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.
Many fundamental phenomena of the physical world are described by partial differential equations. Understanding the behaviour of the solutions of such equations, is therefore of crucial importantce. One of the filelds of mathematics that provides extremely deep and useful tools for this understanding, if the field of Harmonic Analysis. The present project has been devoted to a very natural extension of the so called Calderon Program, which played an important role in Harmonic Analysis in the last fifty years or so. It is interesting that this program has been at the heart of many fundamental discoveries in Analysis and partial differential equations. The so called Calderon commutators, the theory of paraproducts, the Cauchy integral on Lipschitz curves, have all been related to it from the very beginning. To be able to extend this program to functions having arbitray polynomial growth, we had to develop a new method which shed new light also on the original Calderon program. It is natural to expect that this new method will have many other applications in the future. In particular, the results obtained and the methods developed in the current project, are expected to play an important role in the understanding of the water waves and of other problems of mathematical physics.
