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

职业：多线性算子和更一般的非线性算子的时频分析

• 批准号: 9985572
• 负责人: Christoph Thiele
• 金额: $ 20万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9985572&HistoricalAwards=false
• 关键词: CAREER; Time; Frequency; Analysis; Multilinear

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.

研究摘要多线性算子和更一般的非线性算子的时频分析这个建议的重点是研究多线性奇异积分.奇异积分在过去的五十年中在调和分析中扮演了重要的角色，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分解为频率内容位于原点周围同心壳中的片段。当我们讨论多线性奇异积分时，我们发现多线性奇异积分在频率空间中具有平移对称性。因此，研究原点周围的同心壳是不够的，必须研究频率空间中任何点周围的壳。技术已经开发最近来控制壳的相互作用在频率空间中的不同点，这个项目的目的是开发和进一步发展这些技术，从而更好地理解多线性奇异积分。多线性奇异积分例如出现在非线性算子的形式泰勒级数中，或作为迭代格式中的显式校正项。该项目的一个特别的例子将是研究这一理论在偏微分方程和其他数学领域的可能应用。