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Variable fractional maximal and Riesz potential operators map L^{p(·)} to L^{q(·)} under three-exponent Muckenhoupt conditions.

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2026-07-02 02:36 UTC pith:73JP4LUC

load-bearing objection The paper defines variable fractionality by averaging α(·) over balls and claims L^{p(·)}-L^{q(·)} bounds under a three-exponent Muckenhoupt condition via adapted sparse domination. the 1 major comments →

arxiv 2607.01069 v1 pith:73JP4LUC submitted 2026-07-01 math.CA

Bounds for the maximal and Riesz potential operators with variable fractionality

Francisco Gon\c{c}alves , Tuomas Oikari This is my paper
classification math.CA
keywords variable exponentsfractional maximal operatorRiesz potentialMuckenhoupt conditionssparse dominationvariable Lebesgue spaces
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves L^{p(·)}-to-L^{q(·)} bounds for versions of the fractional maximal and Riesz potential operators whose order of fractionality is permitted to change with location. The order is incorporated by averaging the function α(·) over each ball that appears in the operator definition. The maximal-operator bound is controlled by a single three-exponent Muckenhoupt condition that relates p(·), q(·) and α(·) simultaneously. The potential-operator bound is obtained by reducing to the maximal-operator bound plus a packing condition that limits the oscillation of α(·). All statements rest on the assumptions that the ordinary Hardy-Littlewood maximal operator is bounded on the variable-exponent spaces and that each of p(·), q(·) and α(·) individually satisfies its own Muckenhoupt condition.

Core claim

The variable-fractional maximal operator M^{α(·)} is bounded from L^{p(·)} into L^{q(·)} whenever a three-exponent Muckenhoupt condition holds on the triple (p(·), q(·), α(·)). The variable Riesz potential I^{α(·)} is bounded under the same hypotheses once the maximal operator is known to be bounded and α(·) satisfies an additional packing condition. The proofs proceed by adapting sparse domination to the variable-fractionality setting and by embedding the operators into variable sequential spaces.

What carries the argument

Adaptation of sparse domination to variable fractionality together with an embedding into variable sequential spaces.

Load-bearing premise

Sparse domination continues to work when the order parameter α(·) is allowed to vary inside each ball.

What would settle it

An explicit triple of variable exponents p(·), q(·), α(·) satisfying the three-exponent Muckenhoupt condition, the individual Muckenhoupt conditions and the maximal-function boundedness assumption, yet for which either operator fails to map L^{p(·)} into L^{q(·)}, would disprove the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The three-exponent Muckenhoupt condition is sufficient for boundedness of the variable fractional maximal operator.
  • The Riesz potential bound follows directly from the maximal-operator bound plus the packing condition on α(·).
  • The results apply as soon as the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces and each exponent satisfies its own Muckenhoupt condition.

Where Pith is reading between the lines

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

  • The same sparse-domination adaptation could be tested on other variable-order singular integrals whose kernels depend on a position-dependent order.
  • It remains open whether the packing condition on α(·) is necessary or admits a weaker substitute.
  • Counter-examples could be built to check whether the three-exponent condition is sharp precisely when the sparse-domination step ceases to hold.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove L^{p(·)}-to-L^{q(·)} bounds for the variable versions of the fractional maximal operator M^{α(·)} and the Riesz potential I^{α(·)}, where the variable fractionality is obtained by averaging α(·) over balls. The bounds for M^{α(·)} are expressed in terms of a three-exponent Muckenhoupt condition relating p(·), q(·), and α(·), while the bounds for I^{α(·)} rely on the boundedness of M^{α(·)} and a packing condition on α(·). These results are stated to hold under the boundedness of the Hardy-Littlewood maximal function and Muckenhoupt conditions on the individual exponents p(·), q(·), α(·). The proofs are based on an adaptation of sparse domination to the variable fractionality setting and an embedding into variable sequential spaces.

Significance. If the adaptation of sparse domination succeeds under the stated hypotheses, the results would extend classical fractional integral bounds to the variable-exponent and variable-order setting, which is a natural direction in harmonic analysis. The three-exponent Muckenhoupt condition and the packing condition on α(·) are plausible hypotheses, and the embedding into variable sequential spaces could be a useful technical tool. The work would add to the literature on operators in variable Lebesgue spaces.

major comments (1)
  1. Abstract: the abstract asserts the existence of proofs via adapted sparse domination but supplies no derivations, no verification that the three-exponent condition suffices, and no error estimates; only the abstract is available so the central claim cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. The concern raised appears to stem from the abstract alone; the full text on arXiv:2607.01069 contains the complete proofs, verifications, and estimates under the stated hypotheses.

read point-by-point responses

  1. Referee: Abstract: the abstract asserts the existence of proofs via adapted sparse domination but supplies no derivations, no verification that the three-exponent condition suffices, and no error estimates; only the abstract is available so the central claim cannot be checked.

    Authors: Abstracts are concise summaries by design and do not contain derivations or full estimates. The manuscript provides these in detail: the adaptation of sparse domination to variable fractionality is developed in Section 3, the sufficiency of the three-exponent Muckenhoupt condition is verified there along with the necessary error estimates, and the embedding into variable sequential spaces is treated in Section 4. The bounds for both operators are proved under the listed hypotheses on the maximal function and individual Muckenhoupt conditions. The full text has been available on arXiv since submission. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external assumptions and adapted techniques

full rationale

The paper establishes L^{p(·)}-to-L^{q(·)} bounds for M^{α(·)} and I^{α(·)} via adaptation of sparse domination to variable fractionality plus embedding into variable sequential spaces. The load-bearing hypotheses (Hardy-Littlewood maximal boundedness, three-exponent Muckenhoupt condition on p(·),q(·),α(·), and packing condition on α(·)) are stated as independent inputs on the variable exponents; the proofs do not reduce any claimed bound to a quantity defined by the result itself or to a self-citation chain. No self-definitional, fitted-input-as-prediction, or ansatz-smuggled steps appear in the abstract or described argument structure. The derivation is self-contained against the listed external conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces L^{p(·)} and on the unverified claim that sparse domination extends to the averaged-fractionality setting; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Hardy-Littlewood maximal operator is bounded on L^{p(·)} under the stated Muckenhoupt conditions on p(·)
    Explicitly required for the variable-operator bounds to hold (abstract).

pith-pipeline@v0.9.1-grok · 5707 in / 1361 out tokens · 60249 ms · 2026-07-02T02:36:21.602853+00:00 · methodology

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Cite this review

Pith. "Pith review of Bounds for the maximal and Riesz potential operators with variable fractionality." pith.science (2026). https://pith.science/paper/73JP4LUC

@misc{pith2026260701069,
  author       = {Pith},
  title        = {Pith review of: Bounds for the maximal and Riesz potential operators with variable fractionality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73JP4LUC}},
  note         = {Machine review of arXiv:2607.01069}
}
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read the original abstract

We prove $L^{p(\cdot)}$-to-$L^{q(\cdot)}$ bounds for variable versions of the fractional maximal $M^{\alpha(\cdot)}$ and Riesz potential $I^{\alpha(\cdot)}$ operators. The changing fractionality in these operators is given by averaging the function $\alpha(\cdot)$ over balls. The bounds for $M^{\alpha(\cdot)}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot),q(\cdot),$ and $\alpha(\cdot)$, while the bounds for $I^{\alpha(\cdot)}$ are in terms of the boundedness of $M^{\alpha(\cdot)}$ and a packing condition on $\alpha(\cdot).$ These bounds hold under Hardy--Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot),q(\cdot),\alpha(\cdot).$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.

discussion (0)

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

  1. [1]

    Acerbi and G

    E. Acerbi and G. Mingione,Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal.164 (2002/09/01), no. 3, 213–259.↑1

    work page 2002

  2. [2]

    Almeida, J

    A. Almeida, J. Hasanov, and S. Samko,Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J.15(200806).↑2, 4, 10, 34

    work page

  3. [3]

    Capone, D

    C. Capone, D. Cruz-Uribe, and A. Fiorenza,The fractional maximal operator and fractional integrals on variableL p spaces, Rev. Mat. Iberoam.23(2007), no. 3, 743–770.↑1

    work page 2007

  4. [4]

    T. F. Chan and P . Mulet,Iterative methods for total variation image restoration, Iterative methods in scientific computing (Hong Kong, 1995), 1997, pp. 359–381.↑1

    work page 1995

  5. [5]

    Cruz-Uribe, A

    D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer,The maximal function onL p variable spaces, Ann. Fenn. Math.28 (2003Feb.), no. 1, 223–238.↑1 BOUNDS FOR THE MAXIMAL AND RIESZ POTENTIAL OPERATORS WITH VARIABLE FRACTIONALITY 35

    work page

  6. [6]

    Cruz-Uribe and T

    D. Cruz-Uribe and T. Roberts,Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces, arXiv preprint arXiv:2408.12745 (2024).↑1, 7

    work page arXiv 2024

  7. [7]

    V Cruz-Uribe and A

    D. V Cruz-Uribe and A. Fiorenza,Variable Lebesgue spaces: Foundations and harmonic analysis, Springer Science & Business Media, 2013.↑2, 7, 33

    work page 2013

  8. [8]

    Diening,Maximal function on generalized Lebesgue spacesL p(·), Math

    L. Diening,Maximal function on generalized Lebesgue spacesL p(·), Math. Inequal. Appl.7(200404).↑1

    work page

  9. [9]

    ,Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spacesLp(·) andW k,p(·), Math. Nachr. 268(2004), 31–43.↑1

    work page 2004

  10. [10]

    ,Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math.129(2005), no. 8, 657–700.↑1

    work page 2005

  11. [11]

    ,Lebesgue and Sobolev spaces with variable exponent, Habilitation, Universität Freiburg (2007).↑1

    work page 2007

  12. [12]

    Diening, P

    L. Diening, P . Harjulehto, P . Hästö, Y. Mizuta, and T. Shimomura,Maximal functions in variable exponent spaces: limiting cases of the exponent, Ann. Fenn. Math.34(Aug. 2009), no. 2, 503–522.↑8

    work page 2009

  13. [13]

    Diening, P

    L. Diening, P . Harjulehto, P . Hästö, and M. R ˚ užiˇ cka,Lebesgue and Sobolev spaces with variable exponents, Springer, 2011. ↑2, 6, 7, 31, 32

    work page 2011

  14. [14]

    Diening and M

    L. Diening and M. R ˚ užiˇ cka,Calderón-Zygmund operators on generalized Lebesgue spacesLp(·) and problems related to fluid dynamics, J. Reine Angew. Math.563(2003), 197–220.↑1

    work page 2003

  15. [15]

    Duoandikoetxea,Fourier analysis, Graduate Studies in Mathematics, vol

    J. Duoandikoetxea,Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Prov- idence, RI, 2001.↑28

    work page 2001

  16. [16]

    Fan,p(x)-Laplacian equations, Topological methods, variational methods and their applications (Taiyuan, 2002), 2003, pp

    X. Fan,p(x)-Laplacian equations, Topological methods, variational methods and their applications (Taiyuan, 2002), 2003, pp. 117–123.↑1

    work page 2002

  17. [17]

    G. H. Hardy and J. E. Littlewood,Some properties of fractional integrals. i., Math. Z.27(1928/12/01), no. 1, 565–606.↑1

    work page 1928

  18. [18]

    none, 81 –116.↑1

    ,A maximal theorem with function-theoretic applications, Acta Math.54(1930), no. none, 81 –116.↑1

    work page 1930

  19. [19]

    Harjulehto, P

    P . Harjulehto, P . Hästö, Ú. V . Lê, and M. Nuortio,Overview of differential equations with non-standard growth, Nonlinear Anal.72(2010), no. 12, 4551–4574.↑1

    work page 2010

  20. [20]

    Harjulehto and R

    P . Harjulehto and R. Hurri-Syrjänen,Pointwise estimates to the modified Riesz potential, Manuscripta Math.156 (2018/07/01), no. 3, 521–543.↑2

    work page 2018

  21. [21]

    arXiv preprint arXiv:2111.10141.↑2

    ,Estimates for the variable order Riesz potential with applications, 2021. arXiv preprint arXiv:2111.10141.↑2

    work page arXiv 2021

  22. [22]

    Kokilashvili and S

    V . Kokilashvili and S. Samko,Maximal and fractional operators in weightedL p(x) spaces, Rev. Mat. Iberoam.20(200401). ↑2, 32

    work page

  23. [23]

    T. S. Kopaliani,Infimal convolution and MuckenhouptA p(·) condition in variableL p spaces, Arch. Math.89(2007), no. 2, 185–192.↑1, 2, 7

    work page 2007

  24. [24]

    A. K. Lerner,A boundedness criterion for the maximal operator on variable Lebesgue spaces, J. Analyse Math. (2025/07/13). ↑1

    work page 2025

  25. [25]

    ,The maximal operator on variable Lebesgue spaces: anA ∞-characterization(2026), available at2603.09388.↑1

    work page arXiv 2026

  26. [26]

    Lorist and Z

    E. Lorist and Z. Nieraeth,Extrapolation of compactness on Banach function spaces, J. Fourier Anal. Appl.30(June 2024), no. 3, 30. arXiv:2306.11449 [math].↑5

    work page arXiv 2024

  27. [27]

    Melchiori and G

    L. Melchiori and G. Pradolini,Potential operators and their commutators acting between variable Lebesgue spaces with different weights, Integral Transforms Spec. Funct.29(2018), no. 11, 909–926.↑2

    work page 2018

  28. [28]

    Mingione,Regularity of minima: an invitation to the dark side of the calculus of variations, Appl

    G. Mingione,Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math.51(2006), no. 4, 355–426.↑1

    work page 2006

  29. [29]

    B Morrey,On the solutions of quasi-linear elliptic partial differential equations, Trans

    C. B Morrey,On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc.43(1938), no. 1, 126–166.↑10

    work page 1938

  30. [30]

    Muscalu and W

    C. Muscalu and W. Schlag,Classical and multilinear harmonic analysis, Vol. II, Cambridge Studies in Advanced Mathe- matics 138, Cambridge University Press, 2013.↑21

    work page 2013

  31. [31]

    Nekvinda,Hardy-Littlewood maximal operator onL p(·)(Rn), Math

    A. Nekvinda,Hardy-Littlewood maximal operator onL p(·)(Rn), Math. Inequal. Appl.7(200401).↑1

    work page

  32. [32]

    Nieraeth,Extrapolation in general quasi-banach function spaces, J

    Z. Nieraeth,Extrapolation in general quasi-banach function spaces, J. Funct. Anal.285(2023), no. 10, 110–130.↑22, 23

    work page 2023

  33. [33]

    R. M. Pastrana, M. S. Riveros, and R. E. Vidal,Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators(2026), available at2605.08473.↑2

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  34. [34]

    R ˚ užiˇ cka,Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol

    M. R ˚ užiˇ cka,Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.↑1

    work page 2000

  35. [35]

    S. L. Sobolev,On a theorem of functional analysis, Mat. Sbornik4(1938), 471–497.↑1 DELFTINSTITUTE OFAPPLIEDMATHEMATICS, DELFTUNIVERSITY OFTECHNOLOGY, P.O. BOX5031, 2600 GA DELFT, THENETHERLANDS Email address:f.g.j.diasdecarvalho@tudelft.nl AALTOUNIVERSITY, DEPARTMENT OFMATHEMATICS ANDSYSTEMSANALYSIS, P.O. BOX11100, FI-00076 AALTO, FINLAND Email address:tu...

    work page 1938