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Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces

In Memory of Professor Carlos Berenstein

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Abstract

Let X be a ball quasi-Banach function space on \({{\mathbb {R}}}^n\). In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space \(H_X({{\mathbb {R}}}^n)\) associated with X, under some weak assumptions on the Littlewood–Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X. All these results have wide applications. Particularly, when \(X:=M_r^p({{\mathbb {R}}}^n)\) (the Morrey space), \(X:=L^{\vec {p}}({{\mathbb {R}}}^n)\) (the mixed-norm Lebesgue space), \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space), \(X:=L_\omega ^p({{\mathbb {R}}}^n)\) (the weighted Lebesgue space) and \(X:=(E_\Phi ^r)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), the Littlewood–Paley function characterizations of \(H_X({{\mathbb {R}}}^n)\) obtained in this article improve the existing results via weakening the assumptions on the Littlewood–Paley functions and widening the range of \(\lambda \) in the Littlewood–Paley \(g_\lambda ^*\)-function characterization of \(H_X({\mathbb {R}}^n)\).

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The authors would like to thank the referee for her/his useful comments which indeed improve the presentation of this article.

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Authors and Affiliations

  1. Department of Mathematics, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  2. Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City, 242, Taiwan, ROC

    Der-Chen Chang

  3. College of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, People’s Republic of China

    Songbai Wang

  4. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Dachun Yang & Yangyang Zhang

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  1. Der-Chen Chang
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Correspondence to Dachun Yang or Yangyang Zhang.

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Communicated by Irene Sabadini.

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This article is part of the topical collection “In memory of Carlos A. Berenstein (1944-2019)” edited by Irene Sabadini and Daniele Struppa.

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002 and 11671185). Der-Chen Chang is partially supported by an NSF Grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

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Chang, DC., Wang, S., Yang, D. et al. Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces. Complex Anal. Oper. Theory 14, 40 (2020). https://doi.org/10.1007/s11785-020-00998-0

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