Mathematical Sciences: Presidential Young Investigator

这一总统青年研究人员奖将支持与振荡积分和微分理论问题相关的调和分析领域的数学研究。在支助期间将遵循几个重要方向。其中许多问题的根源在于有关偏微分算子及其解的界的度量的问题。其中包括涉及球面上函数的球谐展开式的谱投影范数的界。其他工作涉及与流形的曲率有关的傅立叶分析问题。这包括发展一致的索波列夫不等式和相应的限制定理。沿着这些思路的结果将从局部不平等中提供强有力的全球结论。关于任意紧流形上的尖锐Riesz平均的问题，特别是关于弱型端点结果的工作将继续进行，而弱型端点结果总是最难实现的。其他工作将集中在二阶偏微分算子的唯一延拓，特别是薛定谔算子，双曲和抛物微分算子的唯一延拓以及相关的特征函数展开式。一个新的工作领域将涉及低维曲面上函数的Randon变换或平均的最大估计。有趣的是，在这方面最困难的问题发生在较低的层面，最近有相当大的活动。

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.