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arxiv: 2604.21506 · v1 · submitted 2026-04-23 · 🧮 math.FA · math.AP· math.CV

Recognition: unknown

Boxing inequalities for relative fractional perimeter and fractional Poincar\'e-type inequalities on John domains with the BBM factor

Jiang Li, Manzi Huang, Panu Lahti, Zhuang Wang

Pith reviewed 2026-05-08 13:32 UTC · model grok-4.3

classification 🧮 math.FA math.APmath.CV
keywords John domainsboxing inequalityfractional perimeterPoincaré-Wirtinger inequalityHausdorff contentBBM factortrace inequalityfractional Sobolev inequality
0
0 comments X

The pith

s-John domains satisfy boxing inequalities for relative fractional perimeters that imply Poincaré-Wirtinger trace inequalities carrying the (1-δ) factor.

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

The paper proves that every s-John domain in R^n obeys a boxing inequality bounding the s(n-δ)-dimensional Hausdorff content of a measurable set U of controlled relative measure by a double integral that captures the fractional variation of its characteristic function. This bound incorporates the factor (1-δ) and restricts the integration to pairs where the distance is less than a multiple of the distance to the boundary. The boxing inequality is shown to imply the fractional Poincaré-Wirtinger trace inequality on these domains, with the fractional Sobolev-Poincaré and Hardy-type inequalities obtained as direct special cases. The same factor (1-δ) appears throughout, and the classical Poincaré-Wirtinger trace inequality is recovered in the limit via the Bourgain-Brezis-Mironescu formula. The John condition is proved essentially sharp once the domain is assumed to satisfy an additional separation property, and all stated inequalities are new even when the domain is Lipschitz.

Core claim

For 0 < δ, τ < 1 and 1 ≤ s ≤ n/(n-δ), every s-John domain Ω satisfies the boxing inequality H^{s(n-δ)}_∞(U ∖ N_U) ≤ C(1-δ) ∫_Ω ∫_{|x-y|<τ dist(y,∂Ω)} |χ_U(x)-χ_U(y)| / |x-y|^{n+δ} dx dy whenever |U|/|Ω| ≤ γ < 1. This inequality implies the fractional Poincaré-Wirtinger trace inequality with the BBM factor on s-John domains. The functional formulation of the boxing inequality is equivalent to the set formulation, and the John domain condition is sharp under the separation property.

What carries the argument

The boxing inequality relating the s(n-δ)-Hausdorff content of U minus a null set to the relative fractional perimeter integral restricted by distance to the boundary and scaled by the factor (1-δ).

If this is right

  • The fractional Poincaré-Wirtinger trace inequality holds on every s-John domain and carries the factor (1-δ).
  • The fractional Sobolev-Poincaré inequality and the fractional Hardy-type inequality follow as special cases on the same domains.
  • The classical Poincaré-Wirtinger trace inequality is recovered on s-John domains by passing to the limit with the Bourgain-Brezis-Mironescu formula.
  • Any domain that supports the boxing inequality and satisfies the separation property must itself be a John domain.

Where Pith is reading between the lines

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

  • The explicit (1-δ) factor may permit direct passage to the classical case by taking the limit δ → 1 inside the stated inequalities without additional approximation arguments.
  • The sharpness result suggests that the boxing inequality can serve as a characterization of John domains within the class of domains that satisfy the separation property.
  • The functional formulation of the boxing inequality may extend to vector-valued or higher-order fractional energies on the same domains.

Load-bearing premise

The domain must be an s-John domain, meaning every point inside can be joined to a fixed interior point by a chain of balls whose radii are controlled by a fixed multiple of the distance to the boundary.

What would settle it

Exhibit a domain that obeys the separation property but is not s-John, together with a specific set U of small relative measure, such that the Hausdorff content of U minus a null set strictly exceeds every constant times the fractional integral for that U.

read the original abstract

For $0<\delta,\tau<1$ and $1\le s\le \frac{n}{n-\delta}$, we prove that for a given $s$-John domain $\Omega\subset \mathbb{R}^n$, the following Boxing inequality holds for every Lebesgue measurable set $U\subset\Omega$ with $|U|/|\Omega|\le\gamma<1$: \[ \mathcal{H}^{s(n-\delta)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-\delta)\int_\Omega\int_{|x-y|<\tau\operatorname{dist}(y,\partial\Omega)}\frac{|\chi_U(x)-\chi_U(y)|}{|x-y|^{n+\delta}}\,dx\,dy, \] where $\mathcal{H}^{s(n-\delta)}_{\infty}(U)$ denotes the $s(n-\delta)$-dimensional Hausdorff content of $U$, $\mathcal{N}_U$ is a set of Lebesgue measure zero and the constant $C$ depends only on $n,\tau,s,\gamma$, the John constant and the diameter of $\Omega$. Moreover, we establish the functional formulation of the above Boxing inequality and discuss the equivalence between these two formulations. Based on the Boxing inequality, we prove the fractional Poincar\'e--Wirtinger trace inequality on $s$-John domains, of which the fractional Sobolev--Poincar\'e inequality and fractional Hardy-type inequality are special cases. Notably, we prove all of the aforementioned inequalities with the Bourgain--Brezis--Mironescu (BBM) factor $1-\delta$. Furthermore, with the aid of the Bourgain--Brezis--Mironescu formula, we recover the Poincar\'e--Wirtinger trace inequality. Finally, by showing that, under the separation property, any domain supporting the Boxing inequality is necessarily a John domain, we conclude that the John domain condition is essentially sharp for the above inequalities. All the above inequalities with the BBM factor are new even for Lipschitz domains.

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

0 major / 2 minor

Summary. The paper claims to prove boxing inequalities for relative fractional perimeter on s-John domains, leading to fractional Poincaré-type inequalities with the BBM factor 1-δ. For 0<δ,τ<1 and 1≤s≤n/(n-δ), every s-John domain Ω satisfies H^{s(n-δ)}_∞(U ∖ N_U) ≤ C(1-δ) times the localized fractional perimeter integral for sets U with small measure. This implies the fractional Poincaré-Wirtinger trace inequality, with special cases for Sobolev-Poincaré and Hardy inequalities. The John condition is sharp under separation property, and results are new even for Lipschitz domains. Equivalence between set and functional formulations is discussed, and classical inequalities recovered via BBM formula.

Significance. This provides a valuable contribution to the field of fractional Sobolev inequalities on irregular domains. The inclusion of the BBM factor is a notable strength, as it enables the recovery of classical results and highlights the connection to the Bourgain-Brezis-Mironescu formula. The sharpness result under the separation property adds rigor to the necessity of the John condition. If the proofs are correct, this could influence further research on fractional perimeters and inequalities in John domains.

minor comments (2)
  1. The abstract introduces the functional formulation of the boxing inequality without stating its precise form; a one-sentence description would improve readability.
  2. Notation for N_U (the null set) and the precise localization |x-y|<τ dist(y,∂Ω) could be recalled briefly when the functional version is introduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. The referee's summary correctly reflects the paper's main results on boxing inequalities for relative fractional perimeter on s-John domains, the resulting fractional Poincaré-type inequalities with the BBM factor, and the sharpness of the John condition under the separation property. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes the boxing inequality directly via analysis on s-John domains for the given parameter ranges, then derives the fractional Poincaré-Wirtinger trace inequality (and special cases) from it as a consequence. It recovers the classical Poincaré-Wirtinger inequality using the external Bourgain-Brezis-Mironescu formula and proves sharpness of the John condition via a converse implication under the separation property. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; all constants are stated to depend explicitly on n, τ, s, γ, the John constant, and diam(Ω), with no renaming of known results or smuggling of ansatzes. The central claims remain independent of the target inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proofs rest on the definition of s-John domains, properties of Hausdorff content, and the Bourgain-Brezis-Mironescu formula; no new entities are introduced.

axioms (2)
  • standard math Standard properties of Lebesgue measure, Hausdorff content, and the definition of s-John domains hold in R^n.
    Invoked throughout the statements for the boxing inequality and the converse.
  • standard math The Bourgain-Brezis-Mironescu formula recovers the classical Poincaré-Wirtinger inequality in the limit δ→1.
    Used to recover the non-fractional case from the fractional one.

pith-pipeline@v0.9.0 · 5701 in / 1434 out tokens · 63918 ms · 2026-05-08T13:32:39.391604+00:00 · methodology

discussion (0)

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