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

TL;DR

This paper provides measure-theoretic criteria for embedding and extension domains of Sobolev, Triebel-Lizorkin, and Besov spaces on spaces of homogeneous type, improving existing characterizations especially in metric spaces.

Contribution

It introduces new measure-based characterizations for embedding and extension domains of these function spaces, applicable to general spaces of homogeneous type and doubling metric measure spaces.

Findings

Characterization of embedding and extension domains for Haj{ ext}asz--Triebel--Lizorkin and Besov spaces.

New criteria applicable to quasi-metric and metric spaces.

Enhanced understanding of Sobolev embedding and extension domains.

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 Haj{\l}asz--Triebel--Lizorkin spaces $M^s_{p,q}$ and Haj{\l}asz--Besov spaces $N^s_{p,q}$ 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.