Problems in Harmonic Analysis

谐波分析中的问题

• 批准号: 0801154
• 负责人: Xiaochun Li
• 金额: $ 16.16万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801154&HistoricalAwards=false
• 关键词: Problems; Harmonic; Analysis

The principal investigator is planning to work on a variety of problems in harmonic analysis arising in the study of multilinear and rough linear singular integral operators. Some problems that the PI is planning to work are related to time-frequency analysis, which was used in the celebrated Lacey-Thiele's theorem on the bi-linear Hilbert transform. These problems include uniform bounds for the bi-linear Hilbert transform, the disc and the corresponding maximal bi-linear multiplier, the Hilbert transform along Lipschitz vector fields, and multi-linear Carleson-type operators arising in the almost everywhere convergence of spherical partial sums of Fourier series in higher dimensions. Some problems are related to Szemer\'edi's theorem on arithmetic progression and the (multilinear) oscillatory integrals, in which the time frequency analysis is unvaluable. These type of problems contain the bi-linear Hilbert transform along curves, and the multi-linear oscillatory integrals along curves, the muliti-linear oscillatory integrals with non-degenerate phases which was first studied by Christ, Tao, Thiele and the PI. Some problems are associated to the Kakeya problem such as the Lipschitz maximal functions, initiated by Lacey and the PI, and the Zygmund conjecture. Recently, C. Muscalu and the PI generalized the Carleson-Hunt theorem to the multi-linear case. And M. Lacey and the PI had been able to use the time-frequency analysis to obtain some conditional results for the Hilbert transform along Lipschitz vector fields. A fundamental subject in analysis is the differentiability of the integral of certain functions. Recently M. Lacey and the PI proved some estimates for a Lipschitz Kakeya maximal function by a completely new method based on some crucial geometric observations. It turns out that a complete understanding of the Lipschitz Kakeya maximal function is a key to answer the question on the differentiablity of certain functions in a Lipschitz choice of directions, which was posed by A. Zygmund about seventy years ago. The Hilbert transform along curves had been understood well by work of many people in the last several decades. However, the bi-linear Hilbert transform along curves is a new field. Some partial results such as uniform estimates for some para-products arising in the study of this type of problem were obtained by the PI. As the linear case, the relation of multi-linear singular integrals along curves and multi-linear oscillatory integrals is an interesting and important topic in analysis. This relation is only partially understood. Based on it, D. Fan and the PI obtained an affirmative result for the bi-linear oscillatory integral along parabolas incorporating some oscillatory factors, which is a starting point for understanding the bi-linear Hilbert transform along curves. The main theme in harmonic analysis is disassembling and assembling complicated objects into simpler well-understood pieces, called frequencies, by analogy to decomposing musical pieces into arrangements of a few basic tones. In signal processing, harmonic analysis is used in the detection of irregularities of signals and images, the protection against the loss of information due to a sudden and unexpected interruption, and the retrieval of the original data. These applications provide the main practical motivation for some theoretical research described in this proposal. The research experience and results gained by investigating these difficult problems such as the Stein's and Zygmund's conjectures help the PI better understand a wide range of topics. It also provides the PI with some new visions on mathematics that greatly benefits the PI's teaching. Hopefully it will also benefit fellow mathematical educators, and more importantly dedicated mathematical students, through sharing the new knowledge and visions.

主要研究者计划研究多线性和粗糙线性奇异积分算子研究中出现的调和分析中的各种问题。PI计划工作的一些问题与时频分析有关，该分析用于双线性希尔伯特变换的著名Lacey-Thiele定理。这些问题包括双线性Hilbert变换的一致界、圆盘和相应的最大双线性乘子、沿着Lipschitz向量场的Hilbert变换以及高维Fourier级数球面部分和的几乎处处收敛中的多线性Carleson型算子. 一些问题与Szemer\'edi的等差数列定理和（多线性）振荡积分有关，其中时频分析是没有价值的。这类问题包括沿着曲线的双线性Hilbert变换，沿沿着曲线的多线性振荡积分，Christ，Tao，Thiele和PI首先研究的非退化相位的多线性振荡积分。一些问题是相关的挂谷问题，如Lipschitz极大函数，发起莱西和PI，和Zygmund猜想。最近，C. Muscalu和PI将Carleson-Hunt定理推广到多线性情况。 和M.莱西和PI已经能够使用时频分析来获得沿着Lipschitz向量场的希尔伯特变换的一些条件结果。分析中的一个基本问题是某些函数的积分的可微性。最近M。莱西和PI证明了一些估计Lipschitz Kakeya极大函数的一个全新的方法的基础上，一些关键的几何观察。结果表明，对Lipschitz Kakeya极大函数的完整理解是回答A. Zygmund大约70年前。沿着曲线的Hilbert变换在过去的几十年里已经被许多人的工作所理解。然而，沿沿着曲线的双线性Hilbert变换是一个新的领域。在研究这类问题时，PI得到了一些局部结果，如对某些仿乘积的一致估计。作为线性情形，多线性沿着曲线奇异积分与多线性振荡积分的关系是分析中一个有趣而重要的课题。这种关系只是部分理解。在此基础上，D。Fan和PI得到了包含一些振荡因子的双线性振荡积分沿着抛物线的肯定结果，这是理解双线性Hilbert变换沿着曲线的起点。 谐波分析的主要主题是将复杂的对象分解和组装成更简单的易于理解的片段，称为频率，类似于将音乐片段分解成几个基本音调的排列。在信号处理中，谐波分析用于检测信号和图像的不规则性，防止由于突然和意外中断而导致的信息丢失，以及恢复原始数据。这些应用提供了一些理论研究的主要实际动机，在这个建议。通过调查这些困难的问题，如斯坦和齐格蒙德的理论，获得的研究经验和结果有助于PI更好地理解广泛的主题。它也为PI提供了一些新的数学视野，极大地有利于PI的教学。希望它也将有利于同行的数学教育工作者，更重要的是专门的数学学生，通过分享新的知识和愿景。