Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 503948, 20 pages
Research Article

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

1Department of Mathematics, Ahi Evran University, Kirsehir, Turkey
2Institute of Mathematics and Mechanics, Baku, Azerbaijan

Received 12 July 2009; Accepted 22 October 2009

Academic Editor: Shusen Ding

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 p,ω(n) 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 p,ω1(n) to another p,ω2(n), 1<p<, and from the space 1,ω1(n) to the weak space W1,ω2(n). We also prove a Sobolev-Adams type p,ω1(n)q,ω2(n)-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.