Abstrakt
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.