Boxing inequalities for relative fractional perimeter and fractional Poincaré-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 $|U|/|\Omega |\le \gamma <1$: \[ \mathcal{H}^{s(n-δ)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-δ)\int_Ω\int_{|x-y|<τ\operatorname{dist}(y,\partialΩ)}\frac{|χ_U(x)-χ_U(y)|}{|x-y|^{n+δ}}\,dx\,dy, \] where ${\mathcal{H}}_{\infty }^{s(n-\delta )}(U)$ 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 Poincaré--Wirtinger trace inequality on $s$-John domains, of which the fractional Sobolev--Poincaré 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 Poincaré--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.

35 pages