The investigator will study multi-parameter analogs of the Calderón-Zygmund theory of singular integrals, which significantly generalize the well-known product theory of singular integrals. A critical starting point will be the case of multi-parameter Carnot-Carathéodory (or sub-Riemannian) geometry (a geometry defined by vector fields). There is already a reasonable conjecture as to the analog of a Calderón-Zygmund singular integral in the context of Carnot-Carathéodory geometry: a conjecture which generalizes a number of known and useful types of singular integrals. The Calderón-Zygmund theory of singular integrals has found numerous applications in a wide range of mathematics. However, when the underlying geometry is multi-parameter, there is no known analog of the Calderón-Zygmund theory (outside of the product-type situation). Recent work shows that an analog might be in reach when the underlying geometry is given by a multi-parameter Carnot-Carathéodory geometry.
Harmonic analysis, and more specifically the theory of singular integrals, has found a wide variety of applications in other areas of mathematics, physics, finance, and biology. The diverse methods in harmonic analysis offer the promise of many future applications in the sciences. The research in this project, in particular, has several connections to other areas of mathematics, finance, and mathematical physics. Most directly, it has applications to the theory of several complex variables. More generally, it applies to partial differential equations defined by vector fields: a theory which has implications in mathematical finance and fluid dynamics. One of the main current obstacles in the application of the theory of singular integrals to various questions is that there is no suitable "multi-parameter" theory adapted to the particular application. The main purpose of this project is to develop such a theory, which would be useful in a wide variety of situations--potentially addressing a number of open questions. The project will help continue an active research and training group in harmonic analysis--especially harmonic analysis with applications to partial differential equations--at the University of Wisconsin-Madison. This includes many active discussions and collaborations with graduate students and visiting postdoctoral scholars.
A singular integral operator is defined by an integral which does not converge in the classical sense, but can be made sense of due to some sort of "cancellation." Such operators date back to Hilbert's work (more than 100 years ago) in several complex variables. They gained a stronger foundation due to the work of Calderon and Zgymund, and a very general theory was finally developed by Coifman and Weiss. This general theory showed that to each classical singular integral, there was an underlying "geometry." Such operators proved to be useful in an extremely wide variety of questions, including questions from partial differential equations and several complex varaibles. Despite the great applicability of the Calderon-Zgymund theory of singular integrals, there arose many questions where they did not apply. On main hurdle was that many questions clearly had several underlying geometries, while the Calderon-Zygmund theory can only incorporate one geometry at a time. Thus there was a need for a "multi-parameter" theory of singular integrals--a theory which could incorporate many geometries simultainously. There were a few theories developed in special cases, but no general theory which would apply to many situations simultaniously. This project developed the first, general, multi-parameter theory of singular integrals. The theory was presented in a 400+ page reserach monograph, to appear in the Annals of Mathematics Studies. The monograph developes the theory, shows how it generalizes past theories, and shows how it applies to various situations in PDEs and several complex variables. One main aspect of the monograph is to show that there are four natural definitions one might take for a multi-parameter singular integral. However, under some mild hypotheses, they all turn out to be equivalent. Thus, there is one natural definition, which applies in a wide range of situations, and enjoys many natural properties. It has been shown to be useful in various questions from PDEs and several complex variables, and will likely be useful in many other situations.
