Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type-via Orthonormal Wavelets

It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure mu satisfies the doubling property only. 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. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure mu. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel-Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher-Hytonen on spaces of homogeneous type in the sense of Coifman and Weiss.