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arxiv: 2604.13732 · v1 · submitted 2026-04-15 · 🧮 math.AP · math.FA

A note on Sobolev inequalities in the lower limit case

Petteri Harjulehto , Ritva Hurri-Syrj\"anen This is my paper

Pith reviewed 2026-05-10 12:54 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords Sobolev inequalitiesHausdorff contentquasicontinuous functionsW^{1,1}_0 spacePoincaré inequalitiesChoquet integrabilitysuperlevel inequalities
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The pith

A new Sobolev inequality holds for quasicontinuous functions in W^{1,1}_0(R^n) when gradients satisfy Choquet integrability with respect to Hausdorff content.

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

The paper proves Poincaré-Sobolev inequalities for compactly supported smooth functions in R^n whose gradients are Choquet δ/n-integrable against the δ-dimensional Hausdorff content, with δ ranging from 0 to n. These results directly yield a new Sobolev inequality that applies to the quasicontinuous representatives of functions in the Sobolev space W^{1,1}_0(R^n). A sympathetic reader would care because the standard Lebesgue-integrability version of the inequality excludes many functions whose gradients concentrate on lower-dimensional sets, and the weaker condition enlarges the class of admissible functions. The work also carries an existing superlevel Sobolev inequality over to the same Hausdorff-content setting.

Core claim

For compactly supported smooth functions u whose absolute gradient is Choquet δ/n-integrable with respect to the δ-dimensional Hausdorff content, Poincaré-Sobolev type inequalities hold when n is at least 2 and δ lies in (0,n]. These inequalities imply a new Sobolev inequality for quasicontinuous functions in W^{1,1}_0(R^n). The same framework extends a recently introduced superlevel Sobolev inequality to the Hausdorff-content context.

What carries the argument

Choquet δ/n-integrability of |∇u| with respect to the δ-dimensional Hausdorff content, which replaces ordinary Lebesgue integrability to accommodate gradients that may concentrate on sets of lower dimension.

If this is right

  • A Sobolev inequality holds for every quasicontinuous function belonging to W^{1,1}_0(R^n).
  • The superlevel Sobolev inequality extends directly to the setting of Hausdorff content.
  • Poincaré-Sobolev inequalities remain valid throughout the full range δ in (0,n] for n at least 2.

Where Pith is reading between the lines

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

  • The same technique may permit control of oscillations even when the gradient vanishes almost everywhere with respect to Lebesgue measure but not with respect to Hausdorff content.
  • Further adaptations could treat variational problems whose data are measures supported on lower-dimensional sets.
  • The quasicontinuous version supplies a natural bridge between Sobolev theory and capacity-based estimates.

Load-bearing premise

The functions must be compactly supported and smooth enough that their gradients are defined pointwise and satisfy the Choquet integrability condition with respect to the δ-dimensional Hausdorff content.

What would settle it

A single explicit radial or piecewise linear test function in W^{1,1}_0(R^n) whose gradient meets the Choquet δ/n-integrability condition yet violates the stated Sobolev inequality would disprove the central claim.

read the original abstract

We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $\delta /n$-integrable with respect to the $\delta$-dimensional Hausdorff content, $n\geq 2$, $\delta\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.

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

1 major / 0 minor

Summary. The paper establishes Poincaré-Sobolev-type inequalities for compactly supported smooth functions u in R^n (n ≥ 2) whose gradients satisfy a Choquet δ/n-integrability condition with respect to the δ-dimensional Hausdorff content, for δ ∈ (0,n]. It claims that these inequalities imply a new Sobolev inequality for quasicontinuous representatives of functions in the Sobolev space W^{1,1}_0(R^n), and applies the results to extend a recently introduced superlevel Sobolev inequality to the Hausdorff-content setting.

Significance. If the central claims hold, the work would supply a new family of inequalities in the borderline Sobolev case that incorporate non-additive Hausdorff content, potentially strengthening tools in nonlinear potential theory and capacity estimates. The extension to quasicontinuous functions and the superlevel application are presented as direct consequences, which, if rigorously justified, would be of interest to researchers working on Sobolev embeddings with measure-theoretic constraints.

major comments (1)
  1. The claimed implication from the inequalities on C_c^∞ functions to a Sobolev inequality on all quasicontinuous u ∈ W^{1,1}_0(R^n) (stated in the abstract and presumably proved in the main theorem) rests on density of smooth compactly supported functions. However, L^1 convergence of gradients does not automatically preserve finiteness of the Choquet δ/n-integral with respect to Hausdorff content, as this set function is not lower semi-continuous in the L^1 topology. A concrete justification is required showing that the right-hand side remains controlled in the limit; without it the extension to the full space is not established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for identifying a point that requires clarification in the extension to quasicontinuous functions. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The claimed implication from the inequalities on C_c^∞ functions to a Sobolev inequality on all quasicontinuous u ∈ W^{1,1}_0(R^n) (stated in the abstract and presumably proved in the main theorem) rests on density of smooth compactly supported functions. However, L^1 convergence of gradients does not automatically preserve finiteness of the Choquet δ/n-integral with respect to Hausdorff content, as this set function is not lower semi-continuous in the L^1 topology. A concrete justification is required showing that the right-hand side remains controlled in the limit; without it the extension to the full space is not established.

    Authors: We agree that the Choquet δ/n-integral with respect to Hausdorff content is not lower semi-continuous under L^1 convergence of gradients, so the density argument requires an explicit justification to control the right-hand side in the limit. The manuscript relies on the density of C_c^∞ in W^{1,1}_0 together with quasicontinuity of representatives. In the revised version we will insert a short approximation lemma: given u_k → u in W^{1,1}_0 with u_k ∈ C_c^∞ and u quasicontinuous, we show that the Choquet δ/n-integral of |∇u| is bounded by the liminf of the corresponding integrals for |∇u_k| by using the outer regularity of Hausdorff content, the definition of quasicontinuity (which controls the measure of exceptional sets), and the monotonicity properties of the content. This ensures the limiting inequality holds with the same constant. We thank the referee for highlighting this gap in the exposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard density and Hausdorff content properties

full rationale

The paper proves Poincare-Sobolev inequalities for C_c^∞ functions under the given Choquet integrability condition with respect to δ-Hausdorff content, then states that these imply a new inequality for quasicontinuous representatives in W^{1,1}_0(R^n). This extension is presented as following from the established results for smooth functions via approximation, without any self-definitional construction where the target inequality is used to define the inputs, without fitted parameters renamed as predictions, and without load-bearing self-citations that reduce the central claim to a prior result by the same authors. The derivation chain is self-contained against external benchmarks from classical Sobolev theory and non-additive set functions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the Choquet integral, Hausdorff content, and quasicontinuity in Sobolev spaces; no free parameters or new entities are introduced.

axioms (2)
  • standard math Properties of the Choquet integral with respect to Hausdorff content
    Invoked throughout the abstract as the integrability condition.
  • domain assumption Quasicontinuity of functions in W^{1,1}_0(R^n)
    Used to extend the inequality to the Sobolev space setting.

pith-pipeline@v0.9.0 · 5401 in / 1288 out tokens · 61961 ms · 2026-05-10T12:54:45.975766+00:00 · methodology

discussion (0)

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    R.: A note on Choquet integrals with respect to Hausdorffcapacity

    Adams, D. R.: A note on Choquet integrals with respect to Hausdorffcapacity. In: Cwikel, M., Peetre, J., Sagher, Y ., Wallin, H. (eds.) Function Spaces and Applica- tions (Lund 1986), Lecture Notes in Mathematics vol. 1302, Springer, Berlin (1988) pp. 115–124

    work page 1986

  2. [2]

    R.: Choquet Integrals in Potential Theory,Publ

    Adams, D. R.: Choquet Integrals in Potential Theory,Publ. Mat.42(1998), 3–66

    work page 1998

  3. [3]

    R.: Morrey Spaces, Birkh ¨auser, Cham–Heidelberg–New York, 2015

    Adams, D. R.: Morrey Spaces, Birkh ¨auser, Cham–Heidelberg–New York, 2015

    work page 2015

  4. [4]

    B., Roychowdhury, P.: Uncentered fractional maximal functions and mean oscillation spaces associated with dyadic Hausdorffcontent

    Basak, R., Chen, Y .-W. B., Roychowdhury, P.: Uncentered fractional maximal functions and mean oscillation spaces associated with dyadic Hausdorffcontent. arXiv:2506.23206. 12 PETTERI HARJULEHTO AND RITV A HURRI-SYRJ ¨ANEN

    work page arXiv

  5. [5]

    Math.62 (2011), no

    Cerd ´a, J., Mart ´ın, J., Silvestre, P.: Capacitary function spaces.Collect. Math.62 (2011), no. 1, 95–118

    work page 2011

  6. [6]

    B.: A self-improving property of Riesz potentials in BMO,J

    Chen, Y .-W. B.: A self-improving property of Riesz potentials in BMO,J. Geom. Anal.35(2025), no. 8, Paper No. 237, 23 pp

    work page 2025

  7. [7]

    B., Claros, A.:β-dimensional sharp maximal function and its applica- tions

    Chen, Y .-W. B., Claros, A.:β-dimensional sharp maximal function and its applica- tions. arXiv:2407.04456v3

    work page arXiv

  8. [8]

    B., Claros, A.: Commutators of fractional integrals withBMO β func- tions

    Chen, Y .-W. B., Claros, A.: Commutators of fractional integrals withBMO β func- tions. arXiv:2602.09742

    work page arXiv

  9. [9]

    B., Ooi, K.H., Spector, D.: Capacitary maximal inequalities and appli- cations,J

    Chen, Y .-W. B., Ooi, K.H., Spector, D.: Capacitary maximal inequalities and appli- cations,J. Funct. Anal.286(2024), no. 12, Paper No. 110396, 31 pp

    work page 2024

  10. [10]

    Chen, Y .-W., Spector, D.: On functions of boundedβ-dimensional mean oscillation, Adv. Calc. Var.17(2024), 975–996

    work page 2024

  11. [11]

    O.: A property of Hausdorffmeasure,Proc

    Davies, R. O.: A property of Hausdorffmeasure,Proc. Cambridge Philos. Soc.52 (1956), 30–34

    work page 1956

  12. [12]

    27, Kluwer Academic Publish- ers Group, Dordrecht, 1994

    Denneberg, D.: Non-additive measure and integral, Theory and Decision Library Series B: Mathematical and Statistical Methods vol. 27, Kluwer Academic Publish- ers Group, Dordrecht, 1994

    work page 1994

  13. [13]

    C., Gariepy, R

    Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992

    work page 1992

  14. [14]

    Federer, F.: Geometric Measure Theory, Die Grundlehren der Mathematischen Wis- senschaften, Band 153, Springer-Verlag New York Inc., New York, 1969

    work page 1969

  15. [15]

    Napli7(1958), 102–137

    Gagliardo, E.: Proprieta di alcune classi di funzioni in piu variabili,Ricerche di Mat. Napli7(1958), 102–137

    work page 1958

  16. [16]

    Gilbarg, D., Trudinger, N.:Elliptic partial differential equations of second order, Reprint of the 1998 Edition, Springer-Verlag, Berlin Heidelberg, 2001

    work page 1998

  17. [17]

    Hatano, N., Kawasumi, R., Saito H., Tanaka, H.: Choquet integrals, Hausdorffcon- tent and fractional operators,Bull. Austr. Math. Soc.110(2024), Issue 2, 355–366

    work page 2024

  18. [18]

    Hatano, N., Kawasumi, R., Saito H., Tanaka, H.: Correction to ’Choquet integrals, Hausdorffcontent and fractional operators’,Bull. Austr. Math. Soc.111(2025), Issue 3, 568–570

    work page 2025

  19. [19]

    Harjulehto, P., Hurri-Syrj ¨anen, R.: On Choquet integrals and Poincar´e-Sobolev type inequalities,J. Funct. Anal.284(2023), issue 9, paper no. 109862

    work page 2023

  20. [20]

    On Choquet integrals and pointwise estimates

    Harjulehto, P., Hurri-Syrj ¨anen:, R. On Choquet integrals and pointwise estimates. In: M. Breit-Goodwin et al. (eds.),AWM Research Symbosium, Minneapolis, MN, June 2022,Analysis of Partial Differential Equations in Memory of David R. Adams, Advances in the Mathematical Sciences, Association for Women in Mathematics Series 38, Springer, Cham (2025) pp.147–164

    work page 2022

  21. [21]

    Harjulehto, P., Hurri-Syrj ¨anen, R.: On Choquet integrals and Sobolev type inequal- ities,La Matematica3(2024), 1379–1399

    work page 2024

  22. [22]

    Harjulehto, P., Hurri-Syrj ¨anen, R.: On Hausdorffcontent maximal operator and Riesz potential for non-measurable functions,J. Inequal. Appl.2026(2026), arti- cle number 33, doi: 10.1186/s13660-026-03440-9

    work page doi:10.1186/s13660-026-03440-9 2026

  23. [23]

    Harjulehto, P., Hurri-Syrj ¨anen, R.: On Lebesgue points and measurability with Cho- quet integrals.J. Geom. Anal., doi:10.1007/s12220-026-02419-8

    work page doi:10.1007/s12220-026-02419-8

  24. [24]

    arXiv:2311.15224

    Huang, L., Cao, Y ., Yang, D., Zhuo, C.: Poincar ´e-Sobolev inequalities based on Choquet-Lorentz integrals with respect to Hausdorffcontents on bounded John do- mains. arXiv:2311.15224

    work page arXiv

  25. [25]

    arXiv:2509.20326v2

    Kangasniemi, I., Onninen, J.: Continuity for Sobolev mappings with null Lagrande bounds. arXiv:2509.20326v2

    work page arXiv

  26. [26]

    A NOTE ON SOBOLEV INEQUALITIES IN THE LOWER LIMIT CASE 13

    Mattila, P.:Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press, London, 1995. A NOTE ON SOBOLEV INEQUALITIES IN THE LOWER LIMIT CASE 13

    work page 1995

  27. [27]

    G.: A Theory of Capacities for Potentials of Functions in Lebesgue Classes,Math

    Meyers, N. G.: A Theory of Capacities for Potentials of Functions in Lebesgue Classes,Math. Scand.26(1970), 255–292

    work page 1970

  28. [28]

    Scuola Norm

    Nirenberg, L.: On elliptic partial differential equations,Ann. Scuola Norm. Sup. Pisa (III)13(1959), 1–48

    work page 1959

  29. [29]

    H., Phuc, N

    Ooi, K. H., Phuc, N. C.: The Hardy-Littlewood maximal function, Choquet inte- grals, and embeddings of Sobolev type,Math. Ann.382(2022), 1865–1879

    work page 2022

  30. [30]

    Ponce, A., Spector, D.: A boxing inequality for the fractional perimeter.Ann. Sc. Norm. Super. Pisa Cl. Sci.(5)20(2020), 107–141

    work page 2020

  31. [31]

    Sion, M., Sjerve, D.: Approximation properties of measures generated by continu- ous set functions,Mathematika9(1962), 145–156

    work page 1962

  32. [32]

    Yang, D., Yuan, W.: A note on dyadic Hausdorffcapacities,Bull. Sci. Math.132 (2008), 500–509. (Petteri Harjulehto) Department ofMathematics andStatistics, FI-00014 University ofHelsinki, Finland Email address:petteri.harjulehto@helsinki.fi (Ritva Hurri-Syrj¨anen) Department ofMathematics andStatistics, FI-00014 Univer- sity ofHelsinki, Finland Email addre...

    work page 2008