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Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In thisarticle, the authors introduce the weak Hardy-type space $WH_X({\mathbb R}^n)$,associated with $X$, via the radial maximal function. Assuming that the poweredHardy--Littlewood maximal operator satisfies some Fefferman--Steinvector-valued maximal inequality on $X$ as well as it is bounded on both theweak ball quasi-Banach function space $WX$ and the associated space, theauthors then establish several real-variable characterizations of$WH_X({\mathbb R}^n)$, respectively, in terms of various maximal functions,atoms and molecules. As an application, the authors obtain the boundedness ofCalder\'on--Zygmund operators from the Hardy space $H_X({\mathbb R}^n)$ to$WH_X({\mathbb R}^n)$, which includes the critical case. All these results areof wide applications. Particularly, when $X:=M_q^p({\mathbb R}^n)$ (the Morreyspace), $X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space) and$X:=(E_\Phi^q)_t({\mathbb R}^n)$ (the Orlicz-slice space), which are all ballquasi-Banach function spaces but not quasi-Banach function spaces, all theseresults are even new. Due to the generality, more applications of these resultsare predictable.

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