Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces I: Decompositions with Applications to Boundedness of Calder\'on--Zygmund Operators

Let $X$ be a ball quasi-Banach function space on ${\mathbb{R}}^n$. In this article, the authors introduce the weak Hardy-type space $W{H}_X({\mathbb{R}}^n)$, 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, the authors then establish several real-variable characterizations of $W{H}_X({\mathbb{R}}^n)$, respectively, in terms of various maximal functions, atoms and molecules. As an application, the authors obtain the boundedness of Calder'on–Zygmund operators from the Hardy space $H_X({\mathbb{R}}^n)$ to $W{H}_X({\mathbb{R}}^n)$, 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 $X:=(E_{\mathrm{\Phi }}^q{)}_t({\mathbb{R}}^n)$ (the Orlicz-slice space), which are all ball quasi-Banach function spaces but not quasi-Banach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.