The Boundedness Of Fractional Type Marcinkiewicz Operator On Function Spaces

In this thesis,we systematically studies the weighted endpoint estimates of fractional Marcinkiewicz operator and its commutator generated by BMO function on Lebesgue space,the weighted boundedness on Morrey space and the weighted boundedness of singular integral operator on grand Morrey space.This thesis is divided into four chapters.Chapter 1:Introduce the theoretical background,research significance and research status at home and abroad of singular integral operator,Marcinkiewicz operator and fractional integral operator.Chapter 2:When kernel function Ω(x)belongs to Lipβ(Sn-1),we studied the weighted endpoint estimates of commutator generated by fractional type Marcinkiewicz operator and BMO function on Lebesgue space.Chapter 3:When kernel function Ω(x)belongs to Lipβ(Sn-1),we studied the boundedness of fractional type Marcinkiewicz operator and its commutators generated by BMO function on weighted Morrey space and weak type estimates.Chapter 4:In the setting of homogeneous spaces define a class of grand Morrey space.In this chapter,we studied the weighted boundedness of singular integral commutator on grand Morrey space,summarize the theoretical research results of this paper,and make a prospect.