Methods and Applications for Bilinear Operators

双线性算子的方法和应用

• 批准号: 1500381
• 负责人: Virginia Naibo
• 金额: $ 17.49万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500381&HistoricalAwards=false
• 关键词: Methods; Applications; Bilinear; Operators

The subject matter of this project belongs to the realm of bilinear Fourier analysis. After the pioneering work of Joseph Fourier in the first decades of the nineteenth century, one is now familiar with the process of decomposing a signal or function into its elementary frequency components (analysis) as well as with the reverse process of superposing individual frequency components to form a single signal (synthesis). Fourier analysis thus acts in a way similar to a prism, which allows one to see the individual color components of a beam of light. Along these lines, when two functions or signals coexist, their frequency components interact and this phenomenon plays a key role in the study of certain partial differential equations that arise, for instance, in optics, quantum mechanics, and fluid dynamics. In the field of bilinear Fourier analysis, tools are developed to model the behavior and interaction of two signals by decomposing each one into their constituent frequencies, separating each decomposition into low and high frequencies, and studying the interplay between the low-low, high-low, and high-high frequencies from each decomposition. This project will also contribute to the integration of research and education at the postdoctoral, graduate, and undergraduate levels, to advancing discovery, to forming human resources, and to developing academic curricula.Motivated by the study of commutators, bilinear Leibniz-type rules, paraproducts, and related topics in analysis and partial differential equations, the research activities of this project aim at developing methods in bilinear Fourier analysis to advance the theory of bilinear pseudo-differential operators and their applications. In particular, problems to be addressed include the description of the mapping properties, in the scales of Lebesgue, Besov, and Triebel-Lizorkin spaces, of bilinear pseudodifferential operators with symbols in certain critical classes.

本课题属于双线性傅立叶分析领域。在约瑟夫·傅立叶在19世纪最初几十年的开创性工作之后，人们现在熟悉将信号或函数分解成其基本频率分量的过程（分析）以及叠加各个频率分量以形成单个信号的逆过程（合成）。傅立叶分析的作用类似于棱镜，它允许人们看到光束的各个颜色分量。沿着这些路线，当两个功能或信号共存时，它们的频率分量相互作用，这种现象在研究某些偏微分方程中起着关键作用，例如，在光学，量子力学和流体动力学中。在双线性傅立叶分析领域，开发了一些工具来模拟两个信号的行为和相互作用，方法是将每个信号分解为它们的组成频率，将每个分解分为低频和高频，并研究每个分解的低频-低频、高频-低频和高频-高频之间的相互作用。该项目还将有助于博士后、研究生和本科生水平的研究和教育的整合，促进发现，培养人力资源，并开发学术课程。受双线性莱布尼兹型规则、副积以及分析和偏微分方程相关主题研究的启发，本计画的研究活动旨在发展双线性傅立叶分析的方法，以增进双线性伪微分算子的理论及其应用。特别是，要解决的问题包括描述具有某些临界类符号的双线性伪微分算子在Lebesgue、Besov和Triebel-Lizorkin空间尺度上的映射性质。