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arxiv: 2604.08416 · v1 · submitted 2026-04-09 · 🧮 math.CA · math.AP

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The two-weight fractional Poincar\'e-Sobolev sandwich

Carel Wagenaar, Emiel Lorist

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Pith reviewed 2026-05-10 16:48 UTC · model grok-4.3

classification 🧮 math.CA math.AP
keywords two-weight inequalitiesfractional Poincaré-SobolevTriebel-Lizorkin spacesMuckenhoupt weightssparse dominationSobolev embeddingsdifference norms
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The pith

Two weighted inequalities sandwich the fractional Sobolev seminorm, one bounding it by the gradient and the other embedding the Sobolev space into a difference-norm Triebel-Lizorkin space, with constants asymptotically sharp as the order ne

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

The paper establishes a pair of inequalities that together control the fractional Sobolev seminorm in the presence of two Muckenhoupt weights. The first is a two-weight fractional Poincaré-Sobolev inequality that relates the seminorm to a weighted integral of the gradient. The second embeds the classical first-order Sobolev space into a Triebel-Lizorkin space whose norm is expressed through a difference operator. Both results supply explicit dependence on the weight characteristics and remain valid as the fractional parameter s tends to 1 from below, becoming sharp in that limit. The proofs split into elementary estimates for the subcritical range and sparse domination for the critical range, and include a new sparse domination statement tailored to the difference-norm Triebel-Lizorkin spaces.

Core claim

We establish a two-weight fractional Poincaré-Sobolev sandwich, consisting of a two-weight fractional Poincaré-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches 1. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin

What carries the argument

The two-weight fractional Poincaré-Sobolev sandwich, which pairs a Poincaré-Sobolev inequality with an embedding into a difference-norm Triebel-Lizorkin space and is proved using elementary estimates in the subcritical regime together with a new sparse domination theorem in the critical regime.

If this is right

  • The inequalities supply explicit constants that depend only on the A_p characteristics of the two weights.
  • The constants become asymptotically optimal as the fractional parameter s approaches 1 from below.
  • A new sparse domination principle holds for Triebel-Lizorkin spaces equipped with difference norms.
  • The results cover both subcritical and critical regimes and remain valid when the two weights coincide.
  • The methods unify and extend earlier approaches to one-weight and two-weight fractional inequalities.

Where Pith is reading between the lines

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

  • The quantitative constants could be inserted directly into existence proofs for solutions of fractional PDEs with weighted data.
  • The sparse domination technique developed for the difference norms might adapt to other Besov-type spaces that use similar oscillations.
  • Because the constants are tracked through the A_p constants, one can test the inequalities numerically on power weights to measure the actual gap to optimality for fixed s.

Load-bearing premise

The two weights belong to suitable Muckenhoupt A_p classes whose oscillation can be controlled by a fixed constant.

What would settle it

A concrete pair of Muckenhoupt weights together with a test function for which the ratio between the fractional seminorm and the gradient term exceeds the claimed bound for some s in (0,1), or for which the ratio fails to approach the expected limiting value as s tends to 1.

read the original abstract

We establish a two-weight fractional Poincar\'e-Sobolev sandwich, consisting of a two-weight fractional Poincar\'e-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes a two-weight fractional Poincaré-Sobolev sandwich: a two-weight fractional Poincaré-Sobolev inequality together with a two-weight embedding of the first-order Sobolev space into a Triebel-Lizorkin space defined via a difference norm. Constants are asymptotically sharp as the fractional parameter s approaches 1. Subcritical regimes are handled by elementary estimates and critical regimes by a new sparse-domination result for the difference norms; quantitative dependence on Muckenhoupt characteristics is given throughout. The results are asserted to be new even in the one-weight case.

Significance. If the derivations hold, the work supplies a unified quantitative framework for weighted fractional Sobolev-type inequalities, with explicit control on weight characteristics and a new sparse bound for Triebel-Lizorkin difference norms that may be reusable. The asymptotic sharpness as s→1 and the claimed novelty already in the one-weight setting are concrete strengths that distinguish the contribution from prior literature.

major comments (2)
  1. [§4, Theorem 4.2] §4, Theorem 4.2 (critical regime): the sparse domination for the Triebel-Lizorkin difference norm is the load-bearing step; the proof sketch indicates a reduction to a maximal-function bound, but it is not clear whether the implicit constant remains independent of the Muckenhoupt characteristics when the weight pair (u,v) is allowed to vary arbitrarily within A_p × A_q.
  2. [§5.3, display (5.12)] §5.3, display (5.12): the lower bound for the constant as s→1 is obtained by testing on a specific radial function; this function must be verified to lie in the Sobolev space for every pair of weights in the stated classes, otherwise the claimed asymptotic sharpness is only conditional.
minor comments (3)
  1. [Theorem 1.1] The statement of the main sandwich theorem (Theorem 1.1) should explicitly record the admissible range of s (0,1) and the precise relation between p and q.
  2. [§2 and §4] Notation for the difference norm in the Triebel-Lizorkin space is introduced in §2 but reused with a slightly different normalization in §4; a single displayed definition would improve readability.
  3. [Introduction and §3] Several references to earlier sparse-domination results are cited only by author-year; adding the precise theorem numbers from those papers would help readers locate the exact statements being extended.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. The two major comments are addressed point-by-point below with clarifications and planned revisions.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4, Theorem 4.2 (critical regime): the sparse domination for the Triebel-Lizorkin difference norm is the load-bearing step; the proof sketch indicates a reduction to a maximal-function bound, but it is not clear whether the implicit constant remains independent of the Muckenhoupt characteristics when the weight pair (u,v) is allowed to vary arbitrarily within A_p × A_q.

    Authors: The sparse domination in the proof of Theorem 4.2 is obtained by reducing the difference-norm Triebel-Lizorkin operator to a maximal-function bound whose constants depend only on the Muckenhoupt characteristics [u]_{A_p} and [v]_{A_q} (via the standard weighted maximal-function estimates of Muckenhoupt and Wheeden). The dependence is therefore independent of the particular choice of weights inside the classes. We will insert an explicit remark after the statement of the sparse bound to record this independence and cite the relevant maximal-function theorem. revision: partial

  2. Referee: [§5.3, display (5.12)] §5.3, display (5.12): the lower bound for the constant as s→1 is obtained by testing on a specific radial function; this function must be verified to lie in the Sobolev space for every pair of weights in the stated classes, otherwise the claimed asymptotic sharpness is only conditional.

    Authors: The radial test function used for the lower bound in (5.12) is a standard compactly supported Lipschitz function (essentially a smoothed characteristic function of the unit ball). For any u ∈ A_p and v ∈ A_q this function belongs to the weighted Sobolev space because A_p weights are locally integrable and the function together with its gradient is bounded with compact support. We will add a short verification paragraph in §5.3 making this membership explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its two-weight fractional Poincaré-Sobolev inequalities and embeddings via original elementary estimates in the subcritical regime and a newly established sparse domination result for Triebel-Lizorkin difference norms in the critical regime. Explicit dependence on Muckenhoupt characteristics is provided, with asymptotic sharpness as s → 1 claimed through direct analysis rather than fitting or renaming. No load-bearing steps reduce by construction to self-definitions, fitted parameters presented as predictions, or chains of self-citations that render the central results tautological. The derivation remains self-contained with independent content from prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard background facts from harmonic analysis: Muckenhoupt weight classes control oscillation, Triebel-Lizorkin spaces admit difference-norm characterizations, and sparse operators dominate certain maximal and fractional integrals.

axioms (2)
  • domain assumption Weights belong to Muckenhoupt classes A_p
    Required for the explicit quantitative dependence on weight characteristics in both subcritical and critical regimes.
  • standard math Triebel-Lizorkin spaces admit equivalent difference-norm characterizations
    Background fact used to define the target space in the embedding part of the sandwich.

pith-pipeline@v0.9.0 · 5423 in / 1428 out tokens · 50036 ms · 2026-05-10T16:48:26.552579+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincar\'e-Sobolev inequalities

    math.CA 2026-04 unverdicted novelty 7.0

    Asymptotically sharp A_∞ to A_p embedding for flat weights with [w]_{A_∞} near 1, yielding quantitative weighted Poincaré-Sobolev inequalities that approach the unweighted case.

Reference graph

Works this paper leans on

50 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bourgain, H

    J. Bourgain, H. Brezis, and P. Mironescu. Another look at S obolev spaces. In Optimal control and partial differential equations , pages 439--455. IOS, Amsterdam, 2001

    2001

  2. [2]

    Bourgain, H

    J. Bourgain, H. Brezis, and P. Mironescu. Limiting embedding theorems for W^ s,p when s 1 and applications. J. Anal. Math. , 87:77--101, 2002

    2002

  3. [3]

    Barron, J.M

    A. Barron, J.M. Conde-Alonso , Y. Ou, and G. Rey. Sparse domination and the strong maximal function. Adv. Math. , 345:1--26, 2019

    2019

  4. [4]

    Brazke, A

    D. Brazke, A. Schikorra, and P.-L. Yung. Bourgain- B rezis- M ironescu convergence via T riebel- L izorkin spaces. Calc. Var. Partial Differential Equations , 62(2):Paper No. 41, 33, 2023

    2023

  5. [5]

    Y. Chen, L. Grafakos, D. Yang, and W. Yuan. The ball B anach fractional S obolev inequality and its applications. Studia Math. , 284(1):1--37, 2025

    2025

  6. [6]

    S.-K. Chua. Weighted S obolev inequalities on domains satisfying the chain condition. Proc. Amer. Math. Soc. , 117(2):449--457, 1993

    1993

  7. [7]

    Degenerate P oincar\'e- S obolev inequalities via fractional integration

    Alejandro Claros. Degenerate P oincar\'e- S obolev inequalities via fractional integration. J. Funct. Anal. , 289(6):Paper No. 111000, 28, 2025

    2025

  8. [8]

    Cruz-Uribe and K

    D. Cruz-Uribe and K. Moen. A fractional M uckenhoupt- W heeden theorem and its consequences. Integral Equations Operator Theory , 76(3):421--446, 2013

    2013

  9. [9]

    Cejas, C

    M.E. Cejas, C. Mosquera, C. P\' e rez, and E. Rela. Self-improving P oincar\' e - S obolev type functionals in product spaces. J. Anal. Math. , 149(1):1--48, 2023

    2023

  10. [10]

    Chanillo and R.L

    S. Chanillo and R.L. Wheeden. Weighted P oincar\' e and S obolev inequalities and estimates for weighted P eano maximal functions. Amer. J. Math. , 107(5):1191--1226, 1985

    1985

  11. [11]

    Chanillo and R.L

    S. Chanillo and R.L. Wheeden. Poincar\' e inequalities for a class of non- A_p weights. Indiana Univ. Math. J. , 41(3):605--623, 1992

    1992

  12. [12]

    Drelichman and R.G

    I. Drelichman and R.G. Dur \'a n. Improved Poincar \'e inequalities with weights. J. Math. Anal. Appl. , 347(1):286--293, 2008

    2008

  13. [13]

    F. Dai, L. Grafakos, Z. Pan, D. Yang, W. Yuan, and Y. Zhang. The B ourgain- B rezis- M ironescu formula on ball B anach function spaces. Math. Ann. , 388(2):1691--1768, 2024

    2024

  14. [14]

    V\"ah\"akangas

    Bart omiej Dyda, Lizaveta Ihnatsyeva, and Antti V. V\"ah\"akangas. On improved fractional S obolev- P oincar\'e inequalities. Ark. Mat. , 54(2):437--454, 2016

    2016

  15. [15]

    Dom \' nguez, Y

    O. Dom \' nguez, Y. Li, S. Tikhonov, D. Yang, and W. Yuan. A unified approach to self-improving property via K -functionals. Calc. Var. Partial Differential Equations , 63(9):Paper No. 231, 50, 2024

    2024

  16. [16]

    Dom\' nguez and M

    O. Dom\' nguez and M. Milman. Bourgain- B rezis- M ironescu- M az'ya- S haposhnikova limit formulae for fractional S obolev spaces via interpolation and extrapolation. Calc. Var. Partial Differential Equations , 62(2):Paper No. 43, 37, 2023

    2023

  17. [17]

    Diening, M

    L. Diening, M. Ru z i c ka, and K. Schumacher. A decomposition technique for J ohn domains. Ann. Acad. Sci. Fenn. Math. , 35(1):87--114, 2010

    2010

  18. [18]

    Fackler and T.P

    S. Fackler and T.P. Hyt\" o nen. Off-diagonal sharp two-weight estimates for sparse operators. New York J. Math. , 24:21--42, 2018

    2018

  19. [19]

    Fabes, C.E

    E.B. Fabes, C.E. Kenig, and R.P. Serapioni. The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations , 7:77--116, 1982

    1982

  20. [20]

    Franchi, C

    B. Franchi, C. P\' e rez, and R.L. Wheeden. Self-improving properties of J ohn- N irenberg and P oincar\' e inequalities on spaces of homogeneous type. J. Funct. Anal. , 153(1):108--146, 1998

    1998

  21. [21]

    Grafakos

    L. Grafakos. Classical F ourier analysis , volume 249 of Graduate Texts in Mathematics . Springer, New York, third edition, 2014

    2014

  22. [22]

    a skyl\

    P. Haj asz. Sobolev inequalities, truncation method, and J ohn domains. In Papers on analysis , volume 83 of Rep. Univ. Jyv\" a skyl\" a Dep. Math. Stat. , pages 109--126. Univ. Jyv\" a skyl\" a , Jyv\" a skyl\" a , 2001

    2001

  23. [23]

    Heinonen, T

    J. Heinonen, T. Kilpel \"a inen, and O. Martio. Nonlinear potential theory of degenerate elliptic equations . Mineola, NY: Dover Publications, unabridged republication of the 1993 original edition, 2006

    1993

  24. [24]

    Hyt\"onen and K

    T.P. Hyt\"onen and K. Li. Weak and strong A_p - A_ estimates for square functions and related operators. Proc. Amer. Math. Soc. , 146(6):2497--2507, 2018

    2018

  25. [25]

    P. Hu, Y. Li, D. Yang, and W. Yuan. A sharp localized weighted inequality related to G agliardo and S obolev seminorms and its applications. Adv. Math. , 481:Paper No. 110537, 66, 2025

    2025

  26. [26]

    a nen , J.C. Mart\' nez-Perales , C. P\' e rez, and A.V. V\

    R. Hurri-Syrj\" a nen , J.C. Mart\' nez-Perales , C. P\' e rez, and A.V. V\" a h\" a kangas. On the BBM -phenomenon in fractional P oincar\' e - S obolev inequalities with weights. Int. Math. Res. Not. IMRN , (20):17205--17244, 2023

    2023

  27. [27]

    a nen , J.C. Mart \'i nez-Perales , C. P \'e rez, and A.V. V \

    R. Hurri-Syrj \"a nen , J.C. Mart \'i nez-Perales , C. P \'e rez, and A.V. V \"a h \"a kangas. On the weighted inequality between the G agliardo and S obolev seminorms. Israel Journal of Mathematics , Nov 2025

    2025

  28. [28]

    onen, C. P\'erez, and E. Rela. Sharp reverse H \

    T.P. Hyt\"onen, C. P\'erez, and E. Rela. Sharp reverse H \"older property for A_ weights on spaces of homogeneous type. J. Funct. Anal. , 263(12):3883--3899, 2012

    2012

  29. [29]

    a nen and A.V. V\

    R. Hurri-Syrj\" a nen and A.V. V\" a h\" a kangas. On fractional P oincar\' e inequalities. J. Anal. Math. , 120:85--104, 2013

    2013

  30. [30]

    He and D

    Q. He and D. Yan. Sharp off-diagonal weighted weak type estimates for sparse operators. Math. Inequal. Appl. , 23(4):1479--1498, 2020

    2020

  31. [31]

    Hyt\" o nen

    T.P. Hyt\" o nen. The L^p -to- L^q boundedness of commutators with applications to the J acobian operator. J. Math. Pures Appl. (9) , 156:351--391, 2021

    2021

  32. [32]

    a ck, and A. V\

    J. Kinnunen, J. Lehrb\" a ck, and A. V\" a h\" a kangas. Maximal function methods for S obolev spaces , volume 257 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2021

    2021

  33. [33]

    Kurki and A.V

    E.-K. Kurki and A.V. V\" a h\" a kangas. Weighted norm inequalities in a bounded domain by the sparse domination method. Rev. Mat. Complut. , 34(2):435--467, 2021

    2021

  34. [34]

    Lerner, E

    A.K. Lerner, E. Lorist, and S. Ombrosi. Operator-free sparse domination. Forum Math. Sigma , 10:Paper No. e15, 28, 2022

    2022

  35. [35]

    Lacey, K

    M.T. Lacey, K. Moen, C. P\' e rez, and R.H. Torres. Sharp weighted bounds for fractional integral operators. J. Funct. Anal. , 259(5):1073--1097, 2010

    2010

  36. [36]

    Lorist and Z

    E. Lorist and Z. Nieraeth. Banach function spaces done right. Indag. Math., New Ser. , 35(2):247--268, 2024

    2024

  37. [37]

    Lerner and C

    A.K. Lerner and C. P\' e rez. Self-improving properties of generalized P oincar\' e type inequalities through rearrangements. Math. Scand. , 97(2):217--234, 2005

    2005

  38. [38]

    Lacey, E.T

    M. Lacey, E.T. Sawyer, and I. Uriarte-Tuero . Two weight inequalities for discrete positive operators . arXiv:0911.3437, 2009

    work page arXiv 2009

  39. [39]

    K. Mohanta. Bourgain- B rezis- M ironescu formula for W^ s,p _q -spaces in arbitrary domains. Calc. Var. Partial Differential Equations , 63(2):Paper No. 31, 17, 2024

    2024

  40. [40]

    MacManus and C

    P. MacManus and C. P\' e rez. Generalized P oincar\' e inequalities: sharp self-improving properties. Internat. Math. Res. Notices , (2):101--116, 1998

    1998

  41. [41]

    Myyryl\" a inen, C

    K. Myyryl\" a inen, C. P\' e rez, and J. Weigt. Weighted fractional P oincar\' e inequalities via isoperimetric inequalities. Calc. Var. Partial Differential Equations , 63(8):Paper No. 205, 32, 2024

    2024

  42. [42]

    Maz'ya and T

    V. Maz'ya and T. Shaposhnikova. On the B ourgain, B rezis, and M ironescu theorem concerning limiting embeddings of fractional S obolev spaces. J. Funct. Anal. , 195(2):230--238, 2002

    2002

  43. [43]

    Nieraeth and C.B

    Z. Nieraeth and C.B. Stockdale. Endpoint weak-type bounds beyond C alder\'on-- Z ygmund theory. arXiv:2409.08921, 2024

    work page arXiv 2024

  44. [44]

    Nieraeth, C.B

    Z. Nieraeth, C.B. Stockdale, and B. Sweeting. Weighted weak-type bounds for multilinear singular integrals. arXiv:2401.15725, 2024

    work page arXiv 2024

  45. [45]

    P\' e rez and E

    C. P\' e rez and E. Rela. Degenerate P oincar\' e - S obolev inequalities. Trans. Amer. Math. Soc. , 372(9):6087--6133, 2019

    2019

  46. [46]

    M. Prats. Measuring T riebel- L izorkin fractional smoothness on domains in terms of first-order differences. J. Lond. Math. Soc. (2) , 100(2):692--716, 2019

    2019

  47. [47]

    Z. Pan, D. Yang, W. Yuan, and Y. Zhang. Gagliardo representation of norms of ball quasi- B anach function spaces. J. Funct. Anal. , 286(2):Paper No. 110205, 78, 2024

    2024

  48. [48]

    Sawyer and R.L

    E. Sawyer and R.L. Wheeden. Weighted inequalities for fractional integrals on E uclidean and homogeneous spaces. Amer. J. Math. , 114(4):813--874, 1992

    1992

  49. [49]

    H. Triebel. Theory of function spaces , volume 78 of Monographs in Mathematics . Birkh\"auser Verlag, Basel, 1983

    1983

  50. [50]

    Z. Zeng, D. Yang, and W. Yuan. Extension and embedding of T riebel- L izorkin-type spaces built on ball quasi- B anach spaces. J. Geom. Anal. , 34(11):Paper No. 335, 71, 2024

    2024