Compactness of higher-order Sobolev embeddings

Abstract

We study higher-order compact Sobolev embeddings on a domain in R^n endowed with a probability measure and satisfying certain isoperimetric inequality. We present a condition on a pair of rearrangement-invariant spaces X and Y which suffices to guarantee a compact embedding of the Sobolev space V^mX into Y.

The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of the underlying domain. We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.

Keywords

• Compactness
• Sobolev space
• rearrangement-invariant space
• isoperimetric function
• almost-compact embedding
• John domain
• Maz'ya domain
• product probability space
• integral operator