Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).