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In this article, for an optimal range of the smoothness parameter $s$ thatdepends (quantitatively) on the geometric makeup of the underlying space, theauthors identify purely measure theoretic conditions that fully characterizeembedding and extension domains for the scale of Haj{\l}asz--Triebel--Lizorkinspaces $M^s_{p,q}$ and Haj{\l}asz--Besov spaces $N^s_{p,q}$ in general spacesof homogeneous type. Although stated in the context of quasi-metric spaces,these characterizations improve related work even in the metric setting. Inparticular, as a corollary of the main results in this article, the authorsobtain a new characterization for Sobolev embedding and extension domains inthe context of general doubling metric measure spaces.

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type