Fourier Analysis on Bounded Domains

本提案涉及许多关于傅里叶变换的核心问题，以及它们如何在有界域上或更一般地说，具有边界的黎曼流形上表现出来。在欧几里得空间或平面上的傅里叶变换的背景下，诸如限制定理、Strichartz估计（波和薛定谔方程的时空可积性估计）和Bochner-Riesz means等主题一直是人们非常感兴趣的主题。然而，这些理论如何在有限的领域中发挥作用仍然存在许多问题。这里可以考虑拉普拉斯函数的本征函数，用本征函数簇上的可积性估计代替球的限制定理。在这方面，PI打算探索这些簇对区域内曲线的限制的可积性估计。在Strichartz估计的情况下，可以用这种限制理论或基于参数的方法进行类比。后一种方法目前正被PI和他的合作者用来获得一般有界域的结果。对于具有特定几何结构（如单位圆盘或单位球）的歧管，将对这些估计进行改进。最后，涉及到外部域的Strichartz估计的相关问题将随着非线性方程的应用而被研究。傅里叶分析在数学和物理理论的发展中仍然是一个重要的因素。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。这里所进行的研究有望在波动现象和模拟波动的方程的研究中得到若干应用。这些调查将有助于深入了解障碍物的存在和形状如何影响波浪的发展，这是一个非常有趣的科学课题。

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.