Hardy inequalities for fractional integrals on general domains

TL;DR

This paper establishes a sharp Hardy inequality for fractional integrals on general domains, matching the half-space constant, thereby confirming a recent conjecture and broadening the understanding of fractional integral inequalities.

Contribution

It proves a sharp Hardy inequality for fractional integrals on arbitrary domains, confirming a conjecture and extending previous results beyond specific geometries.

Findings

The inequality constant matches that of the half-space case.

The result applies to general domains, not just specific geometries.

It settles a recent conjecture by Bogdan and Dyda.

Abstract

We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.