Problems in Harmonic Analysis

主要研究人员正计划研究多线性和粗线性奇异积分算子中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，该分析用于著名的关于双线性希尔伯特变换的Lacey-Thiele定理。这些问题包括双线性Hilbert变换、圆盘和相应的最大双线性乘子的一致界、沿着Lipschitz向量场的Hilbert变换以及高维傅里叶级数的球面部分和几乎处处收敛时产生的多线性Carleson型算子。一些问题涉及到Szemer‘EDI关于算术级数的定理和(多线性)振荡积分，其中时频分析是没有价值的。这类问题包括沿曲线的双线性Hilbert变换，以及沿曲线的多线性振荡积分，具有非退化相位的多线性振荡积分，这是由Christian，Tao，Thiele和Pi最先研究的。一些问题与Kakeya问题有关，例如由Lacey和PI提出的Lipschitz极大函数，以及Zygmund猜想。最近，C.Muscalu和Pi将Carleson-Hunt定理推广到多线性情形。而M.Lacey和PI已经能够利用时频分析得到沿着Lipschitz向量场的Hilbert变换的一些条件结果。分析中的一个基本主题是某些函数的积分的可微性。最近，M.Lacey和PI基于一些重要的几何观测，用一种全新的方法证明了Lipschitz Kakeya极大函数的一些估计。对Lipschitz Kakeya极大函数的完整理解是回答A·Zygmund在大约70年前提出的关于某些函数在Lipschitz方向选择中的可微性问题的关键。在过去的几十年里，许多人对沿曲线的希尔伯特变换有了很好的理解。然而，曲线上的双线性Hilbert变换是一个新的领域。在这类问题的研究中，PI得到了一些局部结果，如对某些仿积的一致估计。作为线性情形，沿曲线的多线性奇异积分与多线性振荡积分之间的关系是分析中一个有趣而重要的问题。人们对这种关系的理解还只有一部分。在此基础上，D.Fan和PI得到了包含某些振荡因子的沿抛物线的双线性振荡积分的肯定结果，这是理解沿曲线的双线性Hilbert变换的一个起点。调和分析的主要主题是将复杂的物体分解和组装成更简单、更容易理解的片段，称为频率，类似于将音乐片段分解为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的中断而造成的信息丢失，以及恢复原始数据。这些应用为本提案中描述的一些理论研究提供了主要的实践动机。通过研究斯坦因猜想和齐格蒙德猜想等难题而获得的研究经验和结果，有助于PI更好地理解广泛的主题。这也为高职教育提供了一些新的数学观，极大地促进了高职教育的发展。希望通过分享新的知识和愿景，这也将使数学教育同行受益，更重要的是，将有益于敬业的数学学生。

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.