On the structure of Hardy-Sobolev-Maz'ya inequalities

TL;DR

This paper investigates the structure of Hardy-Sobolev-Maz'ya inequalities in half spaces, providing new improvements, optimal constants, and conditions for their validity by combining Hardy and Sobolev terms.

Contribution

It introduces a comprehensive analysis of the structure of Hardy-Sobolev-Maz'ya inequalities, including the addition of Hardy type terms and the critical Sobolev term with optimal constants.

Findings

Derived new best constants for Hardy inequalities

Established necessary and sufficient conditions for Hardy-Sobolev-Maz'ya inequalities

Revealed the structure of inequalities through linear combinations of Hardy terms

Abstract

In this article we establish new improvements of the optimal Hardy inequality in the half space. We first add all possible linear combinations of Hardy type terms thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya type inequalities.