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Riesz Transform Characterization of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

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Let X be a ball quasi-Banach function space satisfying some mild assumptions and \(H_X(\mathbb {R}^n)\) the Hardy space associated with X. In this article, the authors introduce both the Hardy space \(H_X(\mathbb {R}^{n+1}_+)\) of harmonic functions and the Hardy space \(\mathbb {H}_X(\mathbb {R}^{n+1}_+)\) of harmonic vectors, associated with X, and then establish the isomorphisms among \(H_X(\mathbb {R}^n)\), \(H_{X,2}(\mathbb {R}^{n+1}_+)\), and \(\mathbb {H}_{X,2}(\mathbb {R}^{n+1}_+)\), where \(H_{X,2}(\mathbb {R}^{n+1}_+)\) and \(\mathbb {H}_{X,2}(\mathbb {R}^{n+1}_+)\) are, respectively, certain subspaces of \(H_X(\mathbb {R}^{n+1}_+)\) and \(\mathbb {H}_X(\mathbb {R}^{n+1}_+)\). Using these isomorphisms, the authors establish the first order Riesz transform characterization of \(H_X(\mathbb {R}^n)\). The higher order Riesz transform characterization of \(H_X(\mathbb {R}^n)\) is also obtained. The results obtained in this article have a wide range of generality and can be applied to classical Hardy spaces, weighted Hardy spaces, variable Hardy spaces, Herz–Hardy spaces, Lorentz–Hardy spaces, mixed-norm Hardy spaces, local generalized Herz–Hardy spaces, and mixed-norm Herz–Hardy spaces and all the obtained results on the aforementioned last five Hardy-type spaces are completely new.

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Acknowledgements

The authors would like to thank both referees for their carefully reading and many remarks which indeed improve the quality of this article. This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197 and 12122102).

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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, The People’s Republic of China

    Fan Wang, Dachun Yang & Wen Yuan

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  1. Fan Wang
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Wang, F., Yang, D. & Yuan, W. Riesz Transform Characterization of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces. J Fourier Anal Appl 29, 56 (2023). https://doi.org/10.1007/s00041-023-10036-0

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