An idea on proving weighted Sobolev embeddings

This article contains a characterization of when certain weighted Sobolev spaces on ${\mathbb{R}}^n$ embed compactly into $L^2({\mathbb{R}}^n,\phi )$. The characterization is in terms of derivatives of the weight function $\phi$ and involves the Wiener capacity, as it is obtained from reformulating the problem in terms of resolvent properties of Schr"odinger operators. This reformulation also works for general domains.