Time-Frequency Analysis and Multilinear Singular Integrals

建议：DMS-9970469主要研究人员：Christoph M.Thiel摘要：本项目的目的是进一步发展多线性奇异积分理论。本文的研究始于Michael Lacey和主要研究者对双线性Hilbert变换在各种L^p空间上的有界性的发现。他们对这项工作的方法，回答了阿尔贝托·卡尔德龙的一个长期存在的问题，利用了与Lennart Carleson和Charles Fefferman关于傅里叶级数几乎处处收敛的证明中类似的时频分析方法。最近的结果包括最大截断双线性Hilbert变换的有界性，双次线性极大算子的有界性，以及双线性Hilbert变换的某些一致估计，这些结果给出了Calderon第一交换子有界性的新证明。该项目的研究主题包括但不限于上述结果从双线性奇异积分到多线性奇异积分的推广，以及从一维环境到更高维度的推广。该项目位于分析领域，自上个世纪严格数学出现以来，分析领域一直是数学的中心领域。更准确地说，该项目专注于本地化傅立叶分析，这是一种任何音乐家都很容易描述的理论。就像乐谱通过将乐谱分解为基本单元来描述复杂的乐曲一样，局部傅立叶分析将具有特定位置、持续时间和频率的音符从基本片段合成复杂的数学函数，每个基本片段都是一个相对简单的函数，具有其特有的时间、持续时间和频率。因此，局域傅立叶分析在声学数据的自动处理中得到了如此成功的应用，这并不令人惊讶。本项目中的具体主题源于偏微分方程组中的非线性问题的研究，并将对其产生影响。偏微分方程组是数学的一个分支，在工程和物理科学中具有越来越重要的地位。特别是，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.