COMPACTNESS OF HIGHER-ORDER SOBOLEV EMBEDDINGS

We study higher-order compact Sobolev embeddings on a domain Omega subset of R-n endowed with a probability measure nu and satisfying certain isoperimetric inequality. Given m is an element of N, we present a condition on a pair of rearrangement-invariant spaces X(Omega, nu) and Y(Omega, nu) which suffices to guarantee a compact embedding of the Sobolev space (VX)-X-m(Omega, nu) into Y(Omega, nu). The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of (Omega, nu). 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.