Fourier Multipliers on Noncommutative Lp Spaces

非交换 Lp 空间上的傅里叶乘子

• 批准号: 1700171
• 负责人: Tao Mei
• 金额: $ 15万
• 依托单位: Baylor University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2024-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700171&HistoricalAwards=false
• 关键词: Fourier; Multipliers; Noncommutative; Lp; Spaces

Mathematicians use "functions" to describe and simulate our real world. A useful method to understand "functions" is to decompose them into frequencies, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. This is called the Fourier transform and is a part of the so-called Fourier analysis method. The study of noncommutative objects offers a new point of view on many topics in mathematics reflecting our daily life and offers possibly the "right" language for quantum mechanics. In real life, the order in which certain operations are executed can make a big difference. For example, first boiling water and adding oil is very different from first boiling oil and then adding water. This is an example of a noncommutative process. Noncommutative Fourier analysis is about functions and their properties in the realm of non-commuting variables. In mathematics, the most important examples are matrix-valued functions. This project is devoted to the Fourier analysis on noncommutative Lp spaces associated with von Neumann algebras, including (non-radial) Fourier multipliers, the Mikhlin-multiplier theory, Dirac Operators, and unconditional sequences of group von Neumann algebras. A typical object is the Hilbert transform on free group von Neumann algebras. Major challenges in the proposed research are the lack of geometric/metric structure and the lack of a commutative product in the abstract setting. The proposed research program will strengthen the existing link between Harmonic Analysis and Functional Analysis. Noncommutative harmonic analysis is motivated by quantum mechanics and prediction theory and will make valuable contributions to these areas and more applied topics such as financial modeling and signal processing.

数学家用“函数”来描述和模拟我们的真实的世界。理解“函数”的一个有用方法是将它们分解为频率，类似于和弦如何被表达为其组成音符的频率（或音高）。这被称为傅里叶变换，是所谓的傅里叶分析方法的一部分。对非对易物体的研究为反映我们日常生活的数学中的许多问题提供了一个新的观点，并可能为量子力学提供“正确”的语言。在真实的生活中，某些操作的执行顺序会产生很大的不同。例如，先煮水再加油与先煮油再加水有很大的不同。这是一个非对易过程的例子。非交换傅立叶分析是关于非交换变量领域中的函数及其性质。在数学中，最重要的例子是矩阵值函数。这个项目致力于与冯诺依曼代数相关的非交换Lp空间的傅立叶分析，包括（非径向）傅立叶乘子，Mikhlin乘子理论，Dirac算子和群冯诺依曼代数的无条件序列。一个典型的对象是自由群冯诺伊曼代数上的希尔伯特变换。在拟议的研究中的主要挑战是缺乏几何/度量结构和缺乏交换产品的抽象设置。拟议的研究计划将加强谐波分析和功能分析之间的现有联系。非对易调和分析是由量子力学和预测理论激发的，将对这些领域以及金融建模和信号处理等更多应用主题做出有价值的贡献。

项目成果

H∞-calculus for semigroup generators on BMO

• DOI: 10.1016/j.aim.2019.02.027
• 发表时间: 2017-01
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: T. Ferguson;T. Mei;Brian Simanek

A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

• DOI: 10.1016/j.aim.2022.108394
• 发表时间: 2022-07
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: T. Mei;Éric Ricard;Quanhua Xu

Free Hilbert transforms

自由希尔伯特变换

• DOI: 10.1215/00127094-2017-0007
• 发表时间: 2017
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Mei, Tao;Ricard, Éric
• 通讯作者: Ricard, Éric

Paley’s Inequality for Discrete Groups

• DOI: 10.1007/s00041-022-09971-1
• 发表时间: 2022-09
• 期刊: Journal of Fourier Analysis and Applications
• 影响因子: 1.2
• 作者: C. Chuah;Yazhou Han;Zhen-Chuan Liu;T. Mei

An operator-valued T1 theory for symmetric CZOs

• DOI: 10.1016/j.jfa.2019.108420
• 发表时间: 2019-07
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: G. Hong;Honghai Liu;T. Mei