An idea on proving weighted Sobolev embeddings

Abstract

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 Schrödinger operators. This reformulation also works for general domains.

References in corpus (3)

• Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials
• Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators
• On the resolvent of the Dirac operator in ${\mathbb{R}}^2$