Compactness of Sobolev-type embeddings with measures

Abstract

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in [Formula: see text] with respect to upper Ahlfors regular measures [Formula: see text], that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure [Formula: see text] and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.