Boundedness Of Toeplitz Type Operators Related To Two Classes Of Integral Operators

The boundedness of operators in function space is one of the core problems in harmonic analysis.Toeplitz type operators,as an important generalization of commutators,have important applications in complex analysis and other fields and have attracted the attention of many scholars.θ-type Calderón-Zygmund operators and strongly singular Calderon-Zygmund operators are the generalizations of CalderónZygmund operators,which are closely related to the theory of pseudo-differential operators.This thesis mainly discusses the boundedness of Toeplitz type operators related to these two classes of operators and the Lipschitz or Campanato function.When the function belongs to Campanato space Cp,β（Rn),there is relatively little research on the corresponding Toeplitz type operator,especially in the case of-n/p≤β<0,which has not been studied by scholars.And the main content of this thesis is the study of this range of indicators.The results obtained in this thesis are the extension and improvement of classical results.The thesis is divided into four chapters.In Chapter 1,we mainly introduce the research background and progress ofθ-type Calderón-Zygmund operators and strongly singular Calderón-Zygmund operators as well as their related commutators and Toeplitz type operators and put forward problems to be studied in this thesis.In Chapter 2,we mainly study the boundedness of Toeplitz type operators Tb related to θ-type Calderon-Zygmund operators and obtain two results.One is that when b belongs to the Lipschitz function class,it is proved that Tb is bounded from Lebesgue space to Campanato space.Second,when b belongs to the Campanato function class,the boundedness of Tb from Morrey space to Campanato space is obtained.In Chapter 3,the boundedness of Toeplitz type operators related to the strongly singular Calderón-Zygmund operator is discussed.Tb is used to represent Toeplitz type operators related to strongly singular Calderon-Zygmund operators.In this chapter,we prove the following two results.First,when b is the Lipschitz function,we prove that Tb is bounded from Lebesgue space to Campanato space.Second,when b is the Campanato function,the boundedness of Tb from Morrey space to Campanato space is obtained.In Chapter 4,we summarize the work done in this thesis.