Some Inequalities For Marcinkiewicz Integrals

Marcinkiewicz integral is an important operator in the theory of singular integral operators,and it is also an important operator in the application of differential equations.Many scholars have studied the properties of marcinkiewicz operator in many function spaces.In this paper,we mainly discuss variational inequalities for marcinkiewicz integral operators with odd kernel and norm inequalities for marcinkiewicz integral operators with non-smooth kernel.Specifically,this paper is mainly divided into the following three parts:In chapter 1,we mainly introduces the relevant background knowledge,research status at home and abroad and some relevant definitions.In chapter 2,we use the idea from special to general to prove the variational inequality of marcinkiewicz integral with odd kernel.Firstly,the variational inequality of marcinkiewicz integral operator is proved by using the classical calderón-zygmund decomposition and rotation of function,and it is proved in the case of?=1,then the conclusion is deduced to the case with odd kernel,that is??L^?,and?(at)=?(t).In chapter 3,by using the sparse control theorem of sublinear operators,we prove the norm inequality of marcinkiewicz integral operator with non-smooth kernel,and obtain the marcinkiewicz integral operator?and its local operator M_?with non-smooth kernel are bounded from L¹ to L^1,?.The application of the norm inequality of marcinkiewicz integral operator of non-smooth kernel in different spaces is given.