Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

The reverse H\"older inequality for $\mathcal{A}_{p(\cdot)}$ weights with applications to matrix weights

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves a quantitative reverse Hölder inequality for variable exponent Muckenhoupt weights $A_{p(\cdot)}$, with explicit constants, and uses it to establish right and left openness for scalar and matrix weights.

desk verdict Theorem 1.1's proof has a homogenization gap that breaks the claimed Q-independent constant; the result may be true but this version is not there yet. read the letter →

arxiv 2411.12849 v2 pith:TPZJU2AW submitted 2024-11-19 math.CA

David Cruz-Uribe , Michael Penrod This is my paper · ORCID
classification math.CA MSC 42B2542B35
keywords variableLebesguespacesMuckenhouptweightsmatrixmaximaloperatorsreverseHölderinequalitylog-HöldercontinuityA_{p(·)}rightandleftopenness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a quantitative reverse Hölder inequality for variable exponent Muckenhoupt weights $A_{p(\cdot)}$. It shows that for every log-Hölder continuous exponent function $p(\cdot)$ with $p_+<\infty$ and every scalar weight $w\in A_{p(\cdot)}$, there is an exponent $r>1$ such that the normalized $L^{rp(\cdot)}$ norm of $w$ on a cube is bounded by a constant times the normalized $L^{p(\cdot)}$ norm, with the constant and $r$ expressed explicitly in terms of the $A_{p(\cdot)}$ characteristic. A sympathetic reader would care because this is the first quantitative structural result of this kind in the variable exponent setting, and it directly implies that the scalar $A_{p(\cdot)}$ classes are both right and left open. The same reverse Hölder machinery is then applied to matrix $A_{p(\cdot)}$ weights, proving right and left openness for those classes as well, a result that is new even in the scalar case.

What carries the argument

The proof is carried by a three-step mechanism. First, Lemma 3.5 converts the $A_{p(\cdot)}$ condition into an $A_\infty$-type density estimate: for the measure $W(E)=\int_E w(x)^{p(x)}\,dx$, one gets $|E|/|Q| \le L_2 [w]_{A_{p(\cdot)}}^{1+2C_\infty p_+/(p_\infty p_-)} (W(E)/W(Q))^{1/p_+}$. Second, Lemma 4.1, a quantitative reverse Hölder inequality for constant exponents, turns this density estimate into the modular bound $\int_Q w(x)^{rp(x)}\,dx \le 2 (\int_Q w(x)^{p(x)}\,dx)^r$. Third, log-Hölder continuity enters through the Diening condition (Lemma 2.7) and comparison lemmas (2.8–2.11), which allow the proof to replace $|Q|^{1/p_Q}$ by $\|\chi_Q\|_{L^{p(\cdot)}}$ at every scale; a normalization argument removes any dependence on the weight's norm on a fixed large cube. The same machinery, extended with reducing operators and averaging operators, transfers the scalar estimate to matrix weights.

What would settle it

Take $n=1$, $p(x)=2+\delta/\log(e+|x|)$ (log-Hölder at infinity) and $w(x)=|x|^{-\alpha}$ for small $\alpha$, and test inequality (1.2) on cubes centered at the origin with the $r$ and $C_{p(\cdot)}$ given by Theorem 1.1; a single cube where the ratio exceeds $C_{p(\cdot)}$ refutes the quantitative claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: if $p(\cdot)\in P(\mathbb{R}^n)\cap LH(\mathbb{R}^n)$ with $p_+<\infty$ and $w$ is a scalar $A_{p(\cdot)}$ weight, then there exist $C_{p(\cdot)}$ and $r>1$ such that for every cube $Q$, $$|Q|^{-1/(r p_Q)}\|w\chi_Q\|_{$L^{{rp(\cdot)}}$} \le C_{p(\cdot)} |Q|^{-1/p_Q}\|w\chi_Q\|_{$L^{{p(\cdot)}}$}.$$ The exponent $r$ and the constant $C_{p(\cdot)}$ are given explicitly in terms of the $A_{p(\cdot)}$ characteristic and constants that depend only on the dimension, the exponent function, and its log-Hölder constants. When the exponent function is constant, the inequality reduces to the classical reverse Hölder inequality for $A_p$ weights. The paper then derives scalar right and left openness (Corollaries 1.3 and 1.4) and, via reducing and averaging operators, the same openness properties for matrix $A_{p(\cdot)}$ weights (Theorems 1.5 and 1.6).

Load-bearing premise

The argument depends on the exponent function $p(\cdot)$ being log-Hölder continuous both locally and at infinity, with $p_+<\infty$; without that continuity, the comparisons between $|Q|^{1/p_Q}$ and the $L^{p(\cdot)}$ norm of $\chi_Q$ that carry the proof from modular to norm estimates fail.

Editorial extensions

If this is right

  • Scalar $A_{p(\cdot)}$ classes are right-open: if $w\in A_{p(\cdot)}$, then $w\in A_{sp(\cdot)}$ for all $s\in[1,r]$ with $r>1$ from the reverse Hölder inequality (Corollary 1.3).
  • Scalar $A_{p(\cdot)}$ classes are left-open: with $p_->1$, if $w\in A_{p(\cdot)}$, then $w\in A_{q(\cdot)}$ where $q'(\cdot)=sp'(\cdot)$ for all $s\in[1,r]$ (Corollary 1.4).
  • Matrix $A_{p(\cdot)}$ classes are right-open and left-open (Theorems 1.5 and 1.6), extending the scalar results and providing the first openness results for matrix variable-exponent weights.
  • When $p(\cdot)$ is constant, Theorem 1.1 reduces to the classical reverse Hölder inequality (1.1), so the result is a genuine extension of the constant-exponent theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the theorem is right, the same three-step mechanism should extend to spaces of homogeneous type, since none of the steps is specific to Euclidean cubes beyond the covering estimates.
  • The normalization argument that removes the weight's norm on a fixed cube could be reused to sharpen known weighted norm inequalities for the maximal operator on $L^{p(\cdot)}$.
  • A natural testable extension, not pursued in the paper, is whether the constant in (1.2) can be made to approach $1$ as $r\to 1$, which would align the variable-exponent inequality with the sharp constant-exponent reverse Hölder theory.
  • The matrix results suggest that every scalar $A_{p(\cdot)}$ consequence, such as weighted degenerate Sobolev inequalities, can be lifted to matrix weights via the reduction to the scalar weights $w_e=|W(\cdot)e|$ for unit vectors $e$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper claims a quantitative reverse Hölder inequality for variable-exponent Muckenhoupt weights: for p(·) log-Hölder continuous with p_+<∞ and w∈A_{p(·)}, there is r>1 and C_{p(·)} such that |Q|^{-1/(r p_Q)}∥wχ_Q∥_{L^{r p(·)}} ≤ C_{p(·)} |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} for all cubes Q. The proof strategy is to convert the A_{p(·)} condition into a quantitative A_∞-type condition for the measure w(x)^{p(x)}dx (Lemma 3.5), apply a known classical reverse Hölder inequality (Lemma 4.1), and then pass from modular estimates to norm estimates using log-Hölder continuity. The paper then derives right- and left-openness for scalar and matrix A_{p(·)} classes via reducing operators and averaging operators.

Significance. If the main theorem is correct, this is the first quantitative reverse Hölder inequality for variable-exponent Muckenhoupt weights, and the resulting openness statements for matrix weights would be new even in the scalar case. The manuscript is carefully written and largely self-contained: constants are tracked explicitly in Lemmas 2.11, 3.3, and 3.5, and the reduction to the classical sharp reverse Hölder inequality is conceptually sound. However, the proof of Theorem 1.1 contains a normalization gap that is load-bearing, and since Corollary 4.4 and the matrix results in Section 6 depend on Theorem 1.1, the current version does not establish the stated conclusions.

major comments (2)
  1. [Section 4, proof of Theorem 1.1] The normalization argument has a gap. The reduction "we may assume |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} = 1" is per cube, so when the normalization is undone, every constant that was computed under that assumption must be evaluated at w/λ_Q, where λ_Q = |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}}. In particular, the factor max{1, ∥wχ_{Q_0}∥_{L^{p(·)}}^{p_+/p_-}} appearing in the estimate just before the word "Therefore" should be replaced by max{1, (∥wχ_{Q_0}∥_{L^{p(·)}}/λ_Q)^{p_+/p_-}}. This quantity is not controlled by [w]_{A_{p(·)}}. The subsequent global normalization v = w/∥wχ_{Q_0}∥_{L^{p(·)}} does not repair the issue: in proving inequality (4.14) for v, the per-cube normalization would introduce the factor 1/λ_Q^v, where λ_Q^v = |Q|^{-1/p_Q}∥vχ_Q∥_{L^{p(·)}}, and no bound for this factor is supplied. Thus the claimed Q-independent constants C_{p(·)} and r in Theorem 1.1 are not justified by the proof as written.
  2. [Section 6 and Corollary 4.4] The applications inherit the normalization gap. Corollary 4.4 uses Theorem 1.1 directly, and Lemmas 5.11 and 5.12 use Corollary 4.4, so Theorems 1.5 and 1.6 are conditional on a repaired proof of Theorem 1.1. The manuscript should either supply the missing control of the per-cube normalization or state explicitly the weaker reverse Hölder inequality that the present argument actually proves, with constants depending on the uncontrolled ratio, and then revisit which applications survive.
minor comments (5)
  1. [Abstract and Section 1] The name "Neugebauer" is misspelled as "Neugeabauer" in the abstract and in the first paragraph of Section 1.
  2. [Section 4 heading] The heading "The Reverse H\"older Inequality in V ariable Lebesgue Spaces" contains an erroneous space in "V ariable"; it should read "Variable".
  3. [References] Reference [12] is formatted as "arXiv2408.12745"; it should be "arXiv:2408.12745".
  4. [Lemma 5.11] The statement of Lemma 5.11 says the implicit constant depends only on d, p(·), C_∞, C_* and [W]_{A_{p(·)}}, but the proof invokes Corollary 4.4, whose constant contains [1]_{A_{v(·)}} with v(·) depending on s. Therefore the constant depends on s through v(·). This does not invalidate the openness applications, but the stated uniformity in s is not supported by the proof.
  5. [Lemma 2.7] The explicit constant for C_D is written as exp(C_0(1 + log2√n)), which is ambiguous; it should presumably be exp(C_0(1 + log(2√n))) or the base of the logarithm should be specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is derived from an A-infinity condition via the classical reverse Hölder inequality, with variable-exponent tools proved in the paper or cited from standard references.

full rationale

The paper's main result is not assumed or fitted. The proof of Theorem 1.1 reduces the A_{p(·)} condition to the quantitative A-infinity-type estimate of Lemma 3.5 (proved in the paper), then applies Lemma 4.1, the classical reverse Hölder inequality for A-infinity/A_p weights from [4,20,21], to the measure dW = w^{p(·)} dx. The variable-exponent norm-to-modular comparisons are handled by Lemmas 2.7, 2.8, 2.10, and 2.11; Lemma 2.10 is cited from [12] but is a standard norm estimate for characteristic functions and is not equivalent to the conclusion. No parameter is fitted to the target quantity, and no uniqueness or ansatz is imported from the authors' prior work to force the result. The cited prior papers [8,11,12] are used as tools or context, and the key lemmas from them are re-proved quantitatively in Sections 3 and 4. The homogenization issue raised by a skeptical reader is a possible proof gap in undoing the per-cube normalization, not a circularity, since it concerns the correctness of the constant removal rather than the conclusion being assumed as an input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities; all constants are tracked functions of the input data (p(·), [w]_{A_{p(·)}}, dimension). It relies on standard variable Lebesgue space theory and on the authors' previous characterizations of matrix A_{p(·)} weights.

assumptions (6)
  • standard math Classical quantitative reverse Hölder inequality for A_p weights (Lemma 4.1, based on [4, Theorem 3.2] and [20,21]).
    Used in the proof of Theorem 1.1 to pass from the A-infinity condition (4.1) to the modular estimate (4.3); it is an external theorem, not proved in this paper.
  • standard math Variable Lebesgue space norm inequalities (Lemmas 2.1-2.4 from [7]) and Diening condition (Lemma 2.7 from [7,14]).
    Foundational tools for all estimates involving L^{p(·)} norms and cubes.
  • standard math Norm of characteristic function estimates (Lemma 2.10 from [12], Lemma 2.11 from [14]).
    Used to relate |Q|^{1/p_Q} to ||χ_Q||_{L^{p(·)}} in the proof of Theorem 1.1 and in Corollary 4.4.
  • standard math Existence of reducing operators (Theorem 5.6 from [2]).
    Used in Section 5 to define reducing operators W_Q^{p(·)} and W_Q^{p'(·)}.
  • standard math Characterizations of matrix A_{p(·)} weights via reducing operators and averaging operators (Proposition 5.7 and Proposition 5.8 from [11]).
    Used in Sections 5-6 to reduce matrix openness to scalar reverse Hölder estimates and boundedness of averaging operators.
  • domain assumption p(·)∈LH(R^n) and p_+<∞ (domain assumption).
    Stated as hypothesis in all main theorems; ensures Diening condition, norm comparison lemmas, and finiteness of [1]_{A_{p(·)}}.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The reverse H\"older inequality for $\mathcal{A}_{p(\cdot)}$ weights with applications to matrix weights." pith.science (2026). https://pith.science/paper/TPZJU2AW

@misc{pith2026241112849,
  author       = {Pith},
  title        = {Pith review of: The reverse H\"older inequality for $\mathcalA_p(\cdot)$ weights with applications to matrix weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPZJU2AW}},
  note         = {Machine review of arXiv:2411.12849}
}
abstract

In this paper we prove a reverse H\"{o}lder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse H\"{o}lder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On off-diagonal operators in matrix-weighted spaces

    math.CA 2026-07 conditional novelty 6.0 of 10

    Off-diagonal convex body domination yields improved matrix-weighted bounds for fractional integrals, commutators, and Sobolev inequalities.

Reference graph

Works this paper leans on

27 extracted references · 24 canonical work pages · cited by 1 Pith paper

  1. [1]

    M. Bownik. Inverse volume inequalities for matrix weights.Indiana Univ. Math. J., 50(1):383–410, 2001

    work page 2001

  2. [2]

    Bownik and D

    M. Bownik and D. Cruz-Uribe. Extrapolation and factorization of matrix weights.preprint, 2023. arXiv:2210.09443

    arXiv 2023

  3. [3]

    Coifman and C

    R. Coifman and C. Fefferman. Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51:241–250, 1974

    work page 1974

  4. [4]

    Cruz-Uribe

    D. Cruz-Uribe. Extrapolation and factorization. In J. Lukes and L. Pick, editors,Function spaces, embeddings and extrapolation X, Paseky 2017, pages 45–92. Matfyzpress, Charles University, 2017. arXiv:1706.02620

    arXiv 2017

  5. [5]

    Cruz-Uribe and J

    D. Cruz-Uribe and J. Cummings. Weighted norm inequalities for the maximal operator onL p(·) over spaces of homogeneous type.Ann. Fenn. Math., 47(1):457–488, 2022

    work page 2022

  6. [6]

    Cruz-Uribe, L

    D. Cruz-Uribe, L. Diening, and P. H¨ ast¨ o. The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal., 14(3):361–374, 2011

    work page 2011

  7. [7]

    Cruz-Uribe and A

    D. Cruz-Uribe and A. Fiorenza.Variable Lebesgue spaces. Applied and Numerical Harmonic Analysis. Birkh¨ auser/Springer, Heidelberg, 2013. Foundations and harmonic analysis

    work page 2013

  8. [8]

    Cruz-Uribe, A

    D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces.J. Math. Anal. Appl., 394(2):744–760, 2012

    work page 2012

Show all 27 references
  1. [9]

    Cruz-Uribe, K

    D. Cruz-Uribe, K. Moen, and S. Rodney. MatrixA p weights, degenerate Sobolev spaces, and mappings of finite distortion.J. Geom. Anal., 26(4):2797–2830, 2016

    2016

  2. [10]

    Cruz-Uribe and M

    D. Cruz-Uribe and M. Penrod. The boundedness of the auxiliary maximal operator onL p(·).In prepa- ration

  3. [11]

    Cruz-Uribe and M

    D. Cruz-Uribe and M. Penrod. Convolution operators in matrix weighted, variable lebesgue spaces. Analysis and Applications, 22(07):1133–1157, 2024

    2024

  4. [12]

    Cruz-Uribe and T

    D. Cruz-Uribe and T. Roberts. Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces.preprint, 2024. arXiv2408.12745. 31

    2024 arXiv

  5. [13]

    Cruz-Uribe and L

    D. Cruz-Uribe and L. Wang. Extrapolation and weighted norm inequalities in the variable Lebesgue spaces.Trans. Amer. Math. Soc., 369(2):1205–1235, 2017

    2017

  6. [14]

    Diening, P

    L. Diening, P. Harjulehto, P. H¨ ast¨ o, and M. R ˚ uˇ ziˇ cka.Lebesgue and Sobolev spaces with variable expo- nents, volume 2017 ofLecture Notes in Mathematics. Springer, Heidelberg, 2011

    2017

  7. [15]

    Duoandikoetxea.Fourier analysis, volume 29 ofGraduate Studies in Mathematics

    J. Duoandikoetxea.Fourier analysis, volume 29 ofGraduate Studies in Mathematics. American Mathe- matical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe

    2001

  8. [16]

    Duoandikoetxea, F

    J. Duoandikoetxea, F. Mart ´ ın-Reyes, and S. Ombrosi. On theA ∞ conditions for general bases.Math. Z., 282(3-4):955–972, 2016

    2016

  9. [17]

    Dyn ′kin and B

    E. Dyn ′kin and B. Osilenker. Weighted estimates for singular integrals and their applications. InMath- ematical analysis, Vol. 21, Itogi Nauki i Tekhniki, pages 42–129. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983

    1983

  10. [18]

    Garc ´ ıa-Cuerva and J

    J. Garc ´ ıa-Cuerva and J. Rubio de Francia.Weighted norm inequalities and related topics, volume 116 ofNorth-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985. Notas de Matem´ atica, 104. [Mathematical Notes]

    1985

  11. [19]

    Goldberg

    M. Goldberg. MatrixA p weights via maximal functions.Pacific J. Math., 211(2):201–220, 2003

    2003

  12. [20]

    Hyt¨ onen and C

    T. Hyt¨ onen and C. P´ erez. Sharp weighted bounds involvingA∞.Anal. PDE, 6(4):777–818, 2013

    2013

  13. [21]

    Hyt¨ onen, C

    T. Hyt¨ onen, C. P´ erez, and E. Rela. Sharp reverse H¨ older property forA∞ weights on spaces of homo- geneous type.J. Funct. Anal., 263(12):3883–3899, 2012

    2012

  14. [22]

    A. K. Lerner. On a dual property of the maximal operator on weighted variableL p spaces. InFunctional analysis, harmonic analysis, and image processing: a collection of papers in honor of Bj¨ orn Jawerth, volume 693 ofContemp. Math., pages 283–300. Amer. Math. Soc., Providence...

    2017

  15. [23]

    Muckenhoupt

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

    1972

  16. [24]

    Muckenhoupt and R

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

    1974

  17. [25]

    Nazarov and S

    F. Nazarov and S. Treil. The hunt for a Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis.Algebra i Analiz, 8(5):32–162, 1996

    1996

  18. [26]

    Roudenko

    S. Roudenko. Matrix-weighted Besov spaces.Trans. Amer. Math. Soc., 355(1):273–314, 2003

    2003

  19. [27]

    Treil and A

    S. Treil and A. Volberg. Wavelets and the angle between past and future.J. Funct. Anal., 143(2):269– 308, 1997. David Cruz-Uribe, OFS, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA Email address:dcruzuribe@ua.edu Michael Penrod, Department of Math...

    1997

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.