Problems Related to Fourier Restriction Estimates

本项目围绕调和分析中的几个基本问题展开，调和分析是一个起源于傅里叶级数和傅里叶变换研究的领域，与偏微分方程、数论、几何测量理论以及信号处理和压缩传感等实际应用密切相关。傅里叶变换将时间的函数分解为不同的频率分量，类似于音乐和弦如何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是如何相互作用的。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的几何形状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立叶限制估计，并且被证明是极具挑战性的研究。即使是最简单的几何物体，如球体和抛物面，许多问题仍然悬而未决。限制估计也很有趣，因为它们与许多其他问题有关，无论是在分析内部还是外部。众所周知，在数论中，约束估计可用于研究Kakeya猜想、薛定谔方程和波动方程解的存在性和增长性以及丢番图方程的解的个数等问题。首先，主要研究人员打算用多项式方法进一步研究抛物面和锥面的傅里叶限制猜想。这种方法探索了函数的时频分解的代数结构，并在理论上证明了它在获得许多最先进的结果方面是非常强大的。第二，主要研究人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)，并将它们应用于估计薛定谔方程或波动方程的散度集，以及距离集问题(关于集的大小如何决定其距离集的大小)。这个方向的主要困难是缺乏一个重要的工具(正交性)，这是由于存在分形量造成的。在这里，一种有趣的方法将通过研究分形度量的行为来直接攻击距离集问题。最后，首席研究员想要探索最近发展起来的一种叫做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、展开算子的研究简化为一类简单得多的并矢正局部算子的方法。这种方法在奇异积分理论中是非常有用的，并已成为描述算子的一种现代观点。首席研究员计划进一步研究与约束估计有关的具有Kakeya性质的算子的稀疏界，如Bochner-Riesz乘子、流形上的奇异积分、方向极大算子和多参数奇异积分算子。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.