Problems in harmonic analysis

摘要主要研究多线性和粗糙线性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。莱西和蒂勒关于双线性希尔伯特变换的鼓舞人心的工作取得了重大突破。这项工作激发了人们对多线性奇异积分算子领域的兴趣和活动，以及研究这些问题的基本工具时频分析。该方法与首席研究员的一些工作相关，因为他计划使用时频分析来解决本提案中的一些问题。这些问题包括在所有可能的指数范围内双线性希尔伯特变换的一致界，圆盘和相应的最大双线性乘子的研究，沿Lipschitz向量场的希尔伯特变换的研究，以及在高维傅里叶级数的球面部分和的几乎处处收敛中产生的carleson型算子的研究。最近，M. Lacey和PI已经能够使用时频分析获得沿C^{1+\e}$向量场的希尔伯特变换的肯定结果。这有助于回答齐格蒙德猜想，即L^2({\mathbb R}^2)$函数的积分是否可以在李普希茨选择的方向上微分。虽然时频分析对与标准Carderon-Zygmund核相关的多重线性奇异积分工作得很好，但它不适用于非奇异振荡积分。M. Christ， T. Tao，C。Thiele和PI建立了多线性振荡积分的L^p估计。该证明依赖于Gowers的“二次均匀性”技术，这是Gowers证明长度为4的等差数列Szemeredi定理的一个关键概念。关于具有振荡因子的非奇异多线性算子的一些基本问题仍在研究中。某些函数积分的可微性是分析中一个非常重要而有趣的问题。Besicovitch集的构造说明了L^2函数的积分不能微分。M. Lacey和PI最近获得的结果对理解L^2({\mathbb R}^2)$函数在Lipschitz选择方向上的可微性有重大影响，这是由a . Zygmund在大约70年前提出的。沿向量场的希尔伯特变换的研究也与Kakeya问题有关。多线性算子的研究涉及到偏微分方程中的一些问题，如薛定谔方程

ABSTRACTThe 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. These problems are related, but not limited, to the study of the Carleson operator and the bilinear Hilbert transform. A significant breakthrough was made by the inspiring work of Lacey and Thiele on the bilinear Hilbert transform. This work instigated renewed interest and activity in the area of multilinear singular integral operators and also in the underlying tool of study of these problems, time-frequency analysis. This method is relevant to some of the principal investigator's work as he is planning to work on some problems in this proposal using time-frequencyanalysis. These problems include uniform bounds for the bilinearHilbert transform in the full range of exponents possible, the study of the disc and the corresponding maximal bilinear multiplier, the study of the Hilbert transform along Lipschitz vector fields, and the study ofCarleson-type operators arising in the almost everywhere convergence of spherical partial sums of Fourier series in higher dimensions.Recently, M. Lacey and the PI had been able to use the time-frequencyanalysis to obtain an affirmative result for the Hilbert transform along $C^{1+\e}$ vector fields. And this casts a light on answering the Zygmund conjecture, which asks if the integrals of $L^2({\mathbb R}^2)$ functions could be differentiated in a Lipschitz choice of directions.Although the time frequency analysis works well for multiliner singularintegrals associated with a standard Carderon-Zygmund kernel, it does not work for the nonsingular oscillatory integrals. M. Christ, T. Tao,C. Thiele, and the PI established $L^p$ estimates for the multilinear oscillatory integrals. The proof relies on Gowers ``quadratic uniformity'' technique, which is a crucial concept in Gowers' proof of Szemeredi's theorem on arithmetic progression of length 4. Some basic questions concerning nonsingular multilinear operators with oscillatory factors are still under the investigation.The differentiability of the integral of certain functions is a veryimportant and interesting subject in analysis. The construction of Besicovitch set indicates that integrals of L^2 functions can not be differentiated. The recent results obtained by M. Lacey and the PI have a significant impact on understanding the differentiablity of $L^2({\mathbb R}^2)$ functions in a Lipschitz choice of directions, which was posed by A. Zygmund about seventy years ago. The study of Hilbert transform along vector fields is also relevant to the Kakeya problem. The study of multilinear operators is related to some problems in PDE such as the Schrodinger equation