The PI will conduct basic research on Harmonic Analysis and Partial Differential Equations, focusing on problems in harmonic analysis in Euclidean spaces centered around Lebesgue norm inequalities. One subject of ongoing research (partly joint with M. Christ) is the mapping properties of Generalized Radon Transforms -- a vast class of averaging operators over lower dimensional submanifolds. Satisfactory results in some cases have been obtained by the PI, via the application of techniques developed by Bourgain, Wolff, and others for the Kakeya problem. Generalized Radon transforms are still poorly understood, except in special cases, and Kakeya-related methods have a great deal more to offer. Understanding generalized Radon transforms is an important ingredient in the analysis of summability of multi-dimensional Fourier nseries and integrals, of more general oscillatory integrals, and of linear and nonlinear wave equations. Other related problems, including the Fourier restriction phenomenon and issues in Geometric Measure Theory such as the packing of submanifolds into Euclidean space, will also be investigated.
Fourier analysis has always found wide applications in natural sciences and engineering. It underlies a powerful and diverse array of tools currently widely used in applications, and offers the promise of further applications in the future. The proposed research deals with foundational issues which may ultimately help to underpin such future applications. Summability theory of multi-dimensional Fourier series and related oscillatory integral problems are irreplaceable tools in the study of a wide class of PDEs, in which the current state of knowledge is incomplete. The proposed research will contribute to the general understanding of these problems. The planned research is related to certain discrete problems of interest in Combinatorics and Number Theory, which in most cases remain wide open.
