Restriction Estimates and General Oscillatory Integrals

这项研究项目涉及谐波分析，也称为傅立叶分析。调和分析是一门关于傅里叶变换的学科，它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，我们有效地将函数写成单色波(只有一个频率的波)的叠加。特别令人感兴趣的是以下限制类型问题：如果一个函数的频率位于特定的限制集中，那么在逐点意义上或在某种平均意义上，函数能有多大？人们已经发现，如果限制频率集是“曲线”的，例如单位球面，则可以得到/期望关于该函数的非常不平凡的估计。这对于理解某些自然偏微分方程式(如薛定谔方程或波动方程)所支配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。例如，一旦人们很好地理解了频率为完美10次方的波的性质，他们就可以回答关于将自然数表示为几个完美10次方的和的各种问题。在这个项目中，PI建议研究限制类型问题：如果我们在频率空间中有一个子集M(通常是曲线子流形或分形集)，并且在物理空间中有一些度量，我们希望当一个函数的频率支持在给定的M时，函数关于给定度量的某个范数有界。一个提出的方向是斯坦的限制猜想。其中，度量是勒贝格度量，流形是单位抛物面。PI还对当度量是分形度量、当M是矩流形或分形集时的情况感兴趣。对于大多数这类问题，最优估计远未得到很好的理解。这项研究的目标将是在上述背景下获得新的估计(即具有改进的指数的估计)。特别是，预计当这个项目发展时，可以得到对Stein猜想的改进。对于所提出的方法，PI预期解析(尺度上的归纳、解耦和精化的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)的子集。几何测度论(径向投影论等)工具可以发挥作用。他还预计，谐波分析和附近地区之间将出现新的联系。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 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.