A Boxing Inequality for the Fractional Perimeter

TL;DR

This paper establishes a new inequality linking Hausdorff content and fractional perimeter, leading to trace and Sobolev inequalities in fractional Sobolev spaces, with results extending classical inequalities as  approaches 0 or 1.

Contribution

The paper introduces a novel Boxing inequality for fractional perimeter, connecting geometric measure theory with fractional Sobolev space embeddings and inequalities.

Findings

Proves the Boxing inequality for fractional perimeter.

Derives trace inequalities and Sobolev embeddings from the inequality.

Extends classical inequalities to the fractional setting with asymptotic analysis.

Abstract

We prove the Boxing inequality: $$\mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{\alpha+d}},$$ for every $\alpha \in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-\alpha}_\infty(U)$ 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.

Full text

See pages 1-33 of Boxing-Inequality-Fractional-Perimeter-Ponce-Spector-v3.pdf