Characterizations for fractional Hardy inequality

TL;DR

This paper characterizes when fractional Hardy inequalities hold in terms of fractional capacity and Whitney cubes, providing a new criterion for bounded open sets.

Contribution

It introduces a Maz'ya type characterization for fractional Hardy inequalities based on fractional capacity and Whitney cube quasiadditivity.

Findings

Fractional Hardy inequality holds iff fractional capacity is quasiadditive.

Boundedness of zero extension operator is linked to fractional Hardy inequality.

Provides a new criterion for fractional Hardy inequalities in open sets.

Abstract

We provide a Maz'ya type characterization for a fractional Hardy inequality. As an application, we show that a bounded open set $G$ admits a fractional Hardy inequality if and only if the associated fractional capacity is quasiadditive with respect to Whitney cubes of $G$ and the zero extension operator acting on $C_c(G)$ is bounded in an appropriate manner.