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In this article, the authors establish the wavelet characterization of Besovand Triebel--Lizorkin spaces on a given space $(X,d,\mu)$ of homogeneous typein the sense of Coifman and Weiss. Moreover, the authors introduce almostdiagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, andobtain their boundedness. Using this wavelet characterization and thisboundedness of almost diagonal operators, the authors obtain the molecularcharacterization of Besov and Triebel--Lizorkin spaces. Applying this molecularcharacterization, the authors further establish the Littlewood--Paleycharacterizations of Triebel--Lizorkin spaces on $X$. The main novelty of thisarticle is that all these results get rid of their dependence on the reversedoubling property of $\mu$ and also the triangle inequality of $d$, by fullyusing the geometrical property of $X$ expressed via its equipped quasi-metric$d$, dyadic reference points, dyadic cubes, and wavelets

Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications