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

  • Long Huang
  • Jun Liu
  • Dachun Yang
  • Wen Yuan
Article
  • 73 Downloads

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 \(\mathbb {R}^n\).

Keywords

Anisotropic (mixed-norm) Hardy space Calderón–Zygmund decomposition Discrete Calderón reproducing formula Atom Littlewood–Paley function Calderón–Zygmund operator 

Mathematics Subject Classification

Primary 42B35 Secondary 42B30 42B25 42B20 30L99 

Notes

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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Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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