Mathematical Sciences: Presidential Young Investigator

总统青年研究员奖将支持 数学研究领域的谐波分析有关 振荡积分和微分理论的问题。 在此期间将遵循几个重要方向 支持。 其中许多问题的根源在于 关于偏微分算子和努力衡量 他们的解决方案的界限。 其中包括范数的界， 涉及球面调和展开的谱投影 在一个球体上运行。 其他工作涉及傅立叶问题 与流形曲率有关的分析。 这包括 一致Sobolev不等式的发展及相应的 限制定理 沿着这些路线的结果将提供 从局部不平等中得出强有力的全局结论。 工作将 继续关于锐Riesz的问题意味着任意 紧流形，特别是关于弱型端点 这些结果总是最难实现的。 其他工作将集中在独特的延续二阶 偏微分算子，特别是薛定谔算子， 双曲型和抛物型微分唯一延拓 算子和相关的本征函数展开。 的一个新领域 工作将涉及Randon变换的最大估计或 低维曲面上函数的平均值。 有趣的是，在这种情况下， 在较低的维度， 最近。

This Presidential Young Investigator Award will support mathematical research in the area of harmonic analysis related to problems of oscillatory integrals and differentiation theory. Several important directions will be followed during the period of support. Many of these have their roots in questions concerning partial differential operators and efforts to measure bounds on their solutions. These include bounds on the norm of spectral projections involving spherical harmonic expansions of functions on a sphere. Other work involves problems of Fourier analysis related to curvature of manifolds. This includes development of uniform Sobolev inequalities and corresponding restriction theorems. Results along these lines will provide powerful global conclusions from local inequalities. Work will continue on the question of sharp Riesz means on arbitrary compact manifolds, especially regarding the weak type end point results which are invariably the most difficult to achieve. Other work will focus on unique continuation of second order partial differential operators, especially Schrodinger operators, unique continuation of hyperbolic and parabolic differential operators and related eigenfunction expansions. A new area of work will concern maximal estimates for Randon transforms or averages of functions over lower dimensional surfaces. Interestingly the most difficult questions in this context occur in lower dimensions where there has been considerable activity lately.