The equivalence between pointwise Hardy inequalities and uniform fatness

TL;DR

This paper establishes a new equivalence between pointwise Hardy inequalities and uniform capacity density of the complement, providing insights applicable in both Euclidean and general metric spaces.

Contribution

It proves a novel equivalence between Hardy inequalities and capacity density, and offers a new proof linking capacity density to the classical Hardy inequality in metric spaces.

Findings

Equivalence between pointwise Hardy inequalities and uniform capacity density.

New proof that capacity density implies classical Hardy inequality.

Relations between Hardy inequalities, capacity density, and Hausdorff content.

Abstract

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.