Measure density and Embeddings of Haj{\l}asz-Besov and Haj{\l}asz-Triebel-Lizorkin spaces

TL;DR

This paper explores the relationship between measure density conditions and Sobolev-type embeddings in Haj{ ext}asz-Besov and Haj{ ext}asz-Triebel-Lizorkin spaces on metric measure spaces, establishing equivalences under certain regularity assumptions.

Contribution

It proves that measure density conditions are necessary and sufficient for Sobolev-type embeddings in these function spaces on metric measure spaces.

Findings

Measure density implies Sobolev embeddings on balls.

Sobolev embeddings imply measure density condition.

Results hold on Ahlfors Q-regular, geodesic metric spaces.

Abstract

In this paper, we investigate the relation between Sobolev-type embeddings of Haj{\l}asz-Besov spaces (and also Haj{\l}asz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,\mu)$ and lower bound for the measure $\mu.$ We prove that if the measure $\mu$ satisfies $\mu(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain $\Omega\subset X,$ then we prove that the domain $\Omega$ satisfies the so-called measure density condition, i.e., $\mu(B(x,r)\cap\Omega)\geq cr^Q$ holds for any ball $B(x,r)\subset X,$ where $X=(X,d,\mu)$ is an Ahlfors $Q$-regular and geodesic metric measure space.