975
[35]

Tanaka and Y

H. Tanaka and Y. Terasawa,Positive operators and maximal operators in a filtered measure space, J. Funct. Anal.264(2013), no. 4, 920–946. MR3004953

work page 2013
[36]

Treil,MixedA 2-A∞ estimates of the non-homogeneous vector square function with matrix weights, Proc

S. Treil,MixedA 2-A∞ estimates of the non-homogeneous vector square function with matrix weights, Proc. Amer. Math. Soc.151(2023), no. 8, 3381–3389. MR4591773

work page 2023
[37]

Treil and A

S. Treil and A. Volberg,Wavelets and the angle between past and future, J. Funct. Anal.143 (1997), no. 2, 269–308. MR1428818

work page 1997
[38]

Volberg,MatrixA p weights viaS-functions, J

A. Volberg,MatrixA p weights viaS-functions, J. Amer. Math. Soc.10(1997), no. 2, 445–466. MR1423034

work page 1997
[39]

Weisz,Martingale Hardy spaces and their applications in Fourier analysis, Lecture Notes in Mathematics, vol

F. Weisz,Martingale Hardy spaces and their applications in Fourier analysis, Lecture Notes in Mathematics, vol. 1568, Springer-Verlag, Berlin, 1994. MR1320508

work page 1994
[40]

Wiener and P

N. Wiener and P. Masani,The prediction theory of multivariate stochastic processes. II. The linear predictor, Acta Math.99(1958), 93–137. MR97859

work page 1958
[41]

Wittwer,A sharp estimate on the norm of the martingale transform, Math

J. Wittwer,A sharp estimate on the norm of the martingale transform, Math. Res. Lett.7(2000), no. 1, 1–12. MR1748283

work page 2000
[42]

G. Xie, F. Weisz, D. Yang, and Y. Jiao,New martingale inequalities and applications to Fourier analysis, Nonlinear Anal.182(2019), 143–192. MR3895312 School of Mathematics, Yangzhou University, Yangzhou 225002, China Email address:weichen@yzu.edu.cn School of Mathematics and Statistics, Central South University, HNP-LAMA, Chang- sha 410083, China Email ad...

work page 2019