Mathematical Sciences: Variable Coefficient Fourier Analysis

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This project continues mathematical research on problems in Fourier analysis and partial differential equations which are related to the wave equation and microlocal analysis. Included are efforts to prove sharp norm-local smoothing estimates for the wave equation in two dimensions, improving those obtained for dispersive equations. Newly designed methods of microlocal analysis will play a major role here. A primary application envisioned concerns embedding results for the spherical wave equation, establishing regularity properties of periodic solutions to nonlinear wave equations and possibly proving local decay estimates for solutions to initial value problems for hyperbolic operators with a potential. A second project involves the Lebesgue and Hardy space mapping properties of Fourier-Airy operators and efforts to prove sharp estimates for solutions to the wave equation in manifolds with geodesically concave or convex boundaries. This would extend recent results on Fourier integral operators which are locally a graph. The obstacle which must be overcome is presence of oscillatory integrals with singular phase functions. This makes standard plane wave decompositions of solutions much more delicate. The model for all of these problems is the study of Fourier integral operators having the property that the projections from the canonical relation to the cotangent spaces have Whitney folding singularities. Work on this project concerns fundamental mathematical tools related to the study of differential equations. The equations are used to model natural phenomena and predict their changing characteristics as they evolve in time and space.