Hardy inequalities and nonlocal capacity

TL;DR

This paper introduces nonlocal capacities linked to Sobolev spaces, establishes Hardy inequalities for these spaces, and estimates capacities of geometric objects, advancing the understanding of nonlocal analysis.

Contribution

It develops new Hardy inequalities for nonlocal Sobolev spaces and applies them to estimate capacities of sets, providing novel tools in nonlocal analysis.

Findings

Established Hardy inequalities for nonlocal Sobolev spaces

Derived Sobolev embeddings using these inequalities

Estimated capacities of balls in nonlocal settings

Abstract

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.