Littlewood-Paley Characterizations of Hardy-type Spaces Associated with Ball Quasi-Banach Function Spaces

Abstract

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 $X$ and is bounded on the associated space, the authors establish various Littlewood--Paley function characterizations of the Hardy space $H_X({\mathbb R}^n)$ associated with $X$, under some weak assumptions on the Littlewood--Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with $X$. All these results have wide applications. Particularly, when $X:=M_r^p({\mathbb R}^n)$ (the Morrey space), $X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space), $X:=L^{p(\cdot)}({\mathbb R}^n)$ (the variable Lebesgue space), $X:=L_\omega^p({\mathbb R}^n)$ (the weighted Lebesgue space) and $X:=(E_\Phi^r)_t({\mathbb R}^n)$ (the Orlicz-slice space), the Littlewood--Paley function characterizations of $H_X({\mathbb R}^n)$ obtained in this article improve the existing results via weakening the assumptions on the Littlewood--Paley functions and widening the range of $\lambda$ in the Littlewood--Paley $g_\lambda^*$-function characterization of $H_X(\mathbb R^n)$.