Decoupling Theory, Time-Frequency Analysis and Related Oscillatory Integrals

该项目涉及调和分析和解析数论领域的工作。解析数论是数论的一个分支，它利用数学分析中的工具来解决整数问题。调和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法。它在包括数学物理、信号处理和数字图像处理在内的各个领域都有应用。该项目涉及一系列与Radon变换和X射线变换有关的谐波分析问题。对这些变换的深入理解将在计算机断层扫描（CT）、声纳技术和雷达技术等领域产生许多应用。另一方面，拟议的项目将刺激调和分析和解析数论之间的相互作用。主要研究者建议研究一些著名的开放问题解析数论，通过最近开发的工具调和分析。该项目涉及三个方向的工作，分别侧重于解耦理论，时频分析，并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度的Parsell-Vinogradov系统。我们的目标是建立某些尖锐的解耦不等式，这将意味着这些系统的整数解的数量上界尖锐。在这个问题上的进展将允许通过较少的线性形式来表示给定次数的每个多项式。在时频分析的方向上，主要研究者提出研究著名的Zygmund猜想的一个重要特例。这个猜想指出，与平面Lipschitz向量场相关联的极大算子在平方可积函数空间上是弱有界的。主要研究者建议通过进一步假设向量场沿着平面上的每一条垂直线沿着是常数来研究该猜想。一个问题，提出了在该地区的振荡积分的目的是证明尖锐的Sobolev正则性估计的平均算子沿着的时刻曲线在每一个维度。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NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.