Boundedness of the parametric Marcinkiewicz integral operator and its commutators on generalized Morrey spaces

In this paper we study the boundedness of the parametric Marcinkiewicz operator mu(rho)(Omega) on generalized Morrey spaces M-p,M-phi. We find the sufficient conditions on the pair (phi(1), phi(2)) which ensure the boundedness of the operators mu(rho)(Omega) from one generalized Morrey space A to another M-p,M-phi 2, 1 < p < infinity, and from the space M-1,M-phi 1 to the weak space WM1,phi 2. As an application of the above result, the boundedness of the commutator of Marcinkiewicz operators [a, mu(rho)(Omega)] on generalized Morrey spaces is also obtained. In the case a is an element of BMO(R-n), we find the sufficient conditions on the pair (phi(1), phi(2)) which ensure the boundedness of the operators [a, mu(rho)(Omega)] from one generalized Morrey space M-p,M-phi 1 to another M-p,M-phi 2 1 < p < infinity, and from the space M-1,M-phi 1 to the weak space WM1,phi 2. In all the cases the conditions for boundedness are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)) which do not require any assumption on the monotonicity of phi(1), phi(2).