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

TL;DR

This paper investigates the relationship between Sobolev-type embeddings in various function spaces and lower bounds on measure of balls in metric measure spaces, providing insights into geometric and functional analysis.

Contribution

It establishes new connections between embedding properties of Sobolev, Besov, and Triebel-Lizorkin spaces and measure lower bounds in metric measure spaces.

Findings

Derived lower bounds for measures based on embedding properties

Linked geometric measure conditions with functional space embeddings

Extended results to both doubling and geodesic metric measure spaces

Abstract

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