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

Let X be a ball quasi-Banach function space on Rⁿ and HX (Rⁿ) the Hardy space associated with X, and let (0, n) and (1, ). 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_ can be extended to a bounded linear operator from HX (Rⁿ) to Hₗ^ (Rⁿ) if and only if there exists a positive constant C such that, for any ball B Rⁿ, |B|^{n} C \|1B\|X^-1{}, where X^ denotes the -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_ from HX (Rⁿ) to HY (Rⁿ) via using the extrapolation theorem. All these results have a wide range of applications.