Topics in Fourier Analysis

本数学研究项目涉及调和分析中的几个问题。一个主要的焦点是估计带有平移和旋转的通勤算子的问题。证明这类算子在勒贝格空间中的有界性的有效刻画是很有必要的。这些结果及其证明方法可用于研究数学物理中经典方程，如波动方程、薛定谔方程和其他色散方程解的正则性。Seeger和他的合作者还将研究有关欧几里德空间子流形上的傅里叶变换行为的各种问题。这些问题都与紧致流形上微分算子的特征函数有关的定量问题有关。其他项目涉及平均算子的正则性，奇异积分算子的有界性，广义Radon变换，以及与微分问题和傅里叶级数有关的各种逐点收敛问题。傅立叶分析的方法和技巧在自然科学和工程中一直被广泛应用于理解物理现象。对波动方程和薛定谔方程的解的研究，以及关于奇异积分算子的一些项目，都是由物理问题推动的。所提出的广义Radon变换的研究涉及到医学成像中出现的问题。这个研究项目也有一个教育部分，因为西格将在他们的研究和教学中指导博士后学生和博士后研究人员。西格还将继续参与组织会议，以促进数学分析领域的研究人员和学生之间的互动。

This mathematics research project is concerned with several problems in harmonic analysis. A primary focus is on the problem of estimating operators that commute with translation and rotations. It is desirable to prove effective characterizations for the boundedness of such operators in Lebesgue spaces. Such results and the techniques to prove them can be used to investigate regularity properties of the solutions of classical equations in mathematical physics, such as the wave equation, the Schroedinger equation and other dispersive equations. Seeger and his collaborators will also study various questions about the behavior of the Fourier transform on submanifolds of Euclidean space. Such problems are tied to quantitative questions about eigenfunctions of differential operators on compact manifolds. Other projects concern the regularity properties of averaging operators, the boundedness of singular integral operators, generalized Radon transforms, and various pointwise convergence questions related to differentiation problems and Fourier series.Methods and techniques from Fourier analysis have always found wide applications in understanding physical phenomena in the natural sciences and in engineering. The proposed investigations of the solutions of wave and Schroedinger equations, as well as some of the projects on singular integral operators, are motivated by questions from physics. The proposed research on generalized Radon transforms is relevant to problems that arise in medical imaging. This research project also has an educational component, as Seeger will direct Ph.D. students and mentor postdoctoral researchers both in their research and in teaching. Seeger will also continue to be involved in the organization of conferences to facilitate interactions of researchers and students in the mathematics field of analysis.