Topics in Fourier Analysis

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1. We shall conduct research on various problems in harmonic analysis. The project will focus on local smoothing properties of solutions for wave equations. Results on local smoothing follow from a crucial inequality involving decompositions of cone multipliers which was originally formulated by Wolff. This inequality and several variants of it have a wide range of applications; one of them concerns the Sobolev regularity for averaging operators along curves and boundedness of associated maximal operators. It is of interest to determine the range of these inequalities. Other parts of the proposal deal with Fourier restriction and extension problems, with variational Carleson theorems, with singular maximal functions and with the wave equation on the Heisenberg group.2. A major part of this project is concerned with the regularity properties of some basic linear partial differential operators of mathematical physics. Quantitative results for these operators are applicable to problems in nonlinear equations with a wide range of applications in the physical sciences. They also yield applications within mathematics, namely to problems on expansions in eigenfunctions of the Laplacian on compact manifolds and the regularity of averaging operators. The study of these averaging operators is motivated by problems in the theory of medical imaging.