Boundedness of fractional integrals on ball Campanato-type function spaces

Let X be a ball quasi-Banach function space on R-n satisfying some mild assumptions and let alpha is an element of (0 , n) and beta is an element of (1 , infinity). In this article, when alpha E (0 , 1), the authors first find a reasonable version (I) over tilde (alpha) of the fractional integral I alpha on the ball Campanato-type function space LX,q,s,d(R-n) with q is an element of [1 , oo), s E Zn+ , and d E (0 , oo). Then the authors prove that (I) over tilde (alpha) is bounded from L-X,L-q,L-s,L-d(R-n) to LX,q,s,d(Rn) if and only if there exists a positive constant C such that, for any ball B subset of R-n , |B|(alpha/n) <= C parallel to 1(B)parallel to beta-1/X (beta,) where X-beta denotes the beta-convexification of X. Furthermore, the authors extend the range alpha E (0 , 1) in (I) over tilde (alpha) to the range alpha E (0 , n) and also obtain the corresponding boundedness in this case. Moreover, (I) over tilde (alpha) is proved to be the adjoint operator of I alpha. 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). (C) 2022 Elsevier Masson SAS. All rights reserved.