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Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

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Abstract

Let X be a ball quasi-Banach function space on ℝn. In this article, we introduce the weak Hardy-type space W HX(ℝn), associated with X, via the radial maximal function. Assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, we then establish several real-variable characterizations of W HX (ℝn), respectively, in terms of various maximal functions, atoms and molecules. As an application, we obtain the boundedness of Calderón-Zygmund operators from the Hardy space HX (ℝn) to W HX (ℝn), which includes the critical case. All these results are of wide applications. Particularly, when \(X: = M_q^p({\mathbb{R}^n})\) (the Morrey space), \(X: = {L^{\vec p}}({\mathbb{R}^n})\) (the mixed-norm Lebesgue space) and \(X: = {(E_\Phi ^q)_t}({^n})\) (the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasi-Banach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002, 11671185 and 11871100). The authors thank the referees for their carefully reading and many motivating and helpful comments which indeed improve the quality 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, China

    Yangyang Zhang, Dachun Yang & Wen Yuan

  2. College of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

    Songbai Wang

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  1. Yangyang Zhang
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Correspondence to Dachun Yang.

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Zhang, Y., Yang, D., Yuan, W. et al. Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators. Sci. China Math. 64, 2007–2064 (2021). https://doi.org/10.1007/s11425-019-1645-1

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