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

该项目涉及对当前谐波分析中几个主题的研究。谐波分析的理论结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努力丰富这些学科向前发展所需的数学工具箱，并提供统一的观点，将看似无关的科学和数学领域联系起来。人们对各种积分变换的研究非常感兴趣，比如希尔伯特变换，它可以被理解为作用于给定信号的频率滤波器，是奇异积分算子的一个典型例子。其他对这个项目很重要的数学运算是傅里叶变换，它把信号分解成它的频率分量、极大函数和各种氡型变换。氡型变换的一个基本例子是用于计算机断层扫描应用的x射线变换。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。该计划将进一步包括组织一个在线系列研讨会，撰写一本关于本科生分析的书，改进课程和培训学生。具体的工作重点将是各种主题的实际和离散谐波分析。第一个主题是与欧氏空间和海森堡群上膨胀的限制集相关的球面极大算子。这项工作将集中于建立依赖于相关膨胀集的分形几何的尖锐勒贝格空间映射性质。第二个主题涉及与Stein-Wainger型最大调制奇异积分相关的调和分析中经典算子的离散类似。这是由遍历理论中的基本猜想所激发的，将涉及数论和分析技术的结合。第三个主题涉及具有radon型行为的多线性奇异积分，如带曲率的三角形希尔伯特变换。特别令人感兴趣的是相关的多线性平滑不等式。这种不等式在算术组合中有直接的应用。此外，还将对carleson型算子的有界性、Hamming立方体的分析以及约束解耦理论等相关问题进行研究。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 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.