Topics in Harmonic Analysis: Maximal Functions, Singular Integrals, and Multilinear Inequalities

本计画针对目前谐波分析中的几个热门议题进行研究。谐波分析的理论成果和方法广泛应用于科学、工程和技术的各个领域，如数字信号处理、医学成像、流体力学和数据压缩。谐波分析的基础研究致力于丰富这些学科向前发展所需的数学工具箱，并提供连接科学和数学看似无关的领域的统一观点。相当感兴趣的是各种积分变换的研究，如希尔伯特变换，它可以被理解为作用于给定信号的频率滤波器，是奇异积分算子的典型例子。其他对这个项目很重要的数学运算符是傅里叶变换，它将信号分解为频率分量，极大函数和各种Radon型变换。Radon型变换的基本示例是X射线变换，其用于计算机断层摄影应用中。总之，该项目的科学目标是促进对这些数学对象及其概括的理论理解。该项目还将组织一系列在线研讨会，编写一本关于本科生分析、课程改进和学生培训的书，工作的具体重点将放在真实的和离散谐波分析的各种主题上。第一个主题是关于欧氏空间和海森堡群上与限制扩张集相关的球面极大算子。这项工作将集中在建立尖锐的勒贝格空间映射属性依赖于相关的膨胀集的分形几何。第二个主题涉及离散类似物的经典运营商在调和分析有关的最大调制奇异积分的斯坦-Wainger型。这是由遍历理论的基本原理所激发的，并将涉及数论和分析技术的结合。第三个主题涉及多线性奇异积分与Radon型行为，如三角希尔伯特变换曲率。特别感兴趣的是相关的多线性平滑不等式。这样的不等式在算术组合学中有直接的应用。此外，还将对Carleson型算子的有界性、汉明立方体的分析、限制和解耦理论等相关问题进行研究。该奖项反映了NSF的法定使命，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 concerns research on several topics of current interest in harmonic analysis. Theoretical results and methods from harmonic analysis are widely applicable in a diverse range of areas in science, engineering and technology such as digital signal processing, medical imaging, fluid mechanics, and data compression. Foundational research in harmonic analysis endeavors to enrich the mathematical toolbox that these disciplines require to move forward and to provide unified perspectives that connect seemingly unrelated fields of science and mathematics. Of considerable interest is the study of various integral transforms such as the Hilbert transform, which can be understood as a frequency filter acting on a given signal and is a prototypical example of a singular integral operator. Other mathematical operators important to this project are the Fourier transform, which decomposes a signal into its frequency components, maximal functions and various Radon-type transforms. A basic example of a Radon-type transform is the X-ray transform, which is used in computerized tomography applications. In summary, the scientific goal of the project is to contribute to an improved theoretical understanding of these mathematical objects and their generalizations. The project will further entail organization of an online seminar series, writing a book on undergraduate analysis, curriculum improvement, and training of students.The specific focus of the work will be on a variety of topics in real and discrete harmonic analysis. The first topic concerns spherical maximal operators associated with restricted sets of dilations on Euclidean spaces and Heisenberg groups. This work will be focused on establishing sharp Lebesgue space mapping properties dependent on the fractal geometry of the associated dilation sets. A second topic concerns discrete analogues of classical operators in harmonic analysis related to maximally modulated singular integrals of Stein-Wainger type. This is motivated by fundamental conjectures in ergodic theory and will involve a combination of techniques from number theory and analysis. The third topic concerns multilinear singular integrals with Radon-type behavior such as triangular Hilbert transforms with curvature. Of particular interest are associated multilinear smoothing inequalities. Such inequalities have direct applications in arithmetic combinatorics. In addition, work will be conducted on related questions such as boundedness of Carleson-type operators, analysis on the Hamming cube and questions related to restriction and 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.