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In this article, via certain lower bound conditions on the measures underconsideration, the authors fully characterize the Sobolev embeddings for thescales of Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in thegeneral context of quasi-metric measure spaces for an optimal range of thesmoothness parameter $s$. An interesting facet of this work is how the range of$s$ for which the above characterizations of these embeddings hold true isintimately linked (in a quantitative manner) to the geometric makeup of theunderlying space. Moreover, although stated for Haj{\l}asz-Triebel-Lizorkin andHaj{\l}asz-Besov spaces in the context of quasi-metric spaces, the main resultsin this article improve known work even for Sobolev spaces in the metricsetting.

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