Time-Frequency Analysis and Multilinear Singular Integrals

提案：DMS-9970469主要研究者：Christoph M. Thiele摘要：本项目的目的是进一步发展多线性奇异积分理论。这项研究始于迈克尔·莱西和主要研究者发现的双线性希尔伯特变换在各种L^p空间上的有界性。他们的方法这项工作，回答了一个长期存在的问题阿尔贝托卡尔德龙，利用方法的时间频率分析类似于那些发现伦纳特Carleson的和查尔斯Fundamman的证明几乎处处收敛的傅立叶级数。最近在这方面的结果包括有界性的最大截断双线性希尔伯特变换，有界性的双次线性最大算子，和某些统一估计的双线性希尔伯特变换，这产生了一个新的证明有界性的卡尔德龙的第一个交换子。本项目的研究课题包括但不限于将上述结果从双线性扩展到多线性奇异积分，以及从一维设置到更高维。本项目涉及自上个世纪严格数学出现以来一直是数学中心领域的分析领域。更准确地说，该项目的重点是局部傅立叶分析，这是一个很容易描述给任何音乐家的理论。就像乐谱通过将其分解为具有指定位置、持续时间和频率的基本单元（音符）来描述复杂的作品一样，局部傅里叶分析从基本片段合成复杂的数学函数，其中每个都是一个相对简单的函数，具有其特征时间、持续时间和频率。因此，局部傅立叶分析在声学数据的自动处理中应用得如此成功并不奇怪。在这个项目中的具体主题产生于并将影响到偏微分方程，数学的一个分支，在工程和物理科学的重要性日益增长的非线性问题的研究。特别是，正在调查的问题的解决方案可能会影响到数据压缩，信号处理和医学成像等具体领域。

Proposal: DMS-9970469Principal Investigator: Christoph M. ThieleAbstract: The purpose of this project is to develop further the theory of multilinear singular integrals. This research started with the discovery by Michael Lacey and the Principal Investigator of the boundedness of the bilinear Hilbert transform on various L^p spaces. Their approach to this work, which answered a long-standing question of Alberto Calderon, exploited methods of time-frequency analysis similar to those found in Lennart Carleson's and Charles Fefferman's proofs of almost everywhere convergence of Fourier series. More recent results in this area include boundedness of the maximal truncated bilinear Hilbert transform, boundedness of the bisublinear maximal operator, and certain uniform estimates for the bilinear Hilbert transform, which yield a new proof for the boundedness of Calderon's first commutator. Research topics for this project include, but are by no means limited to, an extension of the aforementioned results from bilinear to multilinear singular integrals, as well as from the one-dimensional setting to higher dimensions.This project lies in the field of analysis, which has been a central area of mathematics ever since the emergence of rigorous mathematics in the last century. More precisely, the project focuses on localized Fourier analysis, a theory that is readily described to any musician. In the same way that a musical score describes an intricate composition by breaking it down into its elementary units, notes that have specified location, duration and frequency, localized Fourier analysis synthesizes complicated mathematical functions from elementary pieces, each of which is a relatively simple function that has its characteristic time, duration, and frequency. It should come as no great surprise, therefore, that localized Fourier analysis has been applied so successfully in the automated processing of acoustic data. The specific topics in this project arose from and will have impact on the study of nonlinear problems in partial differential equations, a branch of mathematics with ever growing importance in engineering and the physical sciences. In particular, solutions to the problems under investigation could have implications for such concrete areas as data compression, signal processing, and medical imaging.