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

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    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.

    LanguageEnglish
    Pages2197-2267
    Number of pages71
    JournalJournal of Fourier Analysis and Applications
    Volume25
    Issue number5
    Early online date20 Nov 2018
    DOIs
    Publication statusPublished - Oct 2019

    Fingerprint

    Real variables
    Space of Homogeneous Type
    Maximal Function
    Hardy Space
    Radial Functions
    Wavelets
    Littlewood-Paley Function
    Sublinear Operator
    Quasi-metric
    Doubling
    Reverse
    Boundedness
    Smoothness
    Necessary
    Range of data

    Keywords

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

    Cite this

    He, Ziyi ; Han, Yongsheng ; Li, Ji ; Liu, Liguang ; Yang, Dachun ; Yuan, Wen. / A complete real-variable theory of Hardy spaces on spaces of homogeneous type. In: Journal of Fourier Analysis and Applications. 2019 ; Vol. 25, No. 5. pp. 2197-2267.
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    A complete real-variable theory of Hardy spaces on spaces of homogeneous type. / He, Ziyi; Han, Yongsheng; Li, Ji; Liu, Liguang; Yang, Dachun; Yuan, Wen.

    In: Journal of Fourier Analysis and Applications, Vol. 25, No. 5, 10.2019, p. 2197-2267.

    Research output: Contribution to journalArticleResearchpeer-review

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