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 α∈(0,n) and β∈(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 Iα can be extended to a bounded linear operator from HX( Rⁿ) to H_Xβ( Rⁿ) if and only if there exists a positive constant C such that, for any ball B⊂ Rⁿ, |B|α/n≤ C \|1B\|Xβ-1/β, where Xβ denotes the β-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 Iα from HX( Rⁿ) to HY( Rⁿ) via using the extrapolation theorem. All these results have a wide range of applications.