Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces

TL;DR

This paper characterizes Sobolev embeddings for Haj{}asz-Triebel-Lizorkin and Besov spaces on quasi-metric measure spaces, revealing how geometric properties influence the range of smoothness parameters for optimal embeddings.

Contribution

It provides a full characterization of embeddings in quasi-metric spaces, extending and improving known results even in metric space settings.

Findings

Characterization of Sobolev embeddings under lower bound measure conditions

Link between the smoothness parameter range and space geometry

Improved results for Sobolev spaces in metric spaces

Abstract

In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in the context of quasi-metric spaces, the main results in this article improve known work even for Sobolev spaces in the metric setting.