Optimal embeddings for Triebel–Lizorkin and Besov spaces on quasi-metric measure spaces

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 is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Importantly, although the main results in this article are stated in the context of quasi-metric spaces, the authors provide several examples illustrating how this range of s is strictly larger than similar ones currently appearing in the literature, even in the metric setting. Moreover, the authors relate these values of s to the (non)triviality of these function spaces.