  In this article, via certain lower bound conditions on the measures under
consideration, the authors fully characterize the Sobolev embeddings for the
scales of Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-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 Haj{\l}asz-Triebel-Lizorkin and
Haj{\l}asz-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.
