Boundedness For The Commutators Of Parametric Marcinkiewicz Integrals

The research effort has focused on two important topics in harmonic analysis-the theories of operators and function spaces. As known as well, the Marcinkiewicz integral is one of the most important operators in it, and more, it plays and key role in PDE. Many papers have been studied the boundedness of Marcinkiewicz integrals and their commutators on different function spaces with suitable conditions of their kernels for a long time. The purpose of this paper is to study the properties of the commutators generated by parametric Marcinkiewicz integrals and BMO functions on Morrey-Herz spaces and of the integrals themselves on weighted Companator spaces.Our paper is organized as follows.In the first chapter, we introduce some necessary definitions, notations and the present research conditions, et. al.In Chapter 2, the boundedness of the commutators generated by parametric Marcinkiewicz integrals and BMO functions is obtained on Morrey-Herz spaces. That is, when Ω∈Lr(Rn)(0<r<∞), associating the decomposition tech of Morrey-Herz spaces with the explicit estimates of kernels, we prove that the parametric Marcinkiewicz integral commutators μΩ,bρ are bounded from MKp,qα,λ(Rn) to MKp,qα,λ(Rn) with suitable indices α,λ, p and q.In Chapter 3 generalizes the results in Chapter 2 to the case of higher-order parametric Marcinkiewicz integral commutators on Morrey-Herz spaces.In the last chapter, we discuss the estimates of parametric Marcinkiewicz integrals in weighted Campanator spaces. In fact, when Ω∈Lr(Rn)(0<r<∞), we show the following inequality, ||μΩρ(f)||α,ρ,ω≤C||f||∝,c,≧,where||·||α,p,ω denotes the norm of weighted Campanato spaces.