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

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.