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。谐波分析是一门关于傅里叶变换的学科，傅里叶变换将函数分解成其组成频率。换句话说，通过傅里叶变换，我们有效地将函数写为单色波（只有一个频率的波）的叠加。特别感兴趣的是以下限制类型的问题：多大可以是一个函数，在逐点意义上或在一些平均意义上，如果它的频率生活在一个特定的限制集？人们已经发现，如果受限频率集是“弯曲的”，例如是单位球面，则可以获得/期望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，限制型问题也是理解自然数某些行为的关键。例如，人们一旦对频率为完全10次方的波的性质有了很好的理解，就可以回答将自然数表示为几个完全10次方之和的各种问题。在本项目中，PI提出研究限制型问题：如果我们有一个子集M（通常是弯曲的子流形或分形集）在频率空间和一些措施在物理空间，我们希望当函数的频率支集在给定的M中时，该函数关于给定的测度的某个范数有界。一个建议的方向是斯坦的限制猜想，其中的措施只是勒贝格措施和流形是单位抛物面。PI也感兴趣的情况下，当措施是一个分形措施，或当M是一个时刻流形或分形集。对于大多数这类问题，最优估计还远未得到很好的理解。本研究的目标是在上述环境中获得新的估计（即具有改进指数的估计）。特别是，预计改进斯坦因猜想可以得到当这个项目的发展。对于所提出的方法，PI预期的子集分析（归纳尺度，解耦和细化的Hohartz型推理），代数（多项式方法，微分几何和真实的代数几何），组合（多线性Kakeya，和积理论等）。和几何测度理论（径向投影理论等）工具可以发挥作用。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.