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Strict rearrangement inequalities: nonexpansivity and periodic Gagliardo seminorms

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Pith's one-line read The periodic Gagliardo seminorm does not increase under periodic or cylindrical rearrangement; equality forces symmetric profiles up to translation.

desk verdict Strong paper completing the periodic Pólya–Szegő program with new equality characterizations, but Theorems 5.1 and 5.2 need a finite-measure-superlevel-set hypothesis as stated. read the letter →

arxiv 2411.15308 v1 pith:SYSKOHZQ submitted 2024-11-22 math.AP

Gyula Csató , Albert Mas This is my paper · ORCID
classification math.AP MSC 35B1035A1535S0526D15
keywords periodicGagliardoseminormPólya–SzegőinequalityrearrangementinequalitiescylindricalnonexpansivityRieszequalitycases
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

The paper proves Pólya–Szegő type inequalities for the periodic Gagliardo seminorm, a nonlocal fractional-energy functional on functions that are 2π-periodic in one coordinate. The first main result says that two natural symmetric rearrangements—the periodic rearrangement, which symmetrizes each period in the first variable, and the cylindrical rearrangement, which spherically symmetrizes the transverse variables for each first-coordinate slice—never increase this seminorm. The second main contribution is a complete description of equality: for p>1, equality forces the function to be a translate of its rearrangement; for p=1 and continuous functions, every superlevel set may be translated independently, a contrast that mirrors the known local versus nonlocal Pólya–Szegő dichotomy. These results complete the rearrangement program for this seminorm, extending earlier partial results for p=1, p=2, and characteristic functions.

What carries the argument

The engine is a pair of nonexpansivity theorems for rearrangements in $\mathbb{R}^n$ and on the circle. Theorem 1.2 says that for a nonnegative convex J and nonnegative kernel g, the double integral of $J(u(x)-v(y))g(x-y)$ does not increase when u, v, and g are replaced by their Schwarz rearrangements, with equality characterized when J is strictly convex; Theorem 1.3 is the analogous statement for Steiner rearrangement on $(-\pi,\pi)$ with a periodic kernel. These generalize a one-function nonexpansivity lemma from the literature and the Riesz rearrangement inequality on the circle. The bridge to the Gagliardo seminorm is the Laplace-transform representation (5.4): writing $|x-y|^{-(n+sp)}$ as a superposition of Gaussians reduces the seminorm to integrals against the periodic heat kernel $g(z,t)=\sum_{k\in\mathbb{Z}} e^{-(z+2k\pi)^2t}$, whose monotonicity on $(0,\pi)$ is what makes the kernel admissible for the circle inequality.

What would settle it

Check the theorems against the nonzero constant function $u\equiv c$, which is 2π-periodic and has seminorm zero. Its superlevel sets have infinite measure, so the layer-cake definition (2.1) leaves $u^*_{\mathrm{per}}$ and $u^*_{n,1}$ undefined; if the hypotheses are read literally, the claimed inequality has no content for this input. The central claim therefore survives only under the additional finite-superlevel-set hypothesis, and the issue is settled by deciding whether the authors intend that implicit assumption.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if 0<s<1, 1≤p<∞, and u is measurable and 2π-periodic in x1, then $[u^*_{\mathrm{per}}]^{\mathrm{per}}_{W^{s,p}} \le [u]^{\mathrm{per}}_{W^{s,p}}$ and $[u^*_{n,1}]^{\mathrm{per}}_{W^{s,p}} \le [u]^{\mathrm{per}}_{W^{s,p}}$. Equality is characterized in Section 5: for p>1 (and p=1 characteristic functions), equality holds exactly when $u = \pm |u|$ and u coincides, after a translation, with its own rearrangement; for p=1 continuous u, equality means that for almost every level τ the superlevel set $\{|u| > \tau\}$ is a translate, by a vector that may depend on τ, of the corresponding superlevel set of the rearranged function. The paper also establishes generalized nonexpansivity inequalities for rearrangements of pairs of functions, with equality cases, that are the engine behind the seminorm results.

Load-bearing premise

That the superlevel sets of the function, and of its sliced sections in the cylindrical case, have finite measure wherever rearrangements are applied; the theorem statements as written assume only a finite seminorm, which a nonzero periodic constant satisfies while having no layer-cake rearrangement at all.

Editorial extensions

If this is right

  • If the inequalities hold, then every extremal or minimizer of these periodic nonlocal energies that is already symmetric under rearrangement must itself be symmetric up to translation, giving a rigidity statement useful in variational problems on the torus.
  • The periodic fractional perimeter of any periodic set is decreased by both periodic and cylindrical rearrangement, extending the p=1 characteristic-function case to the full p-range of the seminorm.
  • The cylindrical inequality holds independently of periodicity, so it supplies a Pólya–Szegő inequality for functions on slabs or cylinders sliced in the first coordinate.
  • The p=1 equality description shows that superlevel sets may be rearranged with level-dependent translations, so equality in the L1-type case does not force full global symmetry—only symmetry of each level set.
  • The nonexpansivity theorems cover $J(t)=|t|$ including equality, a case not handled by previous polarization-based treatments, and thereby complete the equality analysis for the seminorm applications.

Reading between the lines

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

  • Extending beyond the paper, the Laplace-transform route suggests that the same inequalities hold for other nonlocal operators whose kernels are completely monotone in $|x-y|^2$, giving rearrangement inequalities for powers of periodic operators beyond the fractional Laplacian.
  • Extending beyond the paper, the continuity assumption in the p=1 'only if' direction is likely removable: the proof uses continuity only to upgrade a slicewise equality to all slices, and a limiting argument with the level-dependent translations may suffice.
  • Extending beyond the paper, the two-function nonexpansivity theorems with equality for $J(t)=|t|$ could serve as a tool for sharp rearrangement inequalities with more than two functions, connecting to open questions mentioned in the paper's discussion of rearrangement on spheres.
  • Extending beyond the paper, the rigidity characterization could be used to detect symmetry breaking in periodic nonlocal isoperimetric problems: any minimizer with a flat or non-symmetric superlevel set would violate the equality condition.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proves Pólya-Szegő-type inequalities for the periodic Gagliardo seminorm under two rearrangements: the periodic rearrangement in the x1-variable and the cylindrical (Schwarz in the transverse variables) rearrangement. The main results, Theorems 5.1 and 5.2, state that both rearrangements do not increase the seminorm, and they include fairly complete characterizations of the equality cases for p > 1, for p = 1 and characteristic functions, and for p = 1 and continuous functions. The proofs use the Laplace representation of the seminorm, the monotonicity of the periodic heat kernel, and two nonexpansivity results proved in the paper: Theorem 1.2 for Schwarz rearrangement in Rn and Theorem 1.3 for rearrangement on the circle. The latter are generalizations of results of Frank-Seiringer and Burchard-Hajaiej, and the paper also corrects and extends equality statements for convex J, including the case J(t)=|t|. The central derivation is structured in five steps and appears sound for the intended function classes, conditional on the domain hypotheses discussed below.

Significance. If the stated domain issues are repaired, the paper makes a substantial contribution to the periodic nonlocal Pólya-Szegő program. It settles the inequality and equality cases for the periodic rearrangement that was only conjectured for characteristic functions by Dávila, del Pino, Dipierro and Valdinoci, and it extends the cylindrical rearrangement result beyond p=1,2 to all p in [1,∞). The nonexpansivity theorems are of independent interest and are proved in a transparent, largely self-contained way, building on classical Riesz rearrangement inequalities. The equality analysis is notably careful, especially the J(t)=|t| case, where translations may depend on the level. No fitted parameters or ad hoc assumptions appear; the proofs rely on standard external inputs. These are real strengths. The main weakness is a load-bearing hypothesis gap: the main theorems are stated for all finite-seminorm functions, but the rearrangements used are not defined for such functions without an additional finite-measure condition on superlevel sets.

major comments (2)
  1. [§2, Eq. (2.1); §5, Theorems 5.1 and 5.2; Theorem 1.1] The statements of Theorems 5.1 and 5.2, and therefore of Theorem 1.1, assume only that u is measurable, 2π-periodic in x1, and has finite periodic Gagliardo seminorm. However, the rearrangements u*_per and u*_{n,1} are defined via the layer-cake formula (2.1) only when the superlevel sets {|u|>t} have finite measure (finite appropriate measure in the period strip for the periodic rearrangement). Finite seminorm does not imply this. The constant function u≡1 is a concrete counterexample: it is 2π-periodic, measurable, [u]^per_{W^{s,p}}=0, but {u>1/2}=R^n has infinite measure, so u*_per and u*_{n,1} are not defined. Thus the inequalities (1.3), (5.2), and (5.3) are not well-formed for admissible u. The proofs appear to go through if one adds the natural hypothesis that |{|u|>τ} ∩ ((-π,π)×R^{n-1})| is finite for every τ>0 (which also supplies the sectionwise finite-measure condition needed for the cylindrical rearrangement). This hypothesis should be stated explicitly in Theorems 5.1, 5.2, and Theorem 1.1.
  2. [Proof of Theorem 5.2, first paragraph] The proof of Theorem 5.2 applies Theorem 1.2 to the frozen sections x′↦u(x1,x′) and y′↦u(y1,y′) and justifies this by writing “f ∈ L^p((−π,π)×R^{n−1})” for the function u. But no such integrability assumption appears in the theorem statement, and finite seminorm does not imply it; again u≡1 is a counterexample. What is needed to apply Theorem 1.2 is that the relevant superlevel sets of the sections have finite (n−1)-dimensional measure for almost every x1 and y1, a condition that follows from the finite-measure-in-the-strip hypothesis mentioned in the preceding comment but not from the stated hypotheses. The same issue affects the equality analysis in the p=1 case, where superlevel sets of sections are compared. Please make the hypothesis explicit and verify that it is sufficient for each application of Theorem 1.2.
minor comments (3)
  1. [§4, proof of Theorem 1.2, Step 4] There are two small typos: “wich” should be “which”, and “A analogous cutt-off” should be “An analogous cut-off”.
  2. [§3, paragraph before proof of Theorem 1.3] The phrase “some of the hypothesis’ of the theorem” should read “some of the hypotheses of the theorem”.
  3. [§5, Eq. (5.4)] The Laplace representation is introduced with λ=(n+sp)/2 and the kernel g(z,t)=∑_k e^{-(z+2kπ)^2t}. It would help the reader if the statement that g(·,t)=g(·,t)^*_per and is decreasing in (0,π) were stated as a displayed lemma or with an explicit reference to [10, Appendix B] at the point of first use, rather than only in the surrounding text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main inequalities are forward applications of external Riesz rearrangement theorems and the Laplace transform; the one self-citation (heat-kernel monotonicity) is independent and not equivalent to the target.

full rationale

The derivation chain is self-contained. Theorem 5.1 is obtained from the Laplace-transform representation (5.4) plus the periodic nonexpansivity theorem (Theorem 1.3), whose proof is carried out in Section 3 from the external Riesz rearrangement inequality on the circle (Theorem 3.1, credited to Baernstein, Friedberg-Luttinger, and Burchard-Hajaiej). Theorem 5.2 is obtained analogously from the Schwarz nonexpansivity theorem (Theorem 1.2), proved in Section 4 from the classical Riesz rearrangement inequality. There are no fitted parameters and no target inequality is assumed among the inputs. The only self-citation that is load-bearing is the monotonicity of the periodic heat kernel used to verify the decreasing hypothesis on g(·,t) in Theorem 1.3's equality cases ([10, Appendix B]); this is an independent maximum-principle fact, not a restatement of the Pólya-Szegő inequality, and the paper even indicates the proof mechanism ('follows from a maximum principle on its spatial derivative'). The reviewer's well-formedness objection about Theorems 5.1-5.2 (u≡1 has finite seminorm but no layer-cake rearrangement) is a genuine hypothesis/statement gap, but it is a correctness issue, not a circular reduction: the proofs would go through unchanged under an added finite-measure-superlevel-set hypothesis, and no step defines the conclusion in terms of itself. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants and no invented entities: the free-parameter count is zero. The derivations rest on standard rearrangement technology (the Riesz inequality in R^n and its circle version with equality cases from Burchard-Hajaiej and Baernstein, layer-cake representations, the Laplace transform) plus one self-cited published lemma, the monotonicity of the periodic heat kernel from [10, Appendix B], used to verify the hypotheses of Theorem 1.3. The one structural assumption not established in the text is the well-definedness of the rearrangements for the stated function classes, which the definitions in Section 2 require but the theorems do not restate; this is the point a referee should ask to be clarified. The equality characterizations in Theorems 5.1 and 5.2 inherit the hypotheses of the imported Riesz equality theorems, including the 'one factor constant' cases.

assumptions (6)
  • standard math Equality characterization of the Riesz rearrangement inequality on the circle (Theorem 3.1): equality implies either one factor is constant or both factors equal their periodic rearrangements up to a common translation (Burchard-Hajaiej [9, Theorem 2], Baernstein [3]).
    Applied in Step 3.2 of Theorem 1.3 at (3.14) to pairs (I'_S(u_S − τ), χ_{v>τ}); it is the load-bearing input for all equality characterizations in Theorems 5.1 and 5.2.
  • standard math Classical Riesz rearrangement inequality (1.6) and the strict version for equality ([3, Theorem 2.15(b)], [21, Theorem 3.9]): equality forces the two nonconstant factors to be common translates of their rearrangements.
    Used in Section 4, Step 2, equation (4.2), for the Schwarz nonexpansivity theorem; the authors note the missing vanishing-at-infinity hypotheses in [21, Theorem 3.9].
  • standard math The periodic heat kernel g(z,t) := Σ_{k∈Z} e^{−(z+2kπ)²t} is 2π-periodic, even, and decreasing in z on (0,π) for every t > 0, hence g = g*_per.
    Cited from the authors' prior paper [10, Appendix B] (monotonicity via maximum principle on the circle). It verifies the hypotheses of Theorem 1.3 in Step 1 of the proof of Theorem 5.1, representation (5.4)-(5.5). Published, refereed, parameter-free; treated as independent support.
  • standard math Layer-cake representation, identity (2.3) for integrals of absolutely continuous functions of |u| over rearranged domains, and composition Lemma 2.1: (G∘|u|)* = G∘(u*) for nondecreasing lower semicontinuous G.
    Proven in Section 2 and used throughout Steps 2 and 3 of Theorem 1.3 and Section 4; standard measure-theoretic background.
  • standard math Laplace transform identity z^{−λ} = (1/Γ(λ))∫_0^∞ t^{λ−1} e^{−zt} dt, applied with λ = (n+sp)/2 to obtain representation (5.4) of the periodic Gagliardo seminorm and the kernel (5.5).
    Section 5, derivation of (5.4); elementary and self-contained.
  • domain assumption For the cylindrical case, the sections u(x1,·) admit Schwarz rearrangements for almost every x1, i.e., |{x′ : |u(x1,x′)| > τ}| < ∞ for all τ > 0; for the periodic case, {|u| > τ} must have finite measure so u*_per is defined by (2.1).
    Not derived from the finite-seminorm hypothesis in the text; the statements of Theorems 5.1 and 5.2 do not include it, while the proof of Theorem 5.2 refers to f ∈ L^p((−π,π) × R^{n−1}). This is the flagged technical gap.

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Pith. "Pith review of Strict rearrangement inequalities: nonexpansivity and periodic Gagliardo seminorms." pith.science (2026). https://pith.science/paper/SYSKOHZQ

@misc{pith2026241115308,
  author       = {Pith},
  title        = {Pith review of: Strict rearrangement inequalities: nonexpansivity and periodic Gagliardo seminorms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYSKOHZQ}},
  note         = {Machine review of arXiv:2411.15308}
}
read the original abstract

This paper deals with the behavior of the periodic Gagliardo seminorm under two types of rearrangements, namely under a periodic, and respectively a cylindrical, symmetric decreasing rearrangement. Our two main results are P\'olya-Szeg\H{o} type inequalities for these rearrangements. We also deal with the cases of equality. Our method uses, among others, some classical nonexpansivity results for rearrangements for which we provide some slight improvements. Our proof is based on the ideas of [Frank and Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 2008], where a new proof to deal with the cases of equality in the nonexpansivity theorem was given, albeit in a special case involving the rearrangement of only one function.

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