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.

该项目旨在开发一个耐力/BMO空间和傅立叶乘数的抽象理论，尤其是在非共同框架内。运算符的半群将被用来弥补缺乏几何/度量结构，并且在抽象环境中缺乏交换产物。 从这项工作中获得的新见解也应反映出非交通性几何形状，操作员空间理论和经典谐波分析。对非交流对象的研究为反映日常生活的许多数学主题提供了一种新的观点，并且可以说是量子力学的“正确”语言。 在现实生活中，执行某些操作的顺序可以产生很大的不同。例如，首先沸水，然后添加油与首先沸腾的油大不相同，然后加水。非共同分析是关于函数及其在非公共变量领域中的特性。最重要的示例是矩阵值函数。傅立叶分析或谐波分析的理论对函数有很多要说的要说的，并且正如该项目打算显示的矩阵值函数。该项目还将为控制理论，信号和图像处理做出有价值的贡献，在矩阵值下是自然发生的情况。