Decoupling Theory, Time-Frequency Analysis and Related Oscillatory Integrals

这个项目涉及谐波分析和解析数论领域的工作。解析数论是数论的一个分支，它利用数学分析的工具来解决整数问题。谐波分析起源于傅立叶的工作，他研究了一般函数可以用简单三角函数的和来表示的方式。它在数学物理、信号处理和数字图像处理等各个领域都有应用。本课题涉及到氡变换和x射线变换的一系列谐波分析问题。对这些转换的深入理解将在计算机断层扫描（CT）扫描、声纳技术和雷达技术等领域产生许多应用。另一方面，提出的项目将促进谐波分析和解析数论之间的相互作用。首席研究员建议使用调和分析中最近发展起来的工具来研究解析数论中几个著名的开放问题。该项目涉及三个方向的工作，分别关注解耦理论、时频分析和连接前两个主题的某些振荡积分。解耦理论中提出的一个问题涉及所有维的Parsell-Vinogradov系统。目标是建立某些尖锐的解耦不等式，这些不等式将暗示这些系统的整数解的数量的尖锐上界。这个问题的进展将允许用更少的线性形式表示给定次的每个多项式。在时频分析方向上，主要研究者提出研究著名的齐格蒙德猜想的一个重要特例。这个猜想说明了平面Lipschitz向量场的极大算子在平方可积函数空间上是弱有界的。首席研究员建议进一步假设平面上每条垂直线上的向量场都是恒定的，以此来研究这个猜想。在振荡积分领域提出了一个问题，旨在证明沿每维矩曲线的平均算子的尖锐Sobolev正则性估计。主要研究者和合作者建立的尖锐解耦不等式将为解耦理论提供必要的工具。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 involves work in the fields of harmonic analysis and analytic number theory. Analytic number theory is a branch of number theory which uses tools from mathematical analysis to solve problems about the integers. Harmonic analysis has its roots in the work of Fourier, who studied the way general functions may be represented by sums of simpler trigonometric functions. It has applications in various areas including mathematical physics, signal processing, and digital image processing. The project involves a series of problems in harmonic analysis related to Radon transforms and X-ray transforms. A deeper understanding of these transforms will produce a number of applications in areas such as computed tomography (CT) scan, sonar techniques, and radar techniques. On the other hand, the proposed project will stimulate interactions between harmonic analysis and analytic number theory. The principal investigator proposes to study a few famous open problems in analytic number theory, via tools recently developed in harmonic analysis. The project involves work in three directions, focusing separately on decoupling theory, time-frequency analysis, and certain oscillatory integrals that connect the previous two topics. One problem that is proposed in decoupling theory concerns Parsell-Vinogradov systems in all dimensions. The goal is to establish certain sharp decoupling inequalities that would imply sharp upper bounds on the number of integer solutions of these systems. Progress on this problem will allow a representation of every polynomial of a given degree by fewer linear forms. In the direction of time-frequency analysis, the principal investigator proposes to study an important special case of the famous Zygmund conjecture. This conjecture states that the maximal operator associated with a planar Lipschitz vector field is weakly bounded on the space of square-integrable functions. The principal investigator proposes to study the conjecture by further assuming the vector field to be constant along each vertical line on the plane. One problem that is proposed in the area of oscillatory integrals aims at proving sharp Sobolev regularity estimates for an averaging operator along the moment curve in every dimension. Sharp decoupling inequalities established by the principal investigator and collaborator will provide necessary tools from decoupling theory.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.