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Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

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Abstract

Let X be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of \(H_X(\mathbb {R}^n)\), the Hardy space associated with X, via the Littlewood–Paley g-functions and \(g_\lambda ^*\)-functions. Moreover, the authors obtain the boundedness of Calderón–Zygmund operators on \(H_X(\mathbb {R}^n)\). For the local Hardy-type space \(h_X(\mathbb {R}^n)\) associated with X, the authors also obtain the boundedness of \(S^0_{1,0}(\mathbb {R}^n)\) pseudo-differential operators on \(h_X(\mathbb {R}^n)\) via first establishing the atomic characterization of \(h_X(\mathbb {R}^n)\). Furthermore, the characterizations of \(h_X(\mathbb {R}^n)\) by means of local molecules and local Littlewood–Paley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz–Hardy space, the Lorentz–Hardy space, the Morrey–Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the \(g_\lambda ^*\)-function characterization obtained in this article improves the known results via widening the range of \(\lambda \).

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Acknowledgements

Fan Wang and Sibei Yang would like to thank Ciqiang Zhuo, Ziyi He and Yangyang Zhang for many helpful discussions on the subject of this article. Moreover, Dachun Yang (the corresponding author) would like to thank Yoshihiro Sawano and Kwok Pun Ho for several helpful discussions on the subject of this article and they would also like to thank the referee for her/his very careful reading and useful comments which indeed improved the quality of this article and, particularly, motivated the authors to obtain Theorem 3.8, Remark 3.9, Theorems 3.14 and 4.13.

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Authors and Affiliations

  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

    Fan Wang & Dachun Yang

  2. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Sibei Yang

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  1. Fan Wang
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Correspondence to Dachun Yang.

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The first and the second (corresponding) authors were supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002 and 11671185) and the third author was supported by the National Natural Science Foundation of China (Grant Nos. 11871254 and 11571289).

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Wang, F., Yang, D. & Yang, S. Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces. Results Math 75, 26 (2020). https://doi.org/10.1007/s00025-019-1149-x

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