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

Abstract

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 Hajłasz--Triebel--Lizorkin spaces $M_{p,q}^s$ and Hajłasz--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 in this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces.

arXiv admin note: text overlap with arXiv:2202.06389