Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood--Paley Characterizations and Real Interpolation

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, 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 establish various Littlewood--Paley function characterizations of $WH_X({\mathbb R}^n)$ under some weak assumptions on the Littlewood--Paley functions. The authors also prove that the real interpolation intermediate space $(H_{X}({\mathbb R}^n),L^\infty({\mathbb R}^n))_{\theta,\infty}$, between the Hardy space associated with $X$, $H_{X}({\mathbb R}^n)$, and the Lebesgue space $L^\infty({\mathbb R}^n)$, is $WH_{X^{{1}/{(1-\theta)}}}({\mathbb R}^n)$, where $\theta\in (0, 1)$. 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), all these results are even new; when $X:=L_\omega^\Phi({\mathbb R}^n)$ (the weighted Orlicz space), the result on the real interpolation is new and, when $X:=L^{p(\cdot)}({\mathbb R}^n)$ (the variable Lebesgue space) and $X:=L_\omega^\Phi({\mathbb R}^n)$, the Littlewood--Paley function characterizations of $WH_X({\mathbb R}^n)$ obtained in this article improves the existing results via weakening the assumptions on the Littlewood--Paley functions.