Measure density and Embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces

2018/03/01 by Nijjwal Karak, Karak, Nijjwal
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1803.00224

openalex publication_date 2018/03/01 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

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))≥ cr^Q 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)≥ cr^Q holds for any ball B(x,r)⊂ X, where X=(X,d,μ) is an Ahlfors Q-regular and geodesic metric measure space.

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