Harmonic and Complex Analysis

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

ABSTRACT (DMS-0099767) The principal investigator proposes to study the sharp regularity properties of operators associated to domains of finite type in dimensions greater than two, and to apply the results to the study of the boundary behavior of holomorphic functions in these domains. He also proposes to study corresponding properties on certain nilpotent Lie groups. The operators under consideration include the Szego and Bergman projections, and the relative fundamental solutions to the Kohn-Laplacian. In particular, the principal investigator hopes to find an appropriate metric that controls the size of the distribution kernel for the Szego projection on diagonalizable pseudo-convex domains of finite type, and on convex domains. He hopes to study the boundary behavior of holomorphic functions in certain non-diagonalizable domains, and in this context, he proposes to study variants of the Hardy-Littlewood maximal function. Appropriate nilpotent Lie groups can model many of these problems, and the principal investigator proposes to study the corresponding convolution operators on the groups. The distribution kernels of these operators have singularities that are more complicated than the classical Calderon-Zygmund singular integral operators, and the problem thus leads to interesting questions in harmonic analysis. Interesting questions and results in mathematics often arise at the interface of two or more areas of research, since in such situations, deep results in one field can shed light on previously intractable problems in another. The research outlined in this proposal is directed towards questions at the interface of modern harmonic analysis, several complex variables, and linear partial differential equations. For more than forty years these areas have enjoyed a very profitable symbiotic relationship. For example, attempts to solve the Levy problem in complex analysis led to the development of deep results about hypo-elliptic partial differential equations, and these problems, in turn, led to developments in harmonic analysis on nilpotent Lie groups such as the Heisenberg group. Many of these related results dealt with integral operators that are modeled on the classical singular integrals such as the Hilbert transform and the Riesz transforms whose study goes back to the middle of the twentieth century. In recent years, it has become clear that new problems in complex analysis require the understanding of new kinds of operators in which the singularities of the kernel are considerably more complicated than the classical singular integrals. The research described in this proposal is directed toward understanding these more complicated situations.