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Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

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Abstract

Let \(\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n\), \(\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n\) and \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\) be the anisotropic mixed-norm Hardy space associated with \(\vec {a}\) defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón–Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\), respectively, by means of atoms, the Lusin area function, the Littlewood–Paley g-function or \(g_{\lambda }^*\)-function. The obtained Littlewood–Paley g-function characterization of \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\) coincidentally confirms a conjecture proposed by Hart et al. (Trans Am Math Soc, https://doi.org/10.1090/tran/7312, 2017). Applying the aforementioned Calderón–Zygmund decomposition as well as the atomic characterization of \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\), the authors establish a finite atomic characterization of \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\), which further induces a criterion on the boundedness of sublinear operators from \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\) into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\) to itself [or to the mixed-norm Lebesgue space \(L^{\vec {p}}(\mathbb {R}^n)\)]. The obtained atomic characterizations of \(H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)\) and boundedness of anisotropic Calderón–Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. (J Geom Anal 27:2758–2787, 2017). All these results are new even for the isotropic mixed-norm Hardy spaces on \(\mathbb {R}^n\).

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Acknowledgements

The authors would like to thank the referees for their careful reading and useful comments, which indeed improved the presentation of this article.

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  1. 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

    Long Huang, Jun Liu, Dachun Yang & Wen Yuan

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  1. Long Huang
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  2. Jun Liu
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Correspondence to Wen Yuan.

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This project is supported by the National Natural Science Foundation of China (Grant Nos. 11761131002, 11571039, 11726621, and 11471042).

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Huang, L., Liu, J., Yang, D. et al. Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications. J Geom Anal 29, 1991–2067 (2019). https://doi.org/10.1007/s12220-018-0070-y

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