Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Abstract

Let $X$ be a ball quasi-Banach function space on ${\mathbb{R}}^n$ and $H_X({\mathbb{R}}^n)$ the Hardy space associated with $X$, and let $\alpha \in (0,n)$ and $\beta \in (1,\infty )$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_{\alpha }$ can be extended to a bounded linear operator from $H_X({\mathbb{R}}^n)$ to $H_{X^{\beta }}({\mathbb{R}}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset {\mathbb{R}}^n$, $|B{|}^{\frac{\alpha }{n}}\le C\Vert 1_B{\Vert }_X^{\frac{\beta -1}{\beta }}$, where $X^{\beta }$ denotes the $\beta$-convexification of $X$. Moreover, under some different reasonable assumptions on both $X$ and another ball quasi-Banach function space $Y$, the authors also consider the mapping property of $I_{\alpha }$ from $H_X({\mathbb{R}}^n)$ to $H_Y({\mathbb{R}}^n)$ via using the extrapolation theorem. All these results have a wide range of applications. Particularly, when these are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all these results are new. The proofs of these theorems strongly depend on atomic and molecular characterizations of $H_X({\mathbb{R}}^n)$.

38 pages; Submitted. arXiv admin note: text overlap with arXiv:2108.01559