Fractional Hardy-Sobolev-Maz'ya inequality on balls and halfspaces

TL;DR

This paper establishes a fractional Hardy-Sobolev-Maz'ya inequality for geometric domains like balls and half-spaces, addressing an open problem and extending recent results in the field.

Contribution

It proves the inequality for new domains, namely balls and half-spaces, advancing the understanding of fractional inequalities in mathematical analysis.

Findings

Proved fractional Hardy-Sobolev-Maz'ya inequality for balls.

Extended the inequality to half-spaces, completing partial solutions.

Addressed an open problem in fractional analysis.

Abstract

We prove fractional Hardy--Sobolev--Maz'ya inequality for balls and a half-space, partially answering the open problem posed by Frank and Seiringer [arXiv:0906.1561v1 [math.FA], 2009] We note that for half-spaces this inequality has been recently obtained by Sloane [arXiv:1004.4828v1 [math.FA], 2010].