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

@article{Karak2019MeasureDA,
  title={Measure density and embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces},
  author={Nijjwal Karak},
  journal={Journal of Mathematical Analysis and Applications},
  year={2019}
}
  • Nijjwal Karak
  • Published 1 March 2018
  • Mathematics
  • Journal of Mathematical Analysis and Applications
In this paper, we investigate the relation between Sobolev-type embeddings of Haj{\l}asz-Besov spaces (and also Haj{\l}asz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,\mu)$ and lower bound for the measure $\mu.$ We prove that if the measure $\mu$ satisfies $\mu(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ 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 $\Omega… 
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