A Compact Embedding Theorem for Generalized Sobolev Spaces

TL;DR

This paper presents a straightforward proof of a compact embedding theorem applicable to various Sobolev spaces, including degenerate and nondegenerate cases, with applications to classical and non-Euclidean settings.

Contribution

It provides a unified, elementary proof of a compact embedding theorem for generalized Sobolev spaces, extending to degenerate, classical, and quasimetric space contexts.

Findings

Established a general compact embedding theorem for Sobolev spaces.

Extended results to degenerate Sobolev spaces with nonnegative quadratic forms.

Applied the theorem to classical Rellich-Kondrachov and quasimetric space cases.

Abstract

We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms. Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains, the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to Euclidean space and possibly without any notion of gradient.