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 R~n. In this article, we introduce the weak Hardytype space W HX(R~n), associated with X, via the radial maximal function. Assuming that the powered HardyLittlewood 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 W X and the associated space, we then establish several real-variable characterizations of W HX(R~n), 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(R~n) to W HX(R~n), which includes the critical case. All these results are of wide applications.Particularly, when X := Mq~p(R~n)(the Morrey space), X := L~p(R~n)(the mixed-norm Lebesgue space) and X :=(EΦ~q)t(R~n)(the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasiBanach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.