A complete real-variable theory of Hardy spaces on spaces of homogeneous type

Ziyi He, Yongsheng Han, Ji Li, Liguang Liu, Dachun Yang*, Wen Yuan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    36 Citations (Scopus)


    Let (X, d, μ) be a space of homogeneous type, with the upper dimension ω, in the sense of Coifman and Weiss. Assume that η is the smoothness index of the wavelets on X constructed by Auscher and Hytönen. In this article, when p∈ (ω/ (ω+ η) , 1] , for the atomic Hardy spaces Hcwp(X) introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal functions, the radial maximal functions, the non-tangential maximal functions, the various Littlewood–Paley functions and wavelet functions. This completely answers the question of Coifman and Weiss by showing that no additional (geometrical) condition is necessary to guarantee the radial maximal function characterization of Hcw1(X) and even of Hcwp(X) with p as above. As applications, the authors obtain the finite atomic characterizations of Hcwp(X), which further induce some criteria for the boundedness of sublinear operators on Hcwp(X). Compared with the known results, the novelty of this article is that μ is not assumed to satisfy the reverse doubling condition and d is only a quasi-metric, moreover, the range p∈ (ω/ (ω+ η) , 1] is natural and optimal.

    Original languageEnglish
    Pages (from-to)2197-2267
    Number of pages71
    JournalJournal of Fourier Analysis and Applications
    Issue number5
    Early online date20 Nov 2018
    Publication statusPublished - Oct 2019


    • Space of homogeneous type
    • Hardy space
    • Maximal function
    • Atom
    • Littlewood–Paley function
    • Wavelet


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