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

TL;DR

This paper provides new necessary and sufficient conditions for the compactness of embeddings among fractional Sobolev, Besov, and Triebel-Lizorkin spaces in quasi-metric-measure spaces, extending known results even in metric spaces.

Contribution

It introduces novel characterizations of compact embeddings in quasi-metric spaces, covering optimal exponent ranges influenced by the space's geometry.

Findings

Established new criteria for compactness of embeddings

Extended results to quasi-metric spaces beyond metric spaces

Quantified conditions based on geometric properties

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.