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

The authors study the boundedness for a large class of sublinear operator T generated by Calderon-Zygmund operator on generalized Morrey spaces M-p,M-phi 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 is an element of BMO (R-n), 1 < p < infinity and T-a is a sublinear operator, we find the sufficient conditions on the pair phi(1), phi(2) which ensures the boundedness of the operator Ta from one generalized Morrey space M-p,M-phi 1 to another M-p,M-phi 2. In all cases, the conditions for the boundedness of T-a are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)), which do not assume any assumption on monotonicity of phi(1), phi(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.