An idea on proving weighted Sobolev embeddings

This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, \varphi)$. The characterization is in terms of derivatives of the weight function $\varphi$ 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.