CAREER: Time-Frequency Analysis of Multilinear Operators and More General Nonlinear Operators

研究摘要多线性算子及更一般非线性算子的时频分析本文主要研究多线性奇异积分。在过去的五十年中，奇异积分在调和分析中发挥了重要的作用，并且在其他数学学科中也非常有用，例如偏微分方程。该学科的主要工具是Littlewood Paley理论，其主要信息是欧几里得空间上的函数应该被分解成频率内容位于原点周围的同心壳中的块。在讨论多线性奇异积分时，我们遇到了这些多线性奇异积分在频率空间上具有平移对称性的新现象。因此，仅仅研究原点周围的同心壳层是不够的，我们必须研究频率空间中任何一点周围的壳层。最近已经开发了一些技术来控制壳在频率空间中不同点周围的相互作用，本项目的目的是利用和进一步发展这些技术，从而更好地理解多线性奇异积分。例如，多线性奇异积分出现在非线性算子的正式泰勒级数中，或作为迭代格式中的显式校正项。该项目的一个特别重点将是研究该理论在偏微分方程和其他数学领域的可能应用。

Research AbstractTime-Frequency Analysis of Multilinear Operatorsand More General Nonlinear OperatorsThis proposal focusses on the study of multilinear singularintegrals. Singular integrals have played an important rolein harmonic analysis over the past fifty years and are veryuseful in other mathematical disciplines such as for example partial differential equations. The main tool in the subject is Littlewood Paley theory, whose main message is that functions on Euclidean space should be decomposed into pieces whose frequency content lie in concentric shells around the origin. Passing to multilinear singular integrals one encounters the new phenomenon that these multilinear singular integrals have translation symmetries in frequency space. Therefore it doesnt suffice to study concentric shells around the origin, but one has to study shells around any point in frequency space. Techniques have been develloped recentlyto control the interaction of shells around different points infrequency space, and the purpose of this project is to exploit andfurther devellope these techniques and therefore better understand multilinear singular integrals. Multilinear singular integrals appearfor example in formal Taylor series of nonlinear operators, or asexplicit correction terms in an iterative scheme. A particularemphasis of the project will be to investigate possible applicationsof this theory to partial differential equations and other fieldsin mathematics.