调和分析在非交换遍历论中的应用

Motivated by the development of quantum physics, noncommutative mathematics have become one of the most important research fields among modern mathematics. However, as an important constituent part of noncommutative analysis, noncommutative ergodic theory has not been well developed. Actually, the modern period of the development of noncommutative ergodic theory began with the work by Junge and Xu appearing in J. Amer. Math. Soc, where the authors established the noncommutative analogues of the Dunford-Schwartz and the Stein ergodic theorems. After that, as far as we know, there appeared only one important advance. That is, the noncommutative Nevo-Stein ergodic maximal inequality---the ergodic theorem from the action of free group, which was established independently by Anantharaman and Hu...The purpose of the present proposal is to systematically study noncommutative ergodic theory along the history of the development of classical ergodic theory. This proposal includs two main research themes as follows:..The first theme is the noncommutative ergodic theorems from the group action consisting of five problems: 1) The noncommutative Wiener and Jones ergodic theorems from the action of Euclidean group; 2) The noncommutative Nevo-Thangavelu ergodic theorem from the action of Heisenberg group; 3) The noncommutative Nevo ergodic theorems from the action of simple Lie groups; 4) The noncommutative Margulis-Nevo-Stein ergodic theorems from the action of semi-simple Lie groups; 5) The noncommutative Lindenstrauss ergodic theorem from the action of amenable groups...The second theme is the noncommutative analogues of Bourgain's return time theorem and related theory consisting of two problems: 1) The noncommutative Birkhoff ergodic theorem along subsequences of natural numbers; 2) The noncommutative analogue of Bourgain's return time theorem.

随着量子物理的发展，数学在非交换方向的发展已成为现代数学的重要研究领域。作为非交换分析领域的一个重要研究方向，非交换遍历论的研究还不广泛。 本项目的研究目标是沿着经典遍历论的发展历史全面展开对非交换遍历论的研究。本项目包括两个主要研究方向:. 方向一，群作用下的非交换遍历定理。具体包括以下五个问题：1）欧氏群作用下的非交换Wiener和Jones遍历定理；2）海森堡群作用下的非交换Nevo-Thangavelu遍历定理；3）单李群作用下的非交换Nevo遍历定理；4）半单李群作用下的非交换Margulis-Nevo-Stein遍历定理；5）顺从群作用下的非交换Lindenstrauss遍历定理。方向二，Bourgain返时定理及其相关理论的非交换类比。具体包括以下两个问题：1）沿自然数集中多项式子序列的非交换Birkhoff遍历定理；2）Bourgain返时遍历定理的非交换类比。

随着量子物理的发展，数学在非交换方向的发展已成为现代数学的重要研究领域。作为非交换分析领域的一个重要研究方向，非交换遍历论的研究还不广泛。事实上，非交换极大遍历定理的现代研究开始于Junge和Xu在2007年发表于J.Amer.Math.Soc上非交换Dunford-Schwartz和Stein极大遍历定理。之后仅有的一个重大进展是Anantharaman和Hu分别独立建立的非交换Nevo-Stein极大遍历不等式，即自由群作用下的遍历定理。.本项目的研究目标是沿着经典遍历论的发展历史全面展开对非交换遍历论的研究。已取得研究成果意义如下。一、建立了欧氏群作用下的非交换Wiener 和Jones 遍历定理及海森堡群作用下的非交换Nevo-Thangavelu 遍历定理。这俩结果被认为是群作用下的非交换遍历理论的开端，将极大启发人们进一步研究非交换遍历理论。也就是在这个基础上，主持人彻底解决了多项式增长群作用下的长达十多年的球平均公开问题。部分成果已发表于期刊Erg. Theory & Dyn. System。二、证明了非交换多参数Wiener-Wintner型遍历定理。Wiener-Wintner遍历定理可以看成遍历理论中的另一个重要研究方向，是非交换Bourgain返时定理及其相关理论进一步发展的基础，与非交换分析中另一公开问题非交换Carleson极大不等式密切联系。这一结果已发表于J. Funct. Anal，也将启发人们研究一般群作用下的Wiener-Wintner型遍历定理。三、紧密联系于本项目的研究课题，建立了非交换鞅非对称极大不等式、经典调和分析中的变差不等式与Calderon交换子等，为本项目课题研究的进一步开展提供了很多想法与工具。部分结果已发表于期刊J. Funct. Anal，Math. Z及Anal & PDE等。