A compact embedding theorem for generalized Sobolev spaces

Seng-Kee Chua, Scott Rodney and Richard L. Wheeden

Vol. 265 (2013), No. 1, 17–57

DOI: 10.2140/pjm.2013.265.17

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.

Keywords

compact embedding, Sobolev spaces, degenerate quadratic forms

Mathematical Subject Classification 2010

Primary: 46B50, 46E35

Secondary: 35H20

Milestones

Received: 25 July 2012

Accepted: 22 February 2013

Published: 28 August 2013