Boundedness Of Two Kinds Of Operators

It is well known that the estimates for operators have been popular content in harmonic analysis and played an important role in resolving differential equation and practical problems.In this thesis, the author firstly estimate the rough singular integral operators of R.Fefferman defined by where we denote h(x) a bounded radial function andΩis homogeneous of degree zero with mean value zero on the unit sphere Sn-1.The study of rough singular integral operators has a long history and the bounded-ness of those has been established in many spaces, such as LP spaces, Herz spaces and Hardy spaces. With the development of Herz-type spaces, this thesis aims to extend the boundedness of rough singular integral operators to Herz-type Triebel-Lizorkin spaces and Herz-type Besov spaces.In the following part of this thesis, another important operator-Multivariate Hausdorff operator has been estimated, which contains many classical operators as its special cases, such as Cesaro operator and Copson operator. Therefore, Hausdorff operator has more extensive application and the study of it is very meaningful. In this thesis, two weighted estimates for Multivariate Hausdorff operator on LP space have been given.The thesis consists of two chapters.Chapter 1 prove the boundedness of rough singular integral operators on Herz-type Triebel-Lizorkin space and Herz-type Besov space.Chapter 2 prove the boundedness of Multivariate Hausdorff operator on weighted LP space.