Copyright © 2009 Vagif S. Guliyev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 terms of 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.