Optimal Hardy-Sobolev-Maz'ya inequalities with multiple interior singularities

TL;DR

This paper characterizes Hardy inequalities with interior singularities in multi-dimensional spaces and establishes conditions for Hardy-Sobolev-Maz'ya inequalities with optimal Sobolev terms.

Contribution

It provides a complete characterization of Hardy inequalities with multiple interior singularities and necessary and sufficient conditions for optimal Hardy-Sobolev-Maz'ya inequalities.

Findings

Characterization of Hardy inequalities with multiple interior singularities

Necessary and sufficient conditions for Hardy-Sobolev-Maz'ya inequalities

Identification of potentials with strong interior singularities

Abstract

In this article we first establish a complete characterization of Hardy's inequalities in $\mathbb{R}^n$ involving distances to different codimension subspaces. In particular the corresponding potentials have strong interior singularities. We then provide necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya inequalities with optimal Sobolev terms.