A Boxing Inequality for the Fractional Perimeter

Abstract

We prove the Boxing inequality: $${\mathcal{H}}_{\infty }^{d-\alpha }(U)\le C\alpha (1-\alpha ){\int }_U{\int }_{{\mathbb{R}}^d\setminus U}\frac{\mathrm{d}y\mathrm{d}z}{|y-z{|}^{\alpha +d}},$$ for every $\alpha \in (0,1)$ and every bounded open subset $U\subset {\mathbb{R}}^d$, where ${\mathcal{H}}_{\infty }^{d-\alpha }(U)$ is the Hausdorff content of $U$ of dimension $d-\alpha$ and the constant $C>0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{\alpha ,1}({\mathbb{R}}^d)$ that includes Sobolev's $L^{\frac{d}{d-\alpha }}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $\alpha$ tends to $0$ and $1$, recovering in particular the classical inequalities of first order. Their counterparts in the full range $\alpha \in (0,d)$ are also investigated.