A compact embedding theorem for generalized sobolev spaces

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 on Rn. Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegenerate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains in Rn, 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 Rn and possibly without any notion of gradient.