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

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:=Mqp(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = M_q^p({\mathbb{R}^n})$$\end{document} (the Morrey space), X:=Lp→(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = {L^{\vec p}}({\mathbb{R}^n})$$\end{document} (the mixed-norm Lebesgue space) and X:=(EΦq)t(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X: = {(E_\Phi ^q)_t}({^n})$$\end{document} (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.