Boundedness of sublinear operators generated by Calderón-Zygmund operators on generalized weighted Morrey spaces

Abstract

In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,?(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (?1, ?2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,?1 (w) to another Mp,?2 (w) for p > 1 and from M1,?1 (w) to the weak space WMp,?2 (w). In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (?1, ?2), which do not assume any assumption on monotonicity of ?1, ?2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner- Riesz operator.