On a New Class of Singular Integrals

点击翻译按钮获取中文摘要

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.