Boundedness of fractional integrals on ball Campanato-type function spaces

Let X be a ball quasi-Banach function space on Rn satisfying some mild assumptions and let α∈(0,n) β∈(1,∞). In this article, when α∈(0,1), the authors first find reasonable version I˜α of fractional integral Iα Campanato-type LX,q,s,d(Rn) with q∈[1,∞), s∈Z+n, d∈(0,∞). Then prove that is bounded from LXβ,q,s,d(Rn) to if only there exists positive constant C such that, for any B⊂Rn,|B|αn≤C‖1B‖Xβ−1β, where Xβ denotes β-convexification X. Furthermore, extend range α∈(0,1) in also obtain corresponding boundedness case. Moreover, proved adjoint operator Iα. All these results have wide applications. Particularly, even they are applied, respectively, mixed-norm Lebesgue spaces, Morrey local generalized Herz all obtained new. The proofs strongly depend dual theorem special atomic decomposition molecules HX(Rn) (the Hardy-type associated X) which proves predual LX,q,s,d(Rn).