Boundedness of fractional integrals on ball Campanato-type function spaces

Abstract

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