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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--Steinvector-valued maximal inequality on $X$ and is bounded on the associate spaceof $X$, the authors prove that the fractional integral $I_{\alpha}$ can beextended 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 $B\subset \mathbb{R}^n$,$|B|^{\frac{\alpha}{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{\beta-1}{\beta}}$,where $X^{\beta}$ denotes the $\beta$-convexification of $X$. Moreover, undersome different reasonable assumptions on both $X$ and another ball quasi-Banachfunction 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 theextrapolation 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 Herzspaces, all these results are new. The proofs of these theorems strongly dependon atomic and molecular characterizations of $H_X({\mathbb R}^n)$.

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