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

TL;DR

This paper establishes new fractional inequalities on John domains, including Boxing and Poincaré-type inequalities with the BBM factor, extending known results to more general domains and providing functional formulations.

Contribution

It introduces novel fractional Boxing and Poincaré inequalities with the BBM factor on John domains, including their functional forms and sharpness results, even for Lipschitz domains.

Findings

Proved Boxing inequality for Lebesgue measurable sets in John domains.

Established fractional Poincaré–Wirtinger trace inequality with BBM factor.

Showed John domain condition is necessary for these inequalities.

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.