Mathematical Sciences: Variable Coefficient Fourier Analysis

这个项目继续数学研究的问题， 傅立叶分析和偏微分方程， 与波动方程和微局部分析有关。 包括 是努力证明尖锐的范数局部平滑估计的 二维波动方程，改进了 色散方程 新设计的微局部方法 分析将在这方面发挥重要作用。 主应用 所设想的关注球面波的嵌入结果 方程，建立周期的正则性 非线性波动方程的解，并可能证明局部 初值问题解的衰减估计 双曲型算子 第二个项目涉及勒贝格和哈代空间 傅里叶-艾里算子的映射性质和证明 流形上波动方程解的精确估计 具有测地线凹的或凸的边界。 这将 推广了最近关于傅里叶积分算子的结果， 局部图。 必须克服的障碍是在场 具有奇异相位函数的振荡积分 这 使得解的标准平面波分解 很精致 所有这些问题的模型是傅立叶的研究 积分算子具有的性质， 余切空间的典范关系有Whitney 折叠奇点 这个项目的工作涉及基本的数学工具 与微分方程的研究有关。 方程 用来模拟自然现象并预测它们的变化 它们在时间和空间上的演变。

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.