Decoupling Theory, Time-Frequency Analysis and Related Oscillatory Integrals

解耦理论、时频分析及相关振荡积分

• 批准号: 1800274
• 负责人: Shaoming Guo
• 金额: $ 10.9万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800274&HistoricalAwards=false
• 关键词: Decoupling; Theory; Time; Frequency; Analysis

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.

该项目涉及调和分析和解析数理论领域的工作。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有关整数的问题。调和分析源于傅立叶的工作，他研究了一般函数可以用更简单的三角函数的和来表示的方式。它在数学物理、信号处理、数字图像处理等领域有着广泛的应用。该项目涉及与Radon变换和X射线变换相关的一系列调和分析问题。对这些变换的深入了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，拟议的项目将促进调和分析和解析数理论之间的互动。主要研究人员建议利用最近发展起来的调和分析工具来研究解析数论中的几个著名的公开问题。该项目涉及三个方向的工作，分别侧重于解耦理论、时频分析以及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，这意味着这些系统的整数解的数量将有尖锐的上界。在这个问题上的进展将允许用更少的线性形式来表示给定次数的每个多项式。在时频分析的指导下，主要研究者建议研究著名的齐格蒙德猜想的一个重要特例。这一猜想表明与平面Lipschitz向量场相关的极大算子在平方可积函数空间上是弱有界的。首席研究人员建议进一步假设沿平面上每条垂直线的矢量场是恒定的，以研究这一猜想。在振荡积分领域提出的一个问题旨在证明平均算子在每一维的矩曲线上的精确的Soblev正则性估计。由首席研究员和合作者建立的尖锐的脱钩不平等将为脱钩理论提供必要的工具。该奖项反映了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。