Time-Frequency Analysis and Multilinear Singular Integrals

摘要：本课题旨在进一步发展多元线性奇异积分理论。这项研究始于Michael Lacey和首席研究员在各种L^p空间上发现双线性Hilbert变换的有界性。他们对这项工作的方法，回答了阿尔贝托·卡尔德隆长期存在的问题，利用了时频分析的方法，类似于伦纳特·卡尔森和查尔斯·费费曼对傅里叶级数几乎处处收敛的证明中发现的方法。最近的研究成果包括极大截断双线性希尔伯特变换的有界性、双次线性极大算子的有界性以及双线性希尔伯特变换的若干一致估计，给出了Calderon第一对易子有界性的新证明。本项目的研究课题包括，但绝不限于，将上述结果从双线性扩展到多线性奇异积分，以及从一维设置到高维。该项目位于分析领域，自上世纪严谨数学出现以来，分析一直是数学的中心领域。更准确地说，该项目侧重于局部傅立叶分析，这是一个很容易向任何音乐家描述的理论。就像乐谱将复杂的乐曲分解成基本单元，音符有特定的位置、持续时间和频率一样，局部傅立叶分析从基本部分合成复杂的数学函数，每个基本部分都是一个相对简单的函数，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化处理中得到如此成功的应用，应该不足为奇。本项目的具体主题源自并将对偏微分方程中的非线性问题的研究产生影响，偏微分方程是数学的一个分支，在工程和物理科学中日益重要。具体地说，正在调查的问题的解决方案可能会对数据压缩、信号处理和医学成像等具体领域产生影响。

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.