Hardy-Sobolev inequalities and weighted capacities in metric spaces

Abstract

Let Ω 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 Ω and quasiadditivity of a weighted capacity with respect to Whitney covers of Ω. 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.