Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

9401855 Smith该奖项支持专注于调和分析在双曲微分方程理论问题中的应用的数学研究。工作包括对流形上具有狄利克雷条件的波群的估计的发展，类似于在边界为凹的附加条件下得到的估计。在凸边界流形上估计失败，尽管在这种情况下存在多次反射测地线，产生沿边界行进的滑动射线。因此，找到估计失败的最坏情况的例子，并尝试为这种情况获得更新的估计是很有趣的。我们还会对Hardy空间中的傅里叶积分算子进行研究。本文将分析一种新的函数空间，在该空间上，零阶傅里叶积分算子的代数连续作用。某些与振荡问题相关的算子在标准Hardy空间上无界，在新空间上仍有界。首要任务之一是获得空间的交替表征。最后，我们将构造对角化波群的平方可积函数的框架。谐波分析结合了最能体现综合思想的数学元素。人们试图把复杂的问题分解成基本的组成部分。然后分析这些组件的基本特性。最后，通过组件的重组重构解。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，另一个是连续的。最近，小波理论为一些更经典的谐波分析方法增加了新的维度

9401855 Smith This award supports mathematical research focusing on applications of harmonic analysis to problems in the theory of hyperbolic differential equations. Work includes the development of estimates for the wave group with Dirichlet conditions on manifolds similar to that obtained with the additional condition that the boundaries be concave. Estimates fail on manifolds with convex boundaries although in these instances there exist multiply reflected geodesics producing gliding rays which travel along the boundary. It is therefore of interest to find worst case example where the estimates fail and try to obtain newer estimates for this situation. Work will also be done on the Hardy space for Fourier integral operators. Efforts will go into analyzing a new function space on which the algebra of Fourier integral operators of order zero act continuously. Certain operators associated with oscillatory problems are not bounded on the standard Hardy spaces will remain bounded on the new spaces. One of the first tasks is to obtain an alternate characterization of the space. Finally, work will be done constructing frames for square integrable functions which diagonalize the wave group. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.***