Some function spaces via orthonormal bases on spaces of homogeneous type

Let (X,d,) be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of L2(X) constructed recently by Auscher and Hytönen, we develop the Besov and Triebel-Lizorkin spaces on such a general setting. In this paper, we establish the wavelet characterizations and provide the dualities for these spaces. The results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure to the full generality of the theory of these function spaces.