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

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 $\mu \mathrm{.}$ 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 $\mathrm{\Omega }\subset X,$ then we prove that the domain $\mathrm{\Omega }$ satisfies the so-called measure density condition, i.e., $\mu (B(x,r)\cap \mathrm{\Omega })\ge c{r}^Q$ holds for any ball $B(x,r)\subset X,$ where $X=(X,d,\mu )$ is an Ahlfors $Q$-regular and geodesic metric measure space.