Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

  • Songbai Wang
  • Dachun YangEmail author
  • Wen Yuan
  • Yangyang Zhang


Let X be a ball quasi-Banach function space on \({\mathbb R}^n\). In this article, 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, the authors establish various Littlewood–Paley function characterizations of \(WH_X({{\mathbb {R}}}^n)\) under some weak assumptions on the Littlewood–Paley functions. The authors also prove that the real interpolation intermediate space \((H_{X}({{\mathbb {R}}}^n),L^\infty ({{\mathbb {R}}}^n))_{\theta ,\infty }\), between the Hardy space associated with X, \(H_{X}({{\mathbb {R}}}^n)\), and the Lebesgue space \(L^\infty ({\mathbb R}^n)\), is \(WH_{X^{{1}/{(1-\theta )}}}({{\mathbb {R}}}^n)\), where \(\theta \in (0, 1)\). 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 \(X:=(E_\Phi ^q)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), all these results are even new; when \(X:=L_\omega ^\Phi ({\mathbb R}^n)\) (the weighted Orlicz space), the result on the real interpolation is new and, when \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space) and \(X:=L_\omega ^\Phi ({{\mathbb {R}}}^n)\), the Littlewood–Paley function characterizations of \(WH_X({{\mathbb {R}}}^n)\) obtained in this article improves the existing results via weakening the assumptions on the Littlewood–Paley functions.


Ball quasi-Banach function space Weak Hardy space Weak tent space Orlicz-slice space Maximal function Littlewood–Paley function Real interpolation 

Mathematics Subject Classification

Primary 42B30 Secondary 42B25 42B20 42B35 46E30 



The authors would like to thank Professor Sibei Yang for many discussions on the subject of this article. They would also like to thank the referees for their carefully reading and so many motivating and useful comments which indeed improve the quality of this article.


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Authors and Affiliations

  1. 1.College of Mathematics and StatisticsHubei Normal UniversityHuangshiPeople’s Republic of China
  2. 2.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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