Problems Related to Fourier Restriction Estimates

与傅里叶限制估计相关的问题

• 批准号: 1854148
• 负责人: Yumeng Ou
• 金额: $ 13.88万
• 依托单位: CUNY Baruch College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-21 至 2020-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854148&HistoricalAwards=false
• 关键词: Problems; Related; Fourier; Restriction; Estimates

This project revolves around several fundamental questions in harmonic analysis, which is a field stemming from the study of Fourier series and Fourier transform and is closely connected with partial differential equations, number theory, geometric measure theory, and real life applications such as signal processing and compressed sensing. The Fourier transform decomposes a function of time into different frequency components, similarly to how a music chord can be expressed as the pitches of its constituent notes. Harmonic analysis studies how the time information and the frequency information interact with each other. A fundamental question (i.e. Fourier restriction problem) that has been studied for decades is how the geometry of the frequency support of the Fourier transform dictates the size of the original function in time. Various quantifications of such relations are referred to as Fourier restriction estimates and turn out to be extremely challenging to study. Even for the simplest geometric objects such as spheres and paraboloids, many questions are still wide open. Restriction estimates are interesting also due to their connection with many other problems, within or outside analysis. It is well known that restriction estimates can be applied to study the Kakeya conjecture (on the minimum area of a set containing a unit line segment in each direction), the existence and growth of solutions to Schrödinger and wave equations, and the number of solutions to Diophantine equations in number theory.This project studies several problems related to Fourier restriction estimates. First, the principal investigator intends to further the investigation of the Fourier restriction conjecture for the paraboloid and the cone via the polynomial method. This method explores the algebraic structure of the time-frequency decomposition of the function, and has shown to be extremely powerful in obtaining many state-of-the-art results in the theory. Second, the principal investigator proposes to continue the study of weighted restriction estimates (i.e. when the Lebesgue measure is replaced with a fractal measure) and apply them to estimate divergence sets of the Schrödinger or wave equations and to distance set problems (on how the size of a set dictates the size of its distance set). The major difficulty in this direction is the lack of a crucial tool (orthogonality) caused by the presence of the fractal measure. Here an interesting approach will attack the distance set problems directly by studying the behavior of the fractal measure. Last, the principal investigator wants to explore the role in restriction theory of a recently developed tool called sparse domination. This method, arising from singular integral theory, is a way to reduce the study of the original continuous, spread out operator to that of a class of much simpler dyadic, positive, local operators. This method has shown to be extremely useful in singular integral theory and has become a modern view of point in describing operators. The principal investigator plans to further the study of sparse bounds of operators related to restriction estimates and with a Kakeya nature, such as the Bochner-Riesz multipliers, singular integrals along manifolds, directional maximal operators, and multi-parameter singular integral operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目围绕谐波分析中的几个基本问题，谐波分析是源于傅立叶级数和傅立叶变换研究的一个领域，与偏微分方程，数论，几何测量理论以及信号处理和压缩传感等真实的应用密切相关。傅里叶变换将时间函数分解为不同的频率分量，类似于音乐和弦如何被表示为其组成音符的音高。谐波分析研究时间信息和频率信息如何相互作用。一个基本的问题（即傅立叶限制问题），已经研究了几十年，是如何的几何形状的频率支持的傅立叶变换决定的大小的原始功能的时间。这种关系的各种量化被称为傅立叶限制估计，结果是非常具有挑战性的研究。即使是最简单的几何对象，如球体和抛物面，许多问题仍然是开放的。限制估计也很有趣，因为它们与分析内部或外部的许多其他问题有关。众所周知，限制估计可以应用于研究Kakeya猜想（关于在每个方向上包含单位线段的集合的最小面积）、薛定谔方程和波动方程解的存在性和增长性以及数论中丢番图方程解的个数。本项目研究与Fourier限制估计相关的几个问题。首先，主要研究者打算通过多项式方法进一步研究抛物面和圆锥的傅立叶限制猜想。这种方法探讨了函数的时频分解的代数结构，并已被证明是非常强大的，在获得许多国家的最先进的理论结果。其次，主要研究者建议继续研究加权限制估计（即当勒贝格测度被分形测度取代时），并将其应用于估计薛定谔方程或波动方程的发散集和距离集问题（关于集合的大小如何决定其距离集的大小）。在这个方向上的主要困难是缺乏一个关键的工具（正交性）所造成的分形措施的存在。在这里，一个有趣的方法将攻击的距离集问题直接通过研究分形测度的行为。最后，首席研究员希望探索最近开发的称为稀疏支配的工具在限制理论中的作用。这种方法源于奇异积分理论，是一种将原来的连续展开算子的研究减少到一类简单得多的并矢正局部算子的研究的方法。这种方法在奇异积分理论中已被证明是非常有用的，并已成为描述算子的现代观点。主要研究者计划进一步研究与限制估计有关的算子的稀疏边界，并具有Kakeya性质，如Bochner-Riesz乘数，沿沿着流形的奇异积分，方向极大算子，和多个该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查进行评估，被认为值得支持的搜索.

项目成果

Sparse domination and the strong maximal function

稀疏支配和强极大函数

• DOI: 10.1016/j.aim.2019.01.007
• 发表时间: 2019
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Barron, Alex;Conde-Alonso, José M.;Ou, Yumeng;Rey, Guillermo
• 通讯作者: Rey, Guillermo

On Falconer’s distance set problem in the plane

• DOI: 10.1007/s00222-019-00917-x
• 发表时间: 2018-08
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: L. Guth;A. Iosevich;Yumeng Ou;Hong Wang