A Boxing Inequality for the Fractional Perimeter

Abstract

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