A compact embedding theorem for generalized Sobolev spaces

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 ‫ޒ‬ n .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 ‫ޒ‬ n , 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 ‫ޒ‬ n and possibly without any notion of gradient.