Topics in Multi-linear Harmonic Analysis

在项目的第一部分，我们打算重温谐波分析的一些最基本的算子，即卡尔德隆换向子和李普希茨曲线上的柯西积分。我们的目标是对它们众所周知的L^p有界性给出新的和概念上更简单的证明。作为我们技术的副产品，我们也应该能够对Coifman关于将这些重要结果推广到任意维多面盘的多参数设置的可能性的问题给出一个完全肯定的答案。第二部分旨在继续研究数学物理的AKNS系统及其与傅里叶分析的深刻联系。在这个领域中有一个美丽但又极其困难的猜想要求证明，只要这样一个系统的势矩阵的元素属于实数线上的平方可积函数空间，其对应的解都是有界函数。最近观察到，这个猜想的最简单的特殊情况本质上等同于Carleson关于傅里叶级数的部分和几乎处处收敛的著名定理。我们这里的主要任务是给这个开放问题一个肯定的答案，对于一大类严格包括上三角和下三角的势矩阵。我们相信，本提案的研究不仅将扩展和深化我们对经典分析中许多重要算子的认识，而且还将为量子力学和核物理中的各种过程提供更好的数学理解。

In the first part of the Project we intend to revisit some of the most fundamental operators of harmonic analysis, namely Calderon commutators and the Cauchy integral on Lipschitz curves. Our goal is to give new and conceptually simpler proofs to their well known L^p boundedness properties. As a byproduct of our techniques, we should also be able to give a complete affirmative answer to a question of Coifman regarding the possibility of extending these important results to the multi-parameter setting of a polydisc of arbitrary dimension.The second part aims to continue the study of the AKNS systems of mathematical physics and their deep connection with Fourier analysis. A beautiful but extremely hard conjecture in the field asks to prove that as long as the entries of the potential matrix of such a system belong to the space of square integrable functions on the real line, its corresponding solutions are all bounded functions. It has been recently observed that the simplest particular case of this conjecture is essentially equivalent to Carleson's fameous theorem on almost everywhere convergence of partial sums of Fourier series. Our main task here is to give a positive answer to this open question, for a large class of potential matrices which strictly includes the upper triangular and lower triangular ones.We believe that the research of the present Proposal will not only extend and deepen our knowledge about many important operators of classical analysis, but will also provide a better mathematical understanding of various processes in quantum mechanics and nuclear physics.