Problems in harmonic analysis

主要研究人员计划研究多线性和粗线性奇异积分算子在调和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。Lacey和Thiele关于双线性Hilbert变换的鼓舞人心的工作取得了重大突破。这项工作重新引起了人们对多线性奇异积分算子领域的兴趣和活动，也激发了研究这些问题的基本工具--时频分析的兴趣。这种方法与首席调查员的一些工作有关，因为他计划使用时频分析来解决该提案中的一些问题。这些问题包括双线性Hilbert变换在全指数范围内的一致界，圆盘和相应的最大双线性乘子的研究，沿着Lipschitz向量场的Hilbert变换的研究，以及高维傅里叶级数的球面部分和的几乎处处收敛所产生的Carleson型算子的研究。最近，M.Lacey和PI已经能够使用时频分析得到沿着$C^{1+e}$向量场的Hilbert变换的肯定结果。这解释了Zygmund猜想，即$L^2({Mathbb R}^2)$函数的积分是否可以在Lipschitz选择的方向上可微.尽管时频分析对与标准Carderon-Zygmund核相关的多线性奇异积分很好地工作，但它不适用于非奇异振荡积分.M.Christian，T.T.Thiele，C.Thiele和Pi建立了多线性振荡积分的$L^p$估计。这一证明依赖于Gowers的“二次一致”技巧，这是Gowers证明Szmeredi关于长度为4的算术级数定理的一个重要概念。关于带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。某些函数的积分的可微性是一个非常重要而有趣的分析主题。Besicovitch集的构造表明，L^2函数的积分是不可微的。M.Lacey和PI最近得到的结果对于理解$L^2({Mathbb R}^2)$函数在大约70年前由A.Zygmund提出的Lipschitz方向选择中的可微性有重要影响。沿矢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组中的一些问题，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