Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces

TL;DR

This paper investigates the criteria under which weighted Sobolev space embeddings are compact, providing both necessary and sufficient conditions for such embeddings in weighted domains.

Contribution

It introduces new necessary and sufficient conditions for the compactness of weighted Sobolev space embeddings, advancing understanding in functional analysis.

Findings

Established necessary conditions for compactness.

Derived sufficient conditions for compactness.

Enhanced criteria for weighted Sobolev space embeddings.

Abstract

We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(\Omega,w) \to L^p(\Omega,w),$ where $w$ is some weight on a domain $\Omega \subset \Real^n$.