Hardy/BMO spaces and Fourier Multipliers in the Noncommutative Setting

非交换环境中的 Hardy/BMO 空间和傅里叶乘子

• 批准号: 1266042
• 负责人: Tao Mei
• 金额: $ 13.4万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266042&HistoricalAwards=false
• 关键词: Hardy; BMO; spaces; Fourier; Multipliers

The project aims to develop an abstract theory of Hardy/BMO spaces and Fourier multipliers, particularly within a noncommutative framework. Semigroups of operators will be used crucially to make up for the lack of geometric/metric structure and the lack of a commutative product in the abstract setting. The new insights gained from this work should also reflect back on noncommutative geometry, operator space theory, and classical harmonic analysis. The study of noncommutative objects offers a new point of view on many topics in mathematics that reflect daily life, and it also arguably represents 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 afterwards adding oil is very different from first boiling oil and then adding water. Noncommutative analysis is about functions and their properties in the realm of noncommuting variables. The most important example is matrix-valued functions. The theory of Fourier analysis or harmonic analysis has a lot to say about functions, and as this project intends to show, also about matrix-valued functions. The project will also make valuable contributions to control theory and signal and image processing, where matrix-valued functions are natural occurences.

该项目旨在开发 Hardy/BMO 空间和傅里叶乘数的抽象理论，特别是在非交换框架内。算子半群将被至关重要地用来弥补几何/度量结构的缺乏以及抽象环境中交换积的缺乏。 从这项工作中获得的新见解也应该反思非交换几何、算子空间理论和经典调和分析。对非交换对象的研究为反映日常生活的许多数学主题提供了新的观点，并且它也可以说代表了量子力学的“正确”语言。 在现实生活中，某些操作的执行顺序可能会产生很大的差异。例如，先烧水后加油与先烧油后加水有很大不同。非交换分析是关于非交换变量领域的函数及其属性。最重要的例子是矩阵值函数。傅立叶分析或调和分析理论对函数有很多论述，正如本项目打算展示的那样，也有关于矩阵值函数的论述。该项目还将对控制理论以及信号和图像处理做出有价值的贡献，其中矩阵值函数是自然发生的。