Symmetry Parameter Analysis of Singular Integrals

奇异积分的对称参数分析

基本信息

批准号：
2054863
负责人：
Brett Wick
金额：
$ 19.76万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054863&HistoricalAwards=false
关键词：
Symmetry Parameter Analysis Singular Integrals

项目摘要

Harmonic Analysis is the branch of mathematics concerned with the rigorous description of signals (functions) and of their processing (operators). Examples of signals are sound, images, time series and weather data. Such signals are analyzed via the overlaying (superposition) of basic harmonics of well-specified duration, intensity and frequency. These basic harmonics are functions called wavelets. Image or audio denoising, compression, or pattern recognition are accomplished by filter processing, which refers to a suitable superposition of each wavelet after the action of the filter on it. This is also known as the time-frequency method. A particular concrete example of a re-construction process is used in tomographic imaging, where the shape of a solid body is re-composed from samples of the body along one or two-dimensional rays of penetrating waves, which can be mathematically described as lines or planes in three dimensional space. One component of this mathematics research project focuses on a new family of methods for the wavelet description of the class of singular integral operators, arising for instance in the time-frequency analysis of highly oscillatory signals. Another component of this research project is concerned with the mathematical properties of sampling solid objects along lines or planes. The integrated broader impact activities focus on strengthening the pool of socioeconomically disadvantaged, ethnical minority students (underrepresented groups) in graduate degrees in mathematics and improving retention. Activities connected to training and mentoring of graduate students in Analysis and topical dissemination of knowledge will also be carried out. The broad aim of the first circle of questions is to produce representation formulas for classes of singular integrals in terms of so-called model operators conserving the same invariance structure. This paradigm applies to Zygmund-type operators and modulation invariant operators akin to the bilinear Hilbert transform, both of which are out of reach for dyadic-probabilistic methods. Concrete applications come from elliptic and dispersive PDE, operator theory and quasi-conformal mappings. The second related family of questions is motivated by pointwise convergence of bilinear ergodic averages for Banach-valued functions, a celebrated theorem by Bourgain in the scalar case. The approach is based on Banach-valued variational estimates for the truncated bilinear Hilbert transform. The central item in a further set of questions concerning directional singular integrals is a version of the Kakeya maximal estimate where tubular averages are replaced with averages over singular line segments, and more generally, n-dimensional subspaces. One source of motivation is the connection with Fourier restriction in higher codimensions. The methods involve algebra-geometric techniques such as polynomial partitioning on manifolds.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.

谐波分析是与信号（函数）及其处理（运算符）的严格描述有关的数学分支。信号的示例是声音，图像，时间序列和天气数据。通过明确指定的持续时间，强度和频率的基本谐波的叠加（叠加）分析此类信号。这些基本谐波是称为小波的函数。图像或音频降解，压缩或图案识别是通过过滤器处理来完成的，这是指滤波器在其上作用后的每个小波的合适叠加。这也称为时频法。重建过程的一个特定具体示例用于层析成像中，其中固体体的形状沿着一个或二维的穿透性波的射线从体的样品中重新组合，可以在数学上描述为三维空间中的线条或平面。该数学研究项目的一个组成部分集中于一个新方法，用于奇异积分运算符类的小波描述，例如在高度振荡信号的时频分析中出现。该研究项目的另一个组成部分涉及沿线或平面对实体对象进行采样的数学特性。综合的广泛影响活动着重于加强在数学研究生学位和改善保留率的研究生学位的社会经济处境不利的少数民族学生（代表性不足的群体）。还将开展与研究生培训和指导有关分析和知识局部传播的活动。第一个问题圈子的广泛目的是在所谓的模型运算符方面生成奇异积分类别的表示公式。该范式适用于Zygmund型操作员和类似于双线性希尔伯特变换的调制算子，这两种操作员都无法实现二元型良性方法。具体应用来自椭圆形和分散PDE，操作员理论和准符合形式映射。第二个相关的问题家族是由双线性ergodic平均值的Banach值函数的偶然汇合而动机的，Banach值得函数，这是Bourgain在标量案例中被庆祝的定理。该方法基于截短的双线性希尔伯特变换的Banach价值变异估计值。有关定向奇异积分的另一组问题中的中心项目是Kakeya最大估计值的一种版本，其中管状平均值被单个线段段上的平均值替换，更一般而言，n维子空间。动机的一种来源是在更高的编码中与傅立叶限制的联系。这些方法涉及代数几何技术，例如在流形上进行多项式分区。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的审查标准，被认为值得通过评估来获得支持。