Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

We consider generalized Morrey spaces Mp,ω R with a general function ω x, r defining the Morrey-type norm. We find the conditions on the pair ω1, ω2 which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space Mp,ω1 R to another Mp,ω2 R , 1 < p < ∞, and from the space M1,ω1 R to the weak space WM1,ω2 R . We also prove a Sobolev-Adams type Mp,ω1 R → Mq,ω2 R -theorem for the potential operators Iα. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω1, ω2 , which do not assume any assumption on monotonicity of ω1, ω2 in r. As applications, we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.