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

Abstract

We consider generalized Morrey spaces M p(·),ω(Ω) with variable exponent p(x) and a general function ω(x,r) defining the Morrey-type norm. In case of bounded sets Ω ⊂ 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(·),ω(Ω) → Mq(·),ω (Ω)-theorem for the potential operators Iα(·), also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(x, r), which do not assume any assumption on monotonicity of ω(x, r) in r.