Problems in Harmonic Analysis

摘要研究多线奇异积分算子在谐波分析中出现的各种问题。这些问题与Carleson算子和双线性Hilbert变换的研究有关，但不限于此。最近，莱西和蒂勒在双线性希尔伯特变换方面取得了重大突破。结果表明，时频分析是解决多线性算子研究相关问题的有力工具。此外，虽然这仍在研究中，但对这些问题的分析给我们带来了解决该领域其他重要和困难问题的希望，例如沿向量场的希尔伯特变换，Kakeya问题，以及二维傅里叶级数部分和的Carleson极大算子。看看如何使用这种微妙的分析来解决其他领域的一些问题，比如偏微分方程、数论等，也会很有趣。实际上，多元线性算子已经被用于偏微分方程的研究，因为它们自然地出现在许多方程解的级数展开中。谐波分析不仅是理论数学的一个领域，而且是一个应用领域，位于光学，信号处理，气象学和音乐等不同领域交叉的核心。和声分析的主要内容是将复杂的物体拆解并组合成简单易懂的部分，类似于将复杂的音乐作品分解成几个基本音符的排列。在信号处理中，谐波分析用于检测信号和图像的不规则性。傅里叶变换是一个非常有用的工具来定位这些不规则性。在乘数问题的研究中，非光滑符号的出现类似于信号频率因突然操作而改变的物理现象，如气象现象导致无线电通信或电视传输中断。这种突如其来的操作会造成信息的丢失。避免信息丢失或检索原始数据是本文提出的理论研究的主要课题。

Proposal Number: DMS-0140376PI: Xiaochun Li ABSTRACTResearch will be conducted on a variety of problems in harmonic analysis arising in the study of multilineasingular integral operators. These problems are related,but not limited, to the study of the Carleson operator and the bilinear Hilbert transform. Recently a significantbreakthrough on the bilinear Hilbert transform was made by the inspiring work of Lacey and Thiele. It turns out that the time-frequency analysis is a powerful tool to solve problems related to the study of multilinear operators. Moreover, although this is still under investigation, the analysis of these problems gives us hopes to solve other important and difficult problems in the field, such as the Hilbert transform along vector fields, the Kakeya problem, and Carleson's maximal operator of the partial sums of Fourier series in two dimensions. It will also be very interesting to see how to use this delicate analysis to solve some problems in other fields such as partial differential equations, number theory, etc. Actually, multilinear operators have been used in the study of partial differential equations, since they naturally appear in series expansions of solutions of many equations. Harmonic analysis is not only an area of theoretical mathematics, but also an applicable area lying at theheart of the intersection of fields as diverse as optics, signal processing, meteorology, and music. The main themin Harmonic analysis is about disassembling and assemblingcomplicated objects into simpler well-understood pieces, by analogy to decomposition of intricate musical pieces into arrangements of a few basic notes. In signal processing,harmonic analysis is used to detect irregularities of signals and images. The Fouriertransform is a very useful tool to locate these irregularities. The appearance of a nonsmooth symbol inthe study of multiplier problems is analogous to physical phenomena where the frequencies of signals are altered by an abrupt operation, such as the interruption of radio communication or television transmission by meteorological phenomena. Such a sudden and unexpected operation causes the loss of information. To avoid the loss of information or to retrieve the original data is the main topic of the theoretical research proposed here.