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,μ) and lower bound for the measure μ. We prove that if the measure μ satisfies μ(B(x,r))≥crQ for some Q>0 and for any ball B(x,r)⊂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 Ω⊂X, then we prove that the domain Ω satisfies the so-called measure density condition, i.e., μ(B(x,r)∩Ω)≥crQ holds for any ball B(x,r)⊂X, where X=(X,d,μ) is an Ahlfors Q-regular and geodesic metric measure space.