Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

In this paper the authors study the boundedness for a large class of sublinear operators T-alpha, alpha is an element of [0, n) generated by Calderon-Zygmund operators (alpha = 0) and generated by Riesz potential operator (alpha > 0) on generalized Morrey spaces M-p,M-phi As an application of the above result, the boundeness of the commutator of sublinear operators T-b,T-alpha,T- alpha.is an element of [0, n) on generalized Morrey spaces is also obtained. In the case b is an element of BMO and T-b,T-alpha is a sublinear operator, we find the sufficient conditions on the pair (phi(1),phi(2)) which ensures the boundedness of the operators T-b,T- alpha, alpha is an element of [0, n) from one generalized Morrey space M-p,M-phi 1 to another M-q,M-phi 2 with 1/p - 1/q = alpha/n. In all the cases the conditions for the boundedness 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, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.