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

We consider generalized Morrey spaces Mp(·),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 Mp(·),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.