Variable Coefficient Fourier Analysis

变系数傅立叶分析

• 批准号: 1361476
• 负责人: Christopher Sogge
• 金额: $ 36万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361476&HistoricalAwards=false
• 关键词: Variable; Coefficient; Fourier; Analysis

The principal investigator will study several problems in geometric Fourier analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated to a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits. Similar phenomena arise for manifolds, and we wish to study precisely how their shapes, such as how they are curved, affect the resulting eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the project will also study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity.Among the specific problems in the project, the principal investigator desires to obtain improved estimates for the nodal sets (zero sets) of eigenfunctions. There is a conjecture of Yau asserting that the size of this codimension-one set should be comparable to its frequency. Although it has been fully settled in the real analytic setting, less is known for smooth manifolds. The principal investigator has already shown that there are connections between this problem and Lebesgue space estimates for eigenfunctions that can detect certain types of concentration. Several problems in the project involve developing this active field. The principal investigator would also like to obtain improved bounds for so-called period integrals of eigenfunctions under the assumption of negative curvature. This assumption is known to be necessary, and the problem measures the random cancellation of eigenfunctions along geodesics. There are connections with this problem and analytic number theory, but to date the results that the principal investigator has recently obtained using harmonic analysis are the best known. He would like to combine them with number theory techniques to try to obtain improved bounds. There are connections between these problems and other problems that the project will study involving estimates for wave equations ("Strichartz estimates") and microlocal analysis, especially propagation of singularities.

首席研究员将研究几何傅立叶分析中的几个问题。 这些问题的设置涉及二维或二维以上的几何流形。 与给定流形相关的是称为特征函数的基本对象。 这些是流形振动的基本模式，它们是熟悉的圆三角函数的高维类似物。 乐器设计师很清楚，鼓或弦乐器音板的形状会影响其省略的基本音调。 流形也会出现类似的现象，我们希望精确研究它们的形状（例如它们的弯曲方式）如何影响所得的本征函数。 正如在音乐中一样，随着频率变得越来越大，人们特别期望不同的形状和几何形状在基本振动模式的行为中变得更加明显。 这些特征函数是类似于波动方程的微分方程的解，该项目还将研究涉及它的类似问题。 总体主题是研究波动方程的解如何受到其物理背景的影响，例如是否存在黑洞或背景是否变得非常接近无穷大附近的真空。在该项目的具体问题中，主要研究人员希望获得对本征函数的节点集（零集）的改进估计。 丘有一个猜想，认为这个余维一集的大小应该与其频率相当。 尽管它已经完全解决了真实的解析环境，但对于光滑流形却知之甚少。 首席研究员已经表明，这个问题与可以检测某些类型浓度的特征函数的勒贝格空间估计之间存在联系。 该项目中的几个问题涉及开发这个活跃的领域。 主要研究者还希望在负曲率假设下获得所谓的本征函数周期积分的改进界限。 众所周知，这种假设是必要的，并且该问题测量沿着测地线的本征函数的随机抵消。 这个问题和解析数论有联系，但迄今为止，主要研究者最近使用调和分析获得的结果是最著名的。 他希望将它们与数论技术结合起来，试图获得改进的界限。 这些问题与该项目将研究的其他问题之间存在联系，涉及波动方程的估计（“Strichartz 估计”）和微局域分析，特别是奇点的传播。