Boundedness Of Matrix Dilation And Hausdorff Operators On Some Function Spaces

In this thesis,we study the asymptotic estimates for the norms of the matrix dilation operators on the function spaces and the boundedness of Hausdorff operators on the function spaces.The function spaces which we consider are modulation and Wiener amalgam spaces.The thesis is divided into four chapters:In Chapter 1,we introduce the research background of Hausdorff and matrix dilation operators,and give the main results of this thesis.In Chapter 2,we collect the definitions and basic properties of modulation spaces,and then the asymptotic estimates are established for the norms of the matrix dilation operators on modulation spaces.Finally,the sharp conditions for boundedness of Hausdorff operators on modulation spaces are given.In Chapter 3,we list the definitions and basic properties of Wiener amalgam spaces,and show the asymptotic estimates for the norms of the matrix dilation operators on Wiener amalgam spaces.Finally,the sharp conditions are established for boundedness of Hausdorff operators on Wiener amalgam spaces.As applications,we obtain some boundedness and unboundedness properties for Hardy operators and adjoint Hardy operators on Wiener amalgam spaces.In Chapter 4,the research work of this thesis is summarized,and the future work is prospected.