Non-collapsing condition and Sobolev embeddings for Haj{\l}asz-Besov spaces

In this paper we will focus on understanding the relation between Sobolev embedding theorems for Haj{\l}asz-Besov spaces defined on a doubling metric measure space $(\mathrm{\Omega },d,\mu )$ and the non-collapsing condition of the measure, i.e. $$\underset{x\in \mathrm{\Omega }}{\inf }\mu (B(x,1))>0.$$ We will also obtain embedding results for Haj{\l}asz-Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.