Research in harmonic analysis and partial differential equations

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI will undertake research in harmonic analysis, and partial differential equations (PDE). In PDE, the focus is on the dynamical properties of Schrodinger evolution (SE). One subject of on-going research is dispersive estimates for SE. He will also work on mathematical problems on non-linear SE motivated by the numerical studies in fiber optic communication systems. In harmonic analysis, he focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. In particular, he proposes to continue his investigations on restriction estimates relative to fractal measures, and on their applications in PDE and geometric measure theory. He also proposes to continue his research on the mapping properties of generalized Radon transforms (GRT) -- a huge class of averaging operators over lower dimensional submanifolds of Euclidean spaces. By applying the techniques developed for Kakeya problems, he obtained interesting results in some cases. The results on the mapping properties of GRT have important applications in Fourier restriction phenomenon and in general in the summability theory of multi-dimensional Fourier series. Harmonic 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. The proposed research on nonlinear SE are directly motivated by the engineering problems in fiber optic communication systems, and the methods used are likely to be useful in a range of applications. The study of the mapping properties of GRT has various applications in engineering. For example, the X-ray transform (which is a particular GRT) applied to the density function of a patients body is essentially the data obtained by magnetic resonance imaging. The study of Fourier restriction, summability theory of multi-dimensional Fourier series, and dispersive estimates are irreplaceable tools in the study of a wide class of PDE. The proposed research would make a contribution to the general understanding of these problems. The planned research is also related to certain discrete problems of interest in combinatorics and number theory, which in most cases remain wide open.