Non-collapsing condition and Sobolev embeddings for Hajłasz-Besov spaces

Abstract: 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 $(\Omega,d,\mu)$ and the non-collapsing condition of the measure, i.e. [ \inf_{x\in\Omega}\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.