Problems Related to Fourier Restriction Estimates

该项目围绕谐波分析中的几个基本问题展开，谐波分析是一个源于傅立叶级数和傅立叶变换研究的领域，与偏微分方程、数论、几何测量理论以及信号处理和压缩感知等实际应用密切相关。傅里叶变换将时间函数分解为不同的频率分量，类似于将和弦表示为其组成音符的音高。谐波分析研究的是时间信息和频率信息如何相互作用。一个已经研究了几十年的基本问题（即傅里叶限制问题）是傅里叶变换的频率支持几何如何决定原始函数在时间上的大小。这种关系的各种量化被称为傅里叶限制估计，研究起来极具挑战性。即使是最简单的几何物体，如球体和抛物面，仍有许多问题有待解决。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数论中的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），Schrödinger和波动方程解的存在性和生长性，以及Diophantine方程解的个数。本课题研究了与傅里叶限制估计有关的几个问题。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里叶限制猜想。这种方法探索了函数时频分解的代数结构，并在理论上得到了许多最先进的结果。其次，首席研究员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Schrödinger或波动方程的散度集和距离集问题（关于集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由于分形度量的存在而导致缺乏一个关键的工具（正交性）。这里有一个有趣的方法，通过研究分形测度的行为来直接解决距离集问题。最后，首席研究员想要探索最近开发的一种称为稀疏支配的工具在约束理论中的作用。这种方法源于奇异积分理论，是一种将原来的连续、展开算子的研究简化为一类更简单的并进、正、局部算子的研究的方法。这种方法在奇异积分理论中非常有用，已成为现代描述算子的一种观点。首席研究员计划进一步研究与限制估计相关的算子的稀疏边界和Kakeya性质，如Bochner-Riesz乘子，沿流形的奇异积分，方向极大算子和多参数奇异积分算子。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.