Abstract and Applied Analysis
Volume 2011 (2011), Article ID 356041, 18 pages
http://dx.doi.org/10.1155/2011/356041
Research Article

Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces

Vagif S. Guliyev,1,2 Seymur S. Aliyev,2 and Turhan Karaman1

1Department of Mathematics, Ahi Evran University, 40100 Kirsehir, Turkey
2Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku 1141, Azerbaijan

Received 10 April 2011; Revised 23 April 2011; Accepted 18 May 2011

Academic Editor: Irena Lasiecka

Copyright © 2011 Vagif S. Guliyev et al. 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.

Abstract

The authors study the boundedness for a large class of sublinear operator 𝑇 generated by Calderón-Zygmund operator on generalized Morrey spaces 𝑀 𝑝 , 𝜑 . As an application of this result, the boundedness of the commutator of sublinear operators 𝑇 𝑎 on generalized Morrey spaces is obtained. In the case 𝑎 ∈ B M O ( ℝ 𝑛 ) , 1 < 𝑝 < ∞ and 𝑇 𝑎 is a sublinear operator, we find the sufficient conditions on the pair ( 𝜑 1 , 𝜑 2 ) which ensures the boundedness of the operator 𝑇 𝑎 from one generalized Morrey space 𝑀 𝑝 , 𝜑 1 to another 𝑀 𝑝 , 𝜑 2 . In all cases, the conditions for the boundedness of 𝑇 𝑎 are given in terms of Zygmund-type integral inequalities on ( 𝜑 1 , 𝜑 2 ), which do not assume any assumption on monotonicity of 𝜑 1 , 𝜑 2 in 𝑟 . Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.