In between the inequalities of Sobolev and Hardy

TL;DR

This paper investigates Hardy-Sobolev inequalities on open sets in Euclidean space, establishing conditions involving the Assouad dimension of the complement that determine when these inequalities hold.

Contribution

It provides new necessary and sufficient conditions for Hardy-Sobolev inequalities based on the Assouad dimension, bridging Sobolev and Hardy inequalities.

Findings

Conditions involving Assouad dimension determine validity

Hardy-Sobolev inequalities interpolate between Sobolev and Hardy inequalities

Results apply to open sets with various geometric properties

Abstract

We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.