Hardy-Sobolev inequalities and weighted capacities in metric spaces

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.