A measure characterization of embedding and extension domains for Sobolev, Triebel-Lizorkin, and Besov spaces on spaces of homogeneous type

In this article, for an optimal range of the smoothness parameter s that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajlasz-Triebel- Lizorkin spaces M-p,q(s) and Hajlasz-Besov spaces N(p,q)(s )in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results of this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces. (C) 2022 Elsevier Inc. All rights reserved.