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

WOS: 000307090100001In 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).Science Development Foundation under the President of the Republic of AzerbaijanScience Development Foundation (SDF) - Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [110T695]; Ahi Evran University Scientific Research ProjectsAhi Evran University [BAP FBA-11-13]The research of the second author was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1 and by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695) and by grant of 2011-Ahi Evran University Scientific Research Projects (BAP FBA-11-13)