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 geometric makeup underlying space, authors identify purely measure theoretic conditions fully characterize embedding and extension domains scale Haj?asz–Triebel–Lizorkin spaces Mp,qs Haj?asz–Besov Np,qs in general homogeneous type. Although stated context quasi-metric spaces, these characterizations improve related work even metric setting. particular, as a corollary main results obtain new characterization Sobolev doubling spaces.