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

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 $X$ 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 $W{H}_X({\mathbb{R}}^n)$ 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 $W{H}_{X^{1/(1-\theta )}}({\mathbb{R}}^n)$, 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 $W{H}_X({\mathbb{R}}^n)$ obtained in this article improves the existing results via weakening the assumptions on the Littlewood--Paley functions.

60 pages; Submitted

References in corpus (1)

• The Fatou property of block spaces