Wavelet characterization of Triebel-Lizorkin spaces for p = infinity on spaces of homogeneous type and its applications?

Let (X, d, mu) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, eta is an element of (0, 1) the Holder regularity of wavelets on X constructed by P. Auscher and T. Hytonen, s is an element of (-eta, eta), and q is an element of (max{ omega omega+eta+s }, infinity]. In this article, the authors establish the wavelet characterization of the Triebel-Lizorkin space Fs infinity,q(X). Moreover, the authors introduce almost diagonal operators on the Triebel-Lizorkin sequence space fs infinity,q(X) and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors establish the molecular characterization of Fs infinity,q(X). The authors also obtain both the Lusin area function and the Littlewood- Paley g lambda*-function characterizations of Fs infinity,q(X). Besides, the inhomogeneous counterparts are also given. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of mu and also the triangle inequality of d, by fully using the geometrical properties of X expressed via its equipped quasi-metric d, dyadic reference points, dyadic cubes, and wavelets.(c) 2022 Elsevier Inc. All rights reserved.