Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces

Abstract

In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the context of quasi-metric spaces, the main results in this article improve known work even for Sobolev spaces in the metric setting.