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

Abstract

In this article, the authors establish the wavelet characterization of Besov and Triebel-Lizorkin spaces on a given space (X, d, µ) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel-Lizorkin sequence spaces on X, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel-Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood-Paley characterizations of Triebel-Lizorkin spaces on X. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of µ and also the triangle inequality of d, by fully using the geometrical property of X expressed via its equipped quasi-metric d, dyadic reference points, dyadic cubes, and wavelets.