Fourier Analysis on Bounded Domains

这个建议涉及许多关于傅立叶变换的中心问题，以及它们如何在有界域或更一般的有界黎曼流形上表现出来。 在欧氏空间或平坦环面上的傅立叶变换中，限制定理、波函数估计（波方程和薛定谔方程的时空可积性估计）和Bochner-Riesz平均值等主题在相当长的一段时间内一直是人们非常感兴趣的主题。然而，这些理论如何在有界域上发挥作用仍然存在许多问题。 在这里，我们可以考虑拉普拉斯算子的本征函数，用本征函数簇的可积性估计来代替球面的限制定理。 在这方面，PI打算探索这些集群对域中曲线的限制的可积性估计。 在Eschenhartz估计的情况下，类比可以得出一些限制理论或参数为基础的方法可以采用。后一种方法目前正在使用的PI和他的合作者，以获得一般有界域的结果。 对于具有特定几何结构的流形，如单位圆盘或单位球，将对这些估计进行改进。 最后，将沿着应用于非线性equations.Fourier分析仍然是数学和物理理论发展中的一个重要因素。 特别是，它加强了我们对数学物理中出现的偏微分方程的理解。 这里所进行的研究预计有几个应用程序的研究波动现象和方程的模型it. These调查应产生洞察力的存在和形状的障碍物如何影响波的发展，一个具有重大科学意义的主题。

This proposal deals with many of the central questions concerning Fourier transforms and how they manifest themselves on bounded domains or more generally, Riemannian manifolds with boundary. In the context of the Fourier transform on Euclidean space or the flat torus, topics such as restriction theorems, Strichartz estimates (space-time integrability estimates for wave and Schroedinger equations), and Bochner-Riesz means have been subjects of great interest for quite some time. However, many questions remain on how these theories should play out on a bounded domain. Here it may be appropriate to think of eigenfunctions of the Laplacian, and replace restriction theorems for the sphere with integrability estimates on clusters of eigenfunctions. In this regard, the PI intends to explore integrability estimates on restrictions of these clusters to curves in the domain. In the case of Strichartz estimates, analogies can be drawn with some of this restriction theory or a parametrix-based approach can be employed. This latter approach is currently being used by the PI and his collaborators to obtain results for general bounded domains. Improvements on these estimates will be pursued for manifolds with a specific geometric structure such as the unit disk or the unit ball. Finally, related problems involving Strichartz estimates in exterior domains will be examined along with applications to nonlinear equations.Fourier analysis continues to be a significant factor in the development of both mathematical and physical theories. In particular, it strengthens our understanding of the partial differential equations that arise in mathematical physics. The research pursued here expects to have several applications to the study of wave phenomena and the equations which model it. These investigations should yield insight on how the presence and shape of obstacles influence the development of waves, a subject of great scientific interest.