On fractional Hardy inequalities in convex sets

TL;DR

This paper establishes a fractional Hardy inequality for convex sets within fractional Sobolev spaces, leveraging the superharmonicity of the boundary distance function, with results stable as the fractional parameter approaches 1.

Contribution

It introduces a Hardy inequality for fractional Sobolev spaces in convex domains, extending classical results to fractional orders with stable constants.

Findings

Hardy inequality proven for fractional Sobolev spaces in convex sets

The inequality holds for all p between 1 and infinity and s between 0 and 1

Constant remains stable as s approaches 1

Abstract

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<\infty$ and $0<s<1$, with a constant which is stable as $s$ goes to $1$.