{"id":"71263877-b1cc-4c65-9be5-fa44082aaea0","arxiv_id":"2411.18188","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetrization can increase the fractional Orlicz-Sobolev seminorm in domains, extending Li-Wang's power-law counterexample to general Orlicz growth and providing a controlled converse estimate.","lead":"Schwarz symmetrization can increase, rather than decrease, the fractional Orlicz-Sobolev seminorm of a function supported in a bounded domain. This paper proves that failure for general Young functions and adds a reverse inequality up to a constant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 proof omits the case where Ω differs from a ball by a null set; the asserted strict inequality |Ω*ΔΩ|>0 is false, so the central claim is not proven for e.g. B1\\{0}.","rationale":"The central claim of the paper is Theorem 1.1, the failure of Pólya-Szegő for regional fractional Orlicz seminorms. The proof's first step in the non-ball case makes a measure-theoretic assertion that is not true: 'Ω is not a ball' does not imply |Ω*ΔΩ|>0. Open sets obtained by deleting a point from a ball are counterexamples. Since the strictness in Lemma 3.1(ii) and in (3.4) depends on positive symmetric difference, the written proof gives no strict inequality for these admissible domains. I verified that the same construction used for balls should also work for Ω=B1\\{0} because the removed point is null, so the flaw is an omission rather than a counterexample; however, an omission in the proof of the central theorem is a real concern that should be fixed before acceptance. The reader's weakest_assumption concerned Theorem 1.2's reliance on [3]; that is a separate, secondary issue. Hence agreement_with_reader is disagree, and the verdict should be CONDITIONAL rather than a clean ACCEPT.","tokens_in":9079,"tokens_out":47409,"duration_ms":401797,"concrete_test":"Analyze Ω=B1\\{0} (N≥1) using the paper's construction. For G satisfying (1.1), s∈(0,1), take uε(x)=η((x-x̃)/ε) with |x̃|=1/2 as in the ball case, and verify directly that ∫∫_{Ω×Ω} G(...) dxdy/|x-y|^N < ∫∫_{Ω*×Ω*} G(...) dxdy/|x-y|^N by showing tail_Ω(uε)=tail_{B1}(uε) and tail_Ω*(u*ε)=tail_{B1}(u*ε), then applying the strict ball-case comparison (3.6). If the inequality is strict, the theorem survives but the proof needs a revised case split; if equality holds, Theorem 1.1's statement is false for such domains.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3, proof of Theorem 1.1, begins the non-ball case with '|Ω \\ Ω*| = |Ω* \\ Ω| = 1/2 |Ω*∆Ω| > 0 since Ω is not a ball.' This is false: the open set Ω=B1\\{0} is not a ball, yet Ω*=B1 and the symmetric difference is just {0}, a null set. For any such Ω the subsequent application of Lemma 3.1(ii) (which requires |Ω*ΔΩ|>0) is unavailable; indeed H_Ω(0)=H_Ω*(0) up to null sets, so (3.4)/(3.5) give equality rather than strict inequality, and the final chain collapses to equality. The proof therefore does not establish the strict inequality (1.8) for all open sets of finite measure stated in Theorem 1.1. The gap is patchable since the ball-case construction with x̃≠0 should still give a strict comparison because tail integrals over Ω and B1 coincide a.e., but the written proof contains no such argument. This is a load-bearing gap in the central claim, independent of the Theorem 1.2/Hardy-inequality caveat.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":9283,"tokens_out":25799,"duration_ms":231562,"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":[{"comment":"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":"Section 3, proof of Theorem 1.1 (non-ball case)"},{"comment":"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":"Section 3, proof of Theorem 1.1"},{"comment":"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.","section":"Section 3 and Lemma 3.1"}],"minor_comments":[{"comment":"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.","section":"Throughout the proof of Theorem 1.1"},{"comment":"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":"Equation (3.8)"},{"comment":"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":"Section 4, second paragraph"},{"comment":"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.","section":"Section 4"},{"comment":"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.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the central observation is sound. The main reason for major revision is the incomplete proof of Theorem 1.1 for domains that differ from a ball by a null set, together with the normalization issue in the construction of uε. These are local and patchable. I also recommend that the author re-examine the comparison with [11] in Section 4, since the β conditions are not satisfied by all power functions. No concerns about circularity: the paper uses external results as tools and does not assume the target inequalities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nA 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.\n\nTheorem 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.\n\nWho 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.","headline":"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.","tokens_in":9894,"tokens_out":7847,"would_cite":true,"duration_ms":65766,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E30","45G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Pólya-Szegő inequality","symmetric decreasing rearrangement","fractional Orlicz-Sobolev space","Gagliardo seminorm","Young function","Hardy inequality","bounded domain"],"falsifier":"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.","tokens_in":8824,"feed_emoji":"🔄","tokens_out":12287,"duration_ms":95427,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symmetrization increases fractional Orlicz seminorms in domains","feed_subtitle":"For any power-like Young function, rearrangement can strictly enlarge the regional fractional seminorm.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $t^q+t^p$."],"supporting_citations":[{"why":"Supplies the radial-integral comparison lemma and the concentrating-bump proof strategy that Theorem 1.1 adapts from the power case to the Orlicz setting.","marker":"[11]"},{"why":"Provides the whole-space Pólya-Szegő inequality (1.5) that Theorem 1.2 invokes for the first step of the reverse estimate.","marker":"[6]"},{"why":"Provides the fractional Orlicz-Hardy inequality used at (3.9) to bound the boundary-interaction term by the fractional seminorm.","marker":"[3]"},{"why":"Supplies Lemma 2.1, the power-comparison estimates for Young functions used to factor $\\delta(x)^s$ out of the integrand in (3.8).","marker":"[2]"},{"why":"Supplies the boundary-distance estimate $\\int_{\\mathbb{R}^N\\setminus\\Omega} |x-y|^{-N-sp^-_G} dy \\asymp \\delta(x)^{-sp^-_G}$ used between (3.8) and (3.9).","marker":"[8]"}],"fun_headline_variants":["Rearrangement can boost fractional Orlicz seminorms in domains","Schwarz symmetrization may enlarge regional fractional seminorms","Symmetric rearrangement increases Orlicz-Sobolev seminorms on domains","Fractional Orlicz seminorm in domains not always reduced by rearrangement","Bumps near boundary make rearrangement raise Orlicz seminorm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rearrangement can boost fractional Orlicz seminorms in domains","Schwarz symmetrization may enlarge regional fractional seminorms","Symmetric rearrangement increases Orlicz-Sobolev seminorms on domains","Fractional Orlicz seminorm in domains not always reduced by rearrangement","Bumps near boundary make rearrangement raise Orlicz seminorm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1377,"prompt_tokens":868,"completion_tokens":509,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":418}},"tokens_in":484,"tokens_out":509,"duration_ms":4560,"temperature":1.0,"reasoning_tokens":418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:25:57.660908+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Li and K","cited_arxiv_id":null,"evidence_quote":"Supplies the radial-integral comparison lemma and the concentrating-bump proof strategy that Theorem 1.1 adapts from the power case to the Orlicz setting."},{"cited_title":"De N´ apoli, J","cited_arxiv_id":null,"evidence_quote":"Provides the whole-space Pólya-Szegő inequality (1.5) that Theorem 1.2 invokes for the first step of the reverse estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fractional Orlicz-Hardy inequality used at (3.9) to bound the boundary-interaction term by the fractional seminorm."},{"cited_title":"Bahrouni and A","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.1, the power-comparison estimates for Young functions used to factor $\\delta(x)^s$ out of the integrand in (3.8)."},{"cited_title":"Grisvard, Elliptic problems in nonsmooth domains","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-distance estimate $\\int_{\\mathbb{R}^N\\setminus\\Omega} |x-y|^{-N-sp^-_G} dy \\asymp \\delta(x)^{-sp^-_G}$ used between (3.8) and (3.9)."}],"review_version":1}