Restriction Estimates and General Oscillatory Integrals

本研究项目涉及谐波分析，也称为傅立叶分析。谐波分析是关于傅里叶变换的一门学科，傅里叶变换将一个函数分解成它的组成频率。换句话说，通过傅里叶变换，我们有效地将一个函数写成单色波（只有一个频率的波）的叠加。特别有趣的是以下限制类型的问题：如果一个函数的频率存在于一个特定的限制集中，那么它在点向意义上或在某种平均意义上可以有多大？人们已经发现，如果限制频率集是“弯曲的”，例如作为单位球，可以得到/期望得到关于函数的非常非平凡的估计。这对于理解某些自然偏微分方程（如薛定谔方程或波动方程）所决定的物理现象是有用的。此外，约束型问题也是理解自然数某些行为的关键。事实证明，例如，一旦人们很好地理解了频率为完全10次幂的波的性质，他们就可以回答关于将自然数表示为几个完全10次幂的和的各种问题。在这个项目中,π提出研究限制类型的问题:如果我们有一个子集M(通常是一个弯曲的子流形或分形集)的频率空间和物理空间的一些措施,我们想绑定一些函数的规范给定的测量时的频率支持这个函数在给定的M .猜想一个提出的方向是斯坦的限制措施是勒贝格测度和廖单位抛物面。PI还对度量是分形度量，或者M是矩流形或分形集的情况感兴趣。对于这种类型的大多数问题，最佳估计远没有得到很好的理解。本研究的目标是在上述设置中获得新的估计（即具有改进指数的估计）。特别是，预计在这个项目的发展中，可以得到Stein猜想的改进。对于所提出的方法，PI预计解析（尺度归纳，解耦和精细Strichartz类型推理），代数（多项式方法，微分几何和实代数几何），组合（多线性Kakeya，和积理论等）和几何测量理论（径向投影理论等）工具的子集可以发挥作用。他还预计谐波分析和附近地区之间将出现新的联系。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 research project concerns work in harmonic analysis, also known as Fourier analysis. Harmonic analysis is a subject about the Fourier transform, which decomposes a function into its constituent frequencies. In other words, by taking the Fourier transform we effectively write a function as a superposition of monochromatic waves (waves with only one frequency). Of particular interest are the following restriction type problems: How large can a function be, in the pointwise sense or in some averaged sense, if its frequency lives in a particular restricted set? People have found that if the restricted frequency set is "curved", for example being the unit sphere, very nontrivial estimates about the function can be obtained/expected. This is useful in understanding physics phenomena dictated by certain natural partial differential equations such as the Schrodinger equation or the wave equation. Moreover, restriction type problems also turn out to be the key to the understanding of certain behaviors of natural numbers. It turns out that, for example, once people understand well on properties on waves whose frequencies are perfect 10-th powers, they can consequently answer a variety of questions on representing a natural number as a sum of a few perfect 10-th powers.In this project, the PI proposes to study restriction type problems: If we have a subset M (usually a curved submanifold or a fractal set) in the frequency space and some measure in the physical space, we want to bound some norm of a function with respect of the given measure whenever the frequency support of that function is in the given M. One proposed direction is Stein's Restriction Conjecture, where the measure is just the Lebesgue measure and the manifold is the unit paraboloid. The PI is also interested in the situations when the measure is a fractal measure, or when M is a moment manifold or a fractal set. For most questions of this type, the optimal estimates are far from being well understood. The goal of this research would be to obtain new estimates (i.e. estimates with improved exponents) in the above setting. In particular, it is anticipated that improvements on Stein's conjecture can be obtained when this project develops. For the proposed approach, the PI anticipates a subset of analytic (induction on scales, decoupling and refined Strichartz type reasoning), algebraic (the polynomial method, differential geometry and real algebraic geometry), combinatorial (Multilinear Kakeya, sum-product theory, etc.) and geometric measure theoretic (radial projection theory, etc.) tools can come into play. He also anticipates emerging new connections between harmonic analysis and nearby areas will arise.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.