Wavelet characterization of Triebel–Lizorkin spaces for p=∞ on spaces of homogeneous type and its applications

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Let ( X , d , μ ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, η ∈ ( 0 , 1 ) the Hölder regularity of wavelets on X constructed by P. Auscher and T. Hytönen, s ∈ ( − η , η ) , and q ∈ ( max { ω ω + η , ω ω + η + s } , ∞ ] . In this article, the authors establish the wavelet characterization of the Triebel–Lizorkin space F ̇ ∞ , q s ( X ) . Moreover, the authors introduce almost diagonal operators on the Triebel–Lizorkin sequence space f ̇ ∞ , q s ( X ) and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors establish the molecular characterization of F ̇ ∞ , q s ( X ) . The authors also obtain both the Lusin area function and the Littlewood–Paley g λ ∗ -function characterizations of F ̇ ∞ , q s ( 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 μ 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.