Compactness of higher-order Sobolev embeddings

We study higher-order compact Sobolev embeddings on a domain ∩ ⊆Rn endowed with a probability measure ∨ and satisfying certain isoperimetric inequality. Given m ∈ N, 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 m X ( ∩,∨) into Y (∩,∨). The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of ( ∩,∨). 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