Fourier Analysis on Bounded and Exterior Domains

这一建议处理了傅里叶分析中的许多核心问题，以及它们如何在域上或更一般地说，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，限制定理、Strichartz估计（波和薛定谔方程的时空可积性估计）和局部平滑不等式等主题一直是人们非常感兴趣的主题。然而，关于这些理论如何在有限的或外部的领域中发挥作用，仍然存在许多问题。这里可以考虑拉普拉斯函数的特征函数，用特征函数簇上的可积性估计代替环面的限制定理。在这方面，PI打算探索这些簇对域内曲线的限制的估计。在Strichartz估计的情况下，PI和他的合作者正在使用基于参数的方法来获得一般域的结果。在某些情况下，通过使用相对较新的局部平滑估计，预计会有进一步的改进。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。这些研究将有助于深入了解硬边界表面的存在如何影响波的发展，这是一个具有重大科学意义的课题。

This proposal deals with many of the central questions in Fourier analysis and how they manifest themselves on 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 local smoothing inequalities have been subjects of great interest for quite some time. However, many questions remain on how these theories should play out on a bounded or exterior domain. Here it may be appropriate to think of eigenfunctions of the Laplacian, and replace restriction theorems for the torus with integrability estimates on clusters of eigenfunctions. In this regard, the PI intends to explore estimates on the restriction of these clusters to curves in the domain. In the case of Strichartz estimates, the PI and his collaborators are using a parametrix-based approach to obtain results for general domains. Further improvement is expected in certain contexts by making use of a relatively new family of local smoothing estimates.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 of a hard boundary surface influences the development of waves, a subject of great scientific interest.