Compact embeddings of Sobolev, Besov, and Triebel–Lizorkin spaces

Abstract

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel–Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.