On a New Class of Singular Integrals

本课题的目标是研究一种新的奇异积分族，它扩展了经典的对易子Calderón和Lipschitz曲线上的Cauchy积分。当一个人试图发明一种微积分，其中包含具有任意多项式增长的函数的乘法算子时，它们就会很自然地出现。这些算子的一个有趣的特征是，所谓的T1和Tb定理，在经典的“线性增长情况”中是如此成功，这次不能再应用了，因为相应的表达式远非BMO函数。为了理解它们，研究者计划开发的新方法与他最近对Calderón换向器的新方法有关，也与他几年前介绍的所谓的flag副产物的研究有关。本项目属于经典分析领域，但其结果和方法也有潜在的应用于其他数学领域，包括数学物理和偏微分方程。水波研究的最新进展证明了这种与物理世界现象的联系的存在。随着工作的进展，研究者打算开设课程来解释项目发现的想法，并让研究生参与研究。

The goal of the project is to study a new family of singular integrals, which extend the classical commutators of Calderón and the Cauchy integral on Lipschitz curves. They appear very naturally, when one tries to invent a calculus that includes operators of multiplication with functions having arbitrary polynomial growth. One interesting feature of these operators is that the so called T1 and Tb theorems, so successful in the classical "linear growth case," can no longer be applied this time, because the corresponding expressions are far from being BMO functions. The new methods that the investigator plans to develop in order to understand them, are related to his recent new approach on Calderón commutators and also to the study of the so called flag paraproducts that he introduced a few years ago.The present project lies within the realms of classical analysis, but its results and methods have potential applications to other mathematical areas as well, including mathematical physics and partial differential equations. Recent developments in the study of the water waves certify the existence of such connections with phenomena in the physical world. As the work progresses, the investigator intends to give courses to explain the ideas of the project findings and to involve graduate students in the research.