Geometric characterizations of embedding theorems

TL;DR

This paper characterizes when embedding theorems hold on spaces of homogeneous type, linking geometric measure conditions to functional analysis results, and applies these findings to Sobolev, Besov, and Triebel-Lizorkin spaces.

Contribution

It establishes a geometric criterion for embedding theorems on spaces of homogeneous type, providing new insights and sharp results for classical function spaces.

Findings

Embedding theorems hold iff measures of all balls have lower bounds.

Provides new sharp embedding results for Sobolev, Besov, Triebel-Lizorkin spaces.

Links geometric measure conditions to functional analysis embeddings.

Abstract

The embedding theorem arises in several problems from analysis and geometry. The purpose of this paper is to provide a deeper understanding of analysis and geometry with a particular focus on embedding theorems on spaces of homogeneous type in the sense of Coifman and Weiss. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov and Triebel-Lizorkin spaces.