Generalized fractional integrals in the vanishing generalized weighted local and global Morrey spaces

Abstract

In this paper, we prove the boundedness of generalized fractional integral operators I? in the vanishing generalized weighted Morrey-type spaces, such as vanishing generalized weighted local Morrey spaces and vanishing generalized weighted global Morrey spaces by using weighted Lp estimates over balls. In more detail, we obtain the Spanne-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted local Morrey spaces with wq ? A1+ q/p' for 1 < p < q < ?, and from the vanishing generalized weighted local Morrey spaces to the vanishing generalized weighted weak local Morrey spaces with w A1,q for p = 1, 1 < q < ?. We also prove the Adams-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted global Morrey spaces with w Ap,q for 1 < p < q < ? and from the vanishing generalized weighted global Morrey spaces to the vanishing generalized weighted weak global Morrey spaces with w A1,q for p = 1, 1 < q < ?. The our all weight functions belong to Muckenhoupt-Weeden classes Ap,q.