算子及其交换子在函数空间上有界性的研究是调和分析中一个重要的研究课题之一. Hausdorff算子与Hilbert变换, Riesz变换有着密切关系, 其不仅包括Hardy算子, Riemann-Liouville分数次积分算子以及 Ces$\grave{a}$ro算子等经典算子, 而且在函数论, 复分析, 几何分析, 偏微分方程等领域有广泛应用. 对Hausdorff算子进行研究不仅具有理论意义, 同时具有深远的应用价值. 另外, 交换子理论对研究偏微分方程以及刻画函数空间起到关键性作用. 随着线性以及多线性积分算子的发展, 关于其交换子的研究得到广泛关注. 因此, 我们将对多线性Hausdorff算子及其交换子进行研究.

本论文的主要结果分为三部分：首先, 我们研究了stratified群上的多线性Hausdorff算子, 得到了其在双权中心Morrey空间, 双权Herz空间以及双权Morrey-Herz空间上的有界性. 其次, 我们研究了stratified群上多线性Hausdorff算子与加权BMO函数生成的交换子, 得到了其在双权中心Morrey空间, 双权Morrey-Herz 空间上的有界性. 最后, 我们证明了多线性Hausdorff算子与Lipschitz函数生成的交换子在加权变指标Herz空间以及加权变指标Morrey-Herz空间上的有界性.

 本论文具体内容安排如下.

 第一章, 我们介绍了Hausdorff算子的历史背景以及stratified群的定义与基本性质, 给出了双权Morrey空间, Herz空间, Morrey-Herz空间的定义, 并得到了stratified 群上的多线性Hausdorff算子在双权中心Morrey空间, 双权Herz空间以及双权Morrey-Herz空间上有界性的充分必要条件.

 第二章, 我们介绍了Hausdorff算子交换子的历史背景, 给出了stratified群上的多线性Hausdorff算子交换子的定义, 并得到了stratified群上多线性Hausdorff算子与加权BMO函数生成的交换子在双权中心Morrey空间, 双权Morrey-Herz空间上的有界性.

 第三章, 我们介绍了变指标函数空间的历史背景, 给出了加权变指标Herz空间, 加权变指标Morrey-Herz空间的定义, 并证明了多线性Hausdorff算子与Lipschitz函数生成的交换子在加权变指标Herz空间以及加权变指标Morrey-Herz空间上的有界性.

The study of the boundedness of operators and their commutators in function spaces is an important research topic in harmonic analysis. Hausdorff operators are closely related to Hilbert transform and Riesz transform. They not only include Hardy operators, Riemann-Liouville fractional integral operators and Ces$\grave{a}$ro operators, but also have important applications in the fields of function theory, complex analysis, geometric analysis, partial differential equations. The study of Hausdorff operators not only has theoretical significance, but also has application value. In addition, the theory of commutators plays a key role in the study of partial differential equations and the characterization of function spaces. With the development of linear and multilinear integral operators, the study of their commutators has received extensive attention. Therefore, we will study the multilinear Hausdorff operators and their commutators.

 The main results of this paper are divided into three parts. Firstly, we study the multilinear Hausdorff operators on stratified groups and obtain the boundedness of multilinear Hausdorff operators on two-weighted central Morrey spaces, two-weighted Herz spaces and two-weighted Morrey-Herz spaces. Secondly, we study the commutators generated by multilinear Hausdorff operators and weighted BMO functions on stratified groups, and obtain the boundedness of commutators on two-weighted central Morrey spaces and two-weighted Morrey-Herz spaces. Thirdly, we prove the boundedness of commutators generated by multilinear Hausdorff operators and Lipschitz functions on weighted Herz spaces and Morrey-Herz spaces with variable exponent.

 This paper is organized as follows.

 In Chapter 1, we firstly introduce the historical background of the Hausdorff operators and some basic properties of stratified groups, then give the definitions of the two-weighted Morrey, Herz and Morrey-Herz spaces, and then obtain some sufficient and necessary conditions for the boundedness of multilinear Hausdorff operators on stratified groups on two-weighted central Morrey spaces, two-weighted Herz spaces and two-weighted Morrey-Herz spaces.

 In Chapter 2, we introduce the historical background of the commutators of Hausdorff operators and give the definitions of the commutators of multilinear Hausdorff operators on stratified groups, and then obtain the boundedness of commutators generated by multilinear Hausdorff operators and weighted BMO functions on stratified groups on two-weighted central Morrey spaces and two-weighted Morrey-Herz spaces.

 In Chapter 3, we introduce the historical background of variable exponent function spaces and give the definitions of the weighted Herz and Morrey-Herz spaces with variable exponent, then prove the boundedness of commutators generated by multilinear Hausdorff operators and Lipschitz functions on weighted Herz spaces and Morrey-Herz spaces with variable exponent.