Lower bound of measure and embeddings of Sobolev, Besov and Triebel–Lizorkin spaces

In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Hajlasz-Besov spaces or Hajlasz-Triebel-Lizorkin spaces defined on a doubling and geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.