Fourier Analysis on Bounded Domains

这个建议解决了许多关于傅立叶变换的核心问题，以及它们如何在有界域上表现出来，或者更一般地，带边界的黎曼流形。在欧几里德空间或平坦环面上的傅里叶变换的背景下，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平均等主题一直是人们非常感兴趣的主题。然而，关于这些理论应该如何在有界域上发挥作用，仍然存在许多问题。在这里，考虑拉普拉斯的本征函数，并用本征函数簇上的可积性估计来代替球面的限制定理可能是合适的。在这一点上，PI打算探索这些簇对区域中曲线的限制的可积性估计。在Strichartz估计的情况下，可以用一些限制理论进行类比，或者可以采用基于参数线的方法。后一种方法目前被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的流形，如单位盘或单位球，将对这些估计进行改进。最后，我们将研究与外部区域中的Strichartz估计相关的问题以及在非线性方程中的应用。傅立叶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式的理解。在此进行的研究有望在研究波动现象和模拟波动现象的方程方面有几个应用。这些研究应该能洞察障碍物的存在和形状如何影响波浪的发展，这是一个极具科学意义的课题。

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.