A note on P\'{o}lya-Szeg\"{o} inequality for fractional Orlicz-Sobolev seminorm in domains

Remi Yvant Temgoua

REVIEW 3 major objections 5 minor 16 references

A note on P\'{o}lya-Szeg\"{o} inequality for fractional Orlicz-Sobolev seminorm in domains

T0 review · 3 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 that Schwarz symmetrization can strictly increase the fractional Orlicz-Sobolev seminorm on bounded domains, so the domain version of the Pólya-Szegő inequality fails for every Young function with power-type growth.

desk verdict A genuine extension of Li-Wang to fractional Orlicz-Sobolev spaces, but the proof of Theorem 1.1 misses the case where a non-ball domain differs from its rearrangement by a null set. read the letter →

arxiv 2411.18188 v4 pith:7BUXKL7H submitted 2024-11-27 math.AP

Remi Yvant Temgoua This is my paper · ORCID

classification math.AP MSC 46E3045G05

keywords Pólya-SzegőinequalitysymmetricdecreasingrearrangementfractionalOrlicz-SobolevspaceGagliardoseminormYoungfunctionHardyboundeddomain

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

Symmetric decreasing rearrangement, the operation that replaces a function by its radially decreasing equimeasurable rearrangement, is known to shrink the whole-space fractional Orlicz-Sobolev seminorm. This paper shows the opposite can happen in a bounded domain: for any Young function $G$ with power-type growth, there is a smooth compactly supported function whose rearrangement has a strictly larger fractional seminorm on the rearranged domain $\Omega^*$. That is, the Pólya-Szegő principle for the regional fractional seminorm fails in domains, unlike in the whole space. The paper also proves a one-sided reverse estimate: for Lipschitz-type domains, the rearrangement seminorm is bounded by a constant times the original seminorm, provided a fractional Orlicz-Hardy inequality holds.

What carries the argument

The central object is the radial-integral comparison lemma (Lemma 3.1 of [11]): if $f$ is radial and strictly decreasing, then $\int_{\Omega^*} f\,dx > \int_\Omega f\,dx$ whenever $\Omega^*$ and $\Omega$ differ, and the same strict inequality holds for the exterior integrals $\int_{\mathbb{R}^N\setminus\Omega} f\,dx$ and $\int_{\mathbb{R}^N\setminus\Omega^*} f\,dx$. Theorem 1.1 applies this with $f(y)=|y|^{-N}G(|\eta(x)|/|y|^s)$ to the cross-interaction terms in the double integral, after splitting $\Omega\times\Omega$ into $\mathbb{R}^N\times\mathbb{R}^N$ minus the boundary interactions. For the converse, the machinery is the whole-space Pólya-Szegő inequality (1.5) from [6] combined with a boundary-distance estimate from [8] and the fractional Orlicz-Hardy inequality from [3], which together control the boundary-interaction term by the fractional seminorm.

What would settle it

For a fixed non-ball bounded open set $\Omega$ and a Young function $G$ satisfying (1.1), take the family $u_\varepsilon(x)=\eta(x/\varepsilon)$ used in the proof, where $\eta$ is a nonnegative radial decreasing bump supported in a small ball inside $\Omega$, and compute the ratio of the rearranged seminorm on $\Omega^*$ to the original seminorm on $\Omega$. Theorem 1.1 asserts this ratio is strictly greater than $1$ for all sufficiently small $\varepsilon$; finding any such family with ratio $\le 1$ at arbitrarily small $\varepsilon$ would falsify the theorem.

Watch

Extended reading notes

Core claim

For any Young function $G$ satisfying $1 < p^-_G \le tg(t)/G(t) \le p^+_G < \infty$, any $N \ge 1$, $s \in (0,1)$, and any nonempty open set $\Omega$ with $|\Omega|<\infty$, there exists a nonnegative $u \in C^\infty_c(\Omega)$ such that $$\int_{\$\Omega$\times\$\Omega$} G\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right) \frac{dx\,dy}{|x-y|^N} < \int_{\$\Omega$^*\times\$\Omega$^*} G\left(\frac{|u^*(x)-u^*(y)|}{|x-y|^s}\right) \frac{dx\,dy}{|x-y|^N}.$$ In words, the fractional Orlicz-Sobolev seminorm in domains is not decreasing under Schwarz symmetrization, even though the whole-space inequality (1.5) holds. The construction concentrates a bump near the boundary and uses a radial-integral comparison to show the boundary-interaction term dominates, making the rearranged seminorm strictly larger.

Load-bearing premise

The load-bearing premise is the fractional Orlicz-Hardy inequality used at (3.9): on each domain allowed in Theorem 1.2 and for each Young function satisfying (1.1), the integral $\int_\Omega G(|u|/\delta^s)\,dx$ must be bounded by a constant times the fractional seminorm $\int_{\Omega\times\Omega} G(|u(x)-u(y)|/|x-y|^s)\,dx\,dy/|x-y|^N$. If that inequality fails for any permitted domain and function, the reverse estimate (1.9) collapses.

Editorial extensions

If this is right

For every Young function satisfying (1.1), there are smooth compactly supported functions on any finite-measure open set whose Schwarz symmetrization has a strictly larger fractional Orlicz-Sobolev seminorm, so the regional seminorm is not rearrangement-decreasing.
The same statement covers the ball case, where $\Omega^*=\Omega$, via a translation argument, so the failure is not an artifact of irregular domains.
On bounded Lipschitz domains, epigraphs of Lipschitz functions, and exteriors of bounded Lipschitz domains, the rearranged seminorm is bounded above by a constant multiple of the original seminorm whenever the fractional Orlicz-Hardy inequality applies.
The results reproduce the known power-law case $G(t)=t^p$ with $p\ge 2$ and extend it to non-power growths such as $t^p(1+|\log t|)$, $t^p/\log(e+t)$, and the double-phase model $t^q+t^p$.

Reading between the lines

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

As $s\to 1$, the fractional seminorm with the usual normalization converges to the local Dirichlet energy, for which the Pólya-Szegő inequality holds in domains; the non-monotonicity is therefore a genuinely nonlocal phenomenon and should disappear in that limit.
The failure of (1.6) does not settle the paper's open questions about whether balls are extremal for the first eigenvalue or the best Poincaré constant of the regional fractional $g$-Laplacian; it only rules out the rearrangement-based route to a Faber-Krahn theorem.
The concentration mechanism behind Theorem 1.1 uses only that $y\mapsto |y|^{-N}G(|\eta(x)|/|y|^s)$ is radial and strictly decreasing, so the same counterexample should work for other radial nonlocal kernels under mild growth assumptions.

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.

Desk Editor's Note

read the letter

Short version: this is a legitimate extension of Li-Wang's counterexample to the fractional Orlicz-Sobolev setting, and the quantitative reverse estimate is a nice addition. But the proof of Theorem 1.1 has a gap that the current text does not close.

What is actually new: for Young functions satisfying (1.1), Theorem 1.1 shows that symmetric rearrangement can strictly increase the regional seminorm, and Theorem 1.2 gives a matching bound up to a constant. The examples G(t)=t^p(1+|log t|) and double-phase G=t^q+t^p are real new cases beyond powers. The proof strategy is transferred from Li-Wang rather than invented, but the Orlicz growth bookkeeping via Lemma 2.1 is handled cleanly.

The main soft spot is in the first line of the non-ball case in Theorem 1.1. The proof asserts that if Ω is not a ball then |Ω*ΔΩ|>0. That is false: take Ω=B_1\{0} in R^N. This set is open, not a ball, yet Ω*=B_1 and the symmetric difference is just the null set {0}. For such Ω, Lemma 3.1(ii) cannot be applied, H_Ω(0) equals H_Ω*(0), and the strict inequality in (3.4) collapses to equality. The gap is patchable: when |Ω*ΔΩ|=0, the regional seminorms on Ω agree with those on a ball, and the ball-case construction with a shifted bump should give the result. But that patch is not in the paper, and a referee should ask for it to be written out. This is more than a typo; it is an unproved case in the central claim.

A minor issue: the notation around B_R0 and x0 is sloppy. The construction wants a ball centered at the origin (or a translation normalization) for the identity u_ε*=η(x/ε) to hold. This is easily fixed.

Theorem 1.2 is in better shape. The dependence on the fractional Orlicz-Hardy inequality from Bal–Mohanta–Roy–Sk is explicit, and the domain hypotheses are exactly what that external result requires. The estimate is only as general as [3], but that is a stated assumption, not a hidden one.

Who this is for: people working on rearrangement inequalities, nonlocal Orlicz-Sobolev spaces, and shape optimization for regional fractional g-Laplacians. The paper deserves a serious referee. I would send it out with a request to fix the null-symmetric-difference case in Theorem 1.1 and to clean up the normalization issue.

Referee Report

3 major / 5 minor

Summary. The paper studies whether the Pólya-Szegő inequality for the fractional Orlicz-Sobolev seminorm remains valid when the whole space is replaced by a finite-measure domain Ω. The main result, Theorem 1.1, claims that for every Young function G satisfying (1.1), every N≥1, s∈(0,1), and every nonempty open Ω with |Ω|<∞, there is a nonnegative u∈C∞c(Ω) whose seminorm over Ω×Ω is strictly smaller than the seminorm of its symmetric decreasing rearrangement u∗ over Ω∗×Ω∗; in other words, symmetrization can increase the domain seminorm. Theorem 1.2 gives a complementary reverse estimate up to a constant for three classes of domains, under lim-inf conditions on an auxiliary function β_{s,G}, using the fractional Orlicz-Hardy inequality of [3]. The proofs adapt the explicit computation of Li and Wang [11] and combine it with the Pólya-Szegő principle of De Nápoli–Bonder–Salort [6].

Significance. If the gaps in the proofs are repaired, the note provides a broad extension of [11] to Orlicz growth and shows that the failure of the domain Pólya-Szegő principle is not an artifact of power growth. The paper is honest about its reliance on external results, uses those results as tools rather than assuming the target inequalities, and is therefore not circular. The arguments are explicit and checkable, and the reverse estimate in Theorem 1.2 connects the phenomenon to Hardy inequalities in a useful way. The contribution is modest but appropriate for a note.

major comments (3)

[Section 3, proof of Theorem 1.1 (non-ball case)] The assertion '|Ω \ Ω∗| = |Ω∗ \ Ω| = 1/2 |Ω∗∆Ω| > 0 since Ω is not a ball' is false: the open set Ω = B1 \ {0} satisfies Ω∗ = B1 and |Ω∗∆Ω| = 0, although Ω is not a ball. For such Ω, Lemma 3.1(ii) cannot be invoked, HΩ(0) and HΩ∗(0) coincide up to null sets, and the strict inequality (3.4) is not obtained by the stated dominated-convergence argument. This leaves a genuine class of domains in Theorem 1.1 untreated. The gap appears patchable — the ball-case construction with a translated bump should cover these domains — but the proof as written must be amended.
[Section 3, proof of Theorem 1.1] The definition uε(x) = η(x/ε) is not compatible with the preceding choice of η as a radial decreasing function with η = 1 on {|x − x0| ≤ R0/2}. If η is centered at x0, the support of uε lies near εx0 and need not be contained in Ω, and the identity u∗ε(x) = η(x/ε) used in (3.3) is not correct. The proof should either translate coordinates so that x0 = 0 before defining uε, or define uε(x) = η((x − x0)/ε) with η centered at 0; the normalization must be stated explicitly.
[Section 3 and Lemma 3.1] Lemma 3.1 is stated only for bounded open sets, but Theorem 1.1 allows unbounded open sets of finite measure and the proof applies Lemma 3.1(ii) directly to such sets. The layer-cake argument for radial decreasing functions does extend to finite-measure sets, but the manuscript should either state Lemma 3.1 in the needed generality or justify the extension, since the proof as written cites a lemma whose hypotheses are not met.

minor comments (5)

[Throughout the proof of Theorem 1.1] The notation B_{R0} is used sometimes with and sometimes without its center; please fix the notation so that the center is clear in every occurrence.
[Equation (3.8)] The two-sided estimate for ∫_{RN\Ω} dy/|x−y|^{N+sp−G} is quoted from [8] without comment; for the boundary regularity assumed in Theorem 1.2 this is fine, but the uniform dependence on x should be stated explicitly rather than left implicit.
[Section 4, second paragraph] The claim that for G(t)=t^p with p≥2 the results coincide with [11] needs qualification. For example, when p=2, N=1, s=1/2, one has β_{s,G}(λ)≡1 and none of the lim-inf conditions in Theorem 1.2 holds; thus Theorem 1.2 as stated does not cover this power case. Please clarify the exact relation to [11].
[Section 4] The examples in Section 4 should verify the β_{s,G} conditions explicitly, since for G(t)=t^p(1+|log t|) and G(t)=t^q+t^p the verification is not immediate and the reader should be able to check that the hypotheses of Theorem 1.2 are satisfied.
[Various] There are several typos and inconsistencies: 'contibuous' in (1.3), 'inavariant' in §2.1, and inconsistent use of accents in 'Pólya-Szegő' in the title and body; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain rests on external cited results and does not reduce to its inputs.

full rationale

The paper's two main theorems are derived from independent external results rather than from the claims being proved. Theorem 1.1 follows the Li-Wang strategy: after decomposing the domain seminorm into a whole-space term and a tail term, the strict comparison (3.4) is reduced to Lemma 3.1, quoted from Li-Wang [11, Lemma 2.1], which is a standalone rearrangement inequality about integrals over Ω and Ω*. The lemma is not equivalent to the theorem's conclusion; it is an external tool. Theorem 1.2 uses the whole-space Pólya-Szegő principle (1.5) from De Nápoli-Bonder-Salort [6] and the fractional Orlicz-Hardy inequality from Bal-Mohanta-Roy-SK [3] at equation (3.9); neither is an assumption of the conclusion being proved, and no fitted parameter is later renamed as a prediction. There are no self-citations (the author cites no prior work of his own), no imported uniqueness theorem, and no ansatz smuggled in via citation. A separate correctness concern exists: in the proof of Theorem 1.1 the assertion '|Ω \ Ω*| = |Ω* \ Ω| = 1/2 |Ω*∆Ω| > 0 since Ω is not a ball' is false for Ω = B1 \ {0}, so the non-ball case as written does not cover open sets that differ from a ball by a null set. This is a mathematical gap in the proof, not a circularity: it does not make the conclusion an input to the derivation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fit; the only constants appear through external inequalities. The central claims rest on standard Orlicz and rearrangement results plus two external theorems ([3] and [6]).

assumptions (5)

domain assumption Young function G satisfies (2.1): 1 < p^-_G ≤ t g(t)/G(t) ≤ p^+_G < ∞ for all t>0.
Defines the class of growth functions; used throughout Theorems 1.1 and 1.2.
standard math Lemma 3.1 from Li-Wang [11]: for radial strictly decreasing f, ∫_{R^N\Ω} f > ∫_{R^N\Ω*} f when |Ω*ΔΩ|>0.
Quoted external lemma; the engine of Theorem 1.1.
domain assumption Pólya-Szegő inequality (1.5) for fractional Orlicz seminorms on R^N from De Nápoli-Bonder-Salort [6].
First inequality in (3.7), used with M(t)=t^s and N(t)=t^N.
domain assumption Fractional Orlicz-Hardy inequality (Bal-Mohanta-Roy-SK [3, Theorems 1.5 and 1.2]) giving ∫_Ω G(|u|/δ^s) dx ≤ C ∫∫_{Ω×Ω} G(|u(x)-u(y)|/|x-y|^s) dxdy/|x-y|^N under the hypotheses of Theorem 1.2.
Load-bearing for the reverse inequality (1.9); its valid domain classes determine the hypotheses.
standard math Boundary-distance integral estimate ∫_{R^N\Ω} dy/|x-y|^{N+sp^-_G} ≍ δ(x)^{-sp^-_G} for x∈Ω (Grisvard).
Used in (3.8) to bound the y-integral by δ^{-sp^-}.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A note on P\'{o}lya-Szeg\"{o} inequality for fractional Orlicz-Sobolev seminorm in domains." pith.science (2026). https://pith.science/paper/7BUXKL7H

@misc{pith2026241118188,
  author       = {Pith},
  title        = {Pith review of: A note on P\'olya-Szeg\"o inequality for fractional Orlicz-Sobolev seminorm in domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BUXKL7H}},
  note         = {Machine review of arXiv:2411.18188}
}

read the original abstract

In this paper, we study the effect of symmetric radial decreasing rearrangement on fractional Orlicz-Sobolev seminorm in domains. Roughly speaking, we prove that symmetric radial decreasing rearrangement can increase the fractional Orlicz-Sobolev seminorm in domains. Our result extends that of Li-Wang [Commun. Contemp. Math. 21.07 (2019): 1850059.] to the setting of fractional seminorm in domains admitting behaviors more general than powers.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[3]

K. Bal, K. Mohanta, P. Roy, and F. Sk, Hardy and Poincar´ e inequalities in fractional Orlicz-Sobolev spaces. Nonlinear Analysis 216 (2022): 112697

work page 2022
[11]

Li and K

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains. Communications in Contemporary Mathematics 21.07 (2019): 1850059

work page 2019
[6]

De N´ apoli, J

P. De N´ apoli, J. Fernandez Bonder, and A. Salort,A P´ olya-Szeg¨ o principle for general fractional Orlicz-Sobolev spaces. Complex Variables and Elliptic Equations 66.4 (2021): 546-568

work page 2021
[1]

F. J. Almgren Jr and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. Journal of the American Mathematical Society (1989): 683-773

work page 1989
[2]

Bahrouni and A

S. Bahrouni and A. M. Salort, Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): S15

work page 2021
[4]

F. J. Bonder and A. M. Salort, Fractional order orlicz-sobolev spaces. Journal of Functional Analysis 277.2 (2019): 333-367. A NOTE ON P ´OLYA-SZEG¨O INEQUALITY FOR FRACTIONAL ORLICZ-SOBOLEV SEMINORM IN DOMAINS 9

work page 2019
[5]

Colombo and G

M. Colombo and G. Mingione, Regularity for double phase variational problems. Archive for Rational Mechanics and Analysis 215 (2015): 443-496

work page 2015
[7]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. Journal of Functional Analysis 255.12 (2008): 3407-3430

work page 2008

Show all 16 references

[8]

Grisvard, Elliptic problems in nonsmooth domains

P. Grisvard, Elliptic problems in nonsmooth domains. Society for Industrial and Applied Mathematics, 2011

2011
[9]

M. A. Krasnosel’ski˘i and Ja. Rutickii, Convex functions and orlicz spaces, 1961, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd. Groningen

1961
[10]

Kufner, O

A. Kufner, O. John, and S. Fucik, Function Spaces., Vol. 3. Springer Science Business Media (1979)

1979
[12]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Annals of Mathematics (1983): 349-374

1983
[13]

P´ olya and G

G. P´ olya and G. Szeg¨ o,Isoperimetric Inequality in Mathematical Physics. Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N.J., 1951

1951
[14]

A. M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator. Journal of Differential Equations 268.9 (2020): 5413-5439

2020
[15]

Talenti, Best constant in Sobolev inequality

G. Talenti, Best constant in Sobolev inequality. Annali di Matematica pura ed Applicata 110 (1976): 353-372

1976
[16]

Talenti, Inequalities in rearrangement invariant function spaces

G. Talenti, Inequalities in rearrangement invariant function spaces. Nonlinear analysis, function spaces and applications (1994): 177-230. The Fields Institute for research in mathematical sciences, 222 college street, 2nd floor, Toronto, Ontario, m5t 3j1 Canada, and School of...

1994

Pith tools

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

[1] [3]

K. Bal, K. Mohanta, P. Roy, and F. Sk, Hardy and Poincar´ e inequalities in fractional Orlicz-Sobolev spaces. Nonlinear Analysis 216 (2022): 112697

work page 2022

[2] [11]

Li and K

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains. Communications in Contemporary Mathematics 21.07 (2019): 1850059

work page 2019

[3] [6]

De N´ apoli, J

P. De N´ apoli, J. Fernandez Bonder, and A. Salort,A P´ olya-Szeg¨ o principle for general fractional Orlicz-Sobolev spaces. Complex Variables and Elliptic Equations 66.4 (2021): 546-568

work page 2021

[4] [1]

F. J. Almgren Jr and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. Journal of the American Mathematical Society (1989): 683-773

work page 1989

[5] [2]

Bahrouni and A

S. Bahrouni and A. M. Salort, Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): S15

work page 2021

[6] [4]

F. J. Bonder and A. M. Salort, Fractional order orlicz-sobolev spaces. Journal of Functional Analysis 277.2 (2019): 333-367. A NOTE ON P ´OLYA-SZEG¨O INEQUALITY FOR FRACTIONAL ORLICZ-SOBOLEV SEMINORM IN DOMAINS 9

work page 2019

[7] [5]

Colombo and G

M. Colombo and G. Mingione, Regularity for double phase variational problems. Archive for Rational Mechanics and Analysis 215 (2015): 443-496

work page 2015

[8] [7]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. Journal of Functional Analysis 255.12 (2008): 3407-3430

work page 2008