On the Hardy-Sobolev-Maz'ya inequality and its generalizations

TL;DR

This paper explores generalizations of the Hardy-Sobolev-Maz'ya inequality, focusing on optimality, stability, minimizers, and the associated energy space to deepen understanding of these functional inequalities.

Contribution

It introduces new generalizations of the Hardy-Sobolev-Maz'ya inequality and investigates their optimality, stability, and minimizers, advancing theoretical understanding.

Findings

Established conditions for optimality and stability.

Proved existence or non-existence of minimizers.

Characterized the natural energy space for the inequalities.

Abstract

The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.