Compactness of Sobolev imbeddings involving rearrangement-invariant norms
We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space be compactly imbedded into the rearrangement-invariant space , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from into . The results are illustrated with examples in which ϱ and σ are both Orlicz norms...