An idea on proving weighted Sobolev embeddings

TL;DR

This paper characterizes when weighted Sobolev spaces on br^n embed compactly into weighted L^2 spaces, using derivatives of the weight function and Wiener capacity, via a Schrf6dinger operator approach.

Contribution

It introduces a new characterization of compact embeddings of weighted Sobolev spaces using capacity and Schrf6dinger operator techniques, extending to general domains.

Findings

Provides necessary and sufficient conditions for compact embeddings.

Links Sobolev embedding properties to Wiener capacity and Schrf6dinger resolvent behavior.

Extends the characterization to arbitrary domains.

Abstract

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.