Hardy-Sobolev inequalities and weighted capacities in metric spaces

TL;DR

This paper establishes a connection between weighted Hardy-Sobolev inequalities and the quasiadditivity of weighted capacities in metric spaces, providing new characterizations and tools for analysis in such spaces.

Contribution

It introduces a novel equivalence between Hardy-Sobolev inequalities and weighted capacity quasiadditivity, utilizing discrete convolution and Maz'ya type characterizations.

Findings

Equivalence between weighted Hardy-Sobolev inequalities and capacity quasiadditivity.

Use of discrete convolution as a capacity test function.

Maz'ya type characterization of inequalities.

Abstract

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $\Omega$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.