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

Abstract

Abstract We consider generalized Morrey spaces with a general function defining the Morrey-type norm. We find the conditions on the pair which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space to another , , and from the space to the weak space . We also prove a Sobolev-Adams type -theorem for the potential operators . In all the cases the conditions for the boundedness are given it termsof Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in . 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.