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

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

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