Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

9401855史密斯该奖项支持将调和分析应用于双曲型微分方程式理论问题的数学研究。工作包括对流形上具有Dirichlet条件的波群的估计的发展，类似于在边界是凹的附加条件下得到的估计。在具有凸边界的流形上估计失败，尽管在这些情况下存在多次反射的测地线，产生沿边界行进的滑动射线。因此，有兴趣找到估计失败的最坏情况的例子，并试图获得这种情况的最新估计。此外，还将研究傅立叶积分算子的Hardy空间。将努力分析一种新的函数空间，在该空间上零级傅立叶积分算子的代数连续作用。与振动问题相关的某些算子在标准Hardy空间上是不有界的，在新空间上仍然是有界的。首要任务之一是获得空间的另一种特征。最后，我们将为对角化波群的平方可积函数构造框架。调和分析结合了这些数学元素，最好地例证了综合的思想。一种是试图将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征。最后，通过组件的重新组合来重建解决方案。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，另一个代表连续分解。最近，小波理论为一些更经典的调和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.***