Estimates For Commutators Of Parametric Marcinkiewicz Integral With Non-Doubling Measures

The important research objects of Harmonic Analysis are function spaces and some operators. Marcinkiewicz integrals are regarded as one of the classical operators in Harmonic Analysis. In recent years, there are many results about the Marcinkiewicz integrals, parametric Marcinkiewicz integral and commutators on n dimension Euclidean spaces Rn. Furthermore, many mathematicians have obtained the bondedness of Marcinkiewicz integrals and commutators with non-doubling measures. On the basis of the previous results, this paper mainly discusses the boundedness of commutators of the parametric Marcinkiewicz integral with non-doubling measures. Our results will enrich the theory of Marcinkiewicz integral commutators. There are four chapters in this paper:In chapter 1, we will introduce about the background, current research situation and the frame of the paper.In chapter 2, under the condition of non-doubling measures, we will discuss the boundedness of the commutator Mρbgenerated by the parametric Marcinkiewicz integral Mρwith Lipschitz function b in Lebesgue spaces and RBMO(μ) spaces.In chapter 3, we will prove the the boundedness of the commutator Mρbgenerated by the parametric Marcinkiewicz integral Mρwith Lipschitz function b from the Morrey spaces to the Morrey spaces, and from the Morrey spaces to the Lipschitz spaces with non-doubling measures.In chapter 4, under the condition of non-doubling measures, we will establish the boundedness of the commutator Mρbgenerated by the parametric Marcinkiewicz integral Mρwith Lipschitz function b in Hardy spaces.