Topics in Fourier Analysis

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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.