Topics in Fourier Analysis

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Proposal Number: DMS-0200186 PI: Andreas Seeger ABSTRACT The proposed research will be concerned with several interrelated topics in Fourier analysis. In particular it is proposed to work on precise regularity properties of oscillatory integral and Fourier integral operators with degenerate canonical relations. The mapping properties are governed by the geometry of the canonical relations. Here we are, in particular, interested in the cases that come up when studying averaging operators associated to curves in higher dimensions. Other projects include the behavior of singular maximal functions and rough singular integrals on classes of integrable functions, the mapping properties of wave operators on nilpotent groups, and questions concerning the failure of weak amenability of Lie groups. The research project is focused on some fundamental question in Fourier analysis, with an emphasis on estimates for various oscillatory integral operators. Such estimates are in particular crucial for understanding the qualitative and quantitative behavior of Radon transforms and related Fourier integral operators. These operators arise in various areas of mathematics, such as partial differential equations, complex analysis and integral geometry, as well as in non-mathematical applications such as tomography and medical imaging. As an example we mention that X-ray tomography involves inverting a Radon transform operator; i.e. one seeks to determine an image from knowledge about the averages of a function for a prescribed family of lines. The operators involved in inverting this restricted X-ray transform tend to be Fourier integral operators as considered in this proposal.