Boxing inequalities for relative fractional perimeter and fractional Poincaré-type inequalities on John domains with the BBM factor
For $0<δ,τ<1$ and $1\le s\le \frac{n}{n-δ}$, we prove that for a given $s$-John domain $Ω\subset \mathbb{R}^n$, the following Boxing inequality holds for every Lebesgue measurable set $U\subsetΩ$ with $|U|/|Ω|\leγ<1$: \[ \mathcal{H}^{s(n-δ)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-δ)\int_Ω\int_{|x-y|<τ\operatorname{dist}(y,\partialΩ)}\frac{|χ_U(x)-χ_U(y)|}{|x-y|^{n+δ}}\,dx\,dy, \] where $\mathcal{H}^{s(n-δ)}_{\infty}(U)$ denotes the $s(n-δ)$-dimensional Hausdorff content of $U$, $\mathcal{N}_U$ is a set of Lebesgue measure zero and the constant $C$ depends only on $n,τ,s,γ$, the John constant and the diameter of $Ω$. 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 $1-δ$. 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.