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

Abstract

Let $X$ be a ball quasi-Banach function space on ${\\mathbb R}^n$. In this article, the authors introduce the weak Hardy-type space $WH_X({\\mathbb R}^n)$, 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 $WH_X({\\mathbb R}^n)$, respectively, in terms of various maximal functions, atoms and molecules. As an application, the authors obtain the boundedness of Calderon--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 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), 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.