Problems in Harmonic Analysis

提案编号：摘要本文将对多线奇异积分算子研究中出现的调和分析中的各种问题进行研究。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，Lacey和Thiele的工作在双线性Hilbert变换上取得了重大突破。结果表明，时频分析是解决与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，这些问题的分析给我们希望解决该领域的其他重要和困难的问题，如希尔伯特变换沿着向量场，Kakeya问题，和Carleson的最大算子的部分和的Fourier级数在二维。这也将是非常有趣的，看看如何使用这种微妙的分析，以解决一些问题，在其他领域，如偏微分方程，数论等实际上，多线性算子已被用于研究偏微分方程，因为他们自然出现在一系列的扩展解决方案的许多方程。 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中心。谐波分析的主要内容是将复杂的物体分解和组装成简单的易于理解的片段，类似于将复杂的音乐片段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。在乘法器问题的研究中，非光滑符号的出现类似于信号频率被突然操作改变的物理现象，例如无线电通信或电视传输被气象现象中断。这种突然和意外的操作会导致信息丢失。避免信息丢失或恢复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.