Methods and Applications for Bilinear Operators

本项目的主题属于双线性傅立叶分析领域。约瑟夫·傅立叶在19世纪头几十年的开创性工作之后，人们现在熟悉了将信号或函数分解为其基本频率分量(分析)的过程，以及将单个频率分量叠加形成单个信号的反向过程(合成)。因此，傅里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。沿着这些思路，当两个函数或信号共存时，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成它们的组成频率，将每个分解分为低频和高频，并研究每个分解的低-低、高-低和高-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，形成人力资源，并开发学术课程。该项目的研究活动受到交换子、双线性莱布尼茨规则、副积以及分析和偏微分方程相关主题的研究推动，旨在发展双线性傅立叶分析方法，以促进双线性伪微分算子理论及其应用的发展。特别地，要解决的问题包括在LebesgueBesov和Triebel-Lizorkin空间的尺度上描述符号在某些临界类中的双线性拟微分算子的映射性质。

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.