Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

Abstract

We consider generalized Morrey spaces M p(·),?(?) with variable exponent p(x) and a general function ?(x,r) defining the Morrey-type norm. In case of bounded sets ? ? Rn we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type M p(·),?(?) › Mq(·),? (?)-theorem for the potential operators I?(·), also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ?(x, r), which do not assume any assumption on monotonicity of ?(x, r) in r.