Measure density and Embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces

Abstract

In this paper, we investigate the relation between Sobolev-type embeddings of Hajłasz-Besov spaces (and also Hajłasz-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 $\mu (B(x,r))\ge c{r}^Q$ 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., $\mu (B(x,r)\cap \Omega )\ge c{r}^Q$ holds for any ball $B(x,r)\subset X,$ where $X=(X,d,\mu )$ is an Ahlfors $Q$-regular and geodesic metric measure space.

30 pages