Topics in Harmonic Analysis

数学分析学科的方法在理解自然科学和工程中的物理现象方面得到了广泛的应用。该项目关注谐波分析的主题，旨在为这些学科提供有效的数学工具，并有望促进看似不相关的领域的统一。特别有趣的是研究各种积分变换，如傅里叶变换和x射线变换，以及它们的一些相关变换。傅里叶变换将一个信号（数学上是一个函数）分解成组成它的频率。它已被发现与解决科学和工程中出现的许多数学问题有关，并且可以被视为一类更大的振荡积分算子的特殊情况。x射线变换是一个算子，它赋予一个函数在直线上的积分。它与医学成像中的问题有关，可以被视为一类更大的Radon型变换和平均算子的特殊情况。该项目的一个主要目标是扩展谐波分析中现有的数学工具箱，以促进对这些积分变换及其推广的更深入的理论理解。首席研究员将从事谐波分析方面的几个项目。第一个项目是将解耦理论应用于平均算子和广义Radon变换的正则性问题，以及相关极大函数的有界性问题。第二个项目处理Hardy-Sobolev型空间中Haar展开式的一个新的乘数问题，其中Haar基不是无条件的。第三个项目研究海森堡群上Kohn Laplacian的谱乘子和相应的勒贝格空间乘子变换的行为。提出了一种新的时空估计的证明方法，用于证明与Kohn拉普拉斯算子相关的波动方程解的时空估计，并用它们来约束乘子算子。主要研究者还将从事其他项目，涉及Bochner-Riesz means的几乎处处收敛性，应用于稀疏控制结果的极大函数的局部改进不等式，以及多线性奇异积分算子的有界性。这个项目包括指导研究生。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。

Methods from the mathematical discipline of analysis have found wide applications in understanding physical phenomena in the natural sciences and engineering. This project is concerned with topics in harmonic analysis that are designed to provide efficient mathematical tools for these disciplines and that are expected to contribute to the unification of seemingly unrelated areas. Of particular interest is the study of various integral transforms, such as the Fourier and the X-ray transforms, and some of their relatives. The Fourier transform decomposes a signal (mathematically a function) into the frequencies that make it up. It has been found to be relevant for solving many mathematical problems arising in science and engineering and can be regarded as a special case of a larger class of oscillatory integral operators. The X-ray transform is an operator that assigns to a function its integral over lines. It is relevant to problems in medical imaging and can be considered as a special case of a larger class of Radon type transforms and averaging operators. A main goal of the project is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding of these integral transforms and their generalizations.The principal investigator will work on several projects in harmonic analysis. The first project is concerned with the application of decoupling theory to regularity questions for averaging operators and generalized Radon transforms, and to boundedness problems for associated maximal functions. The second project deals with a new multiplier problem for Haar expansions in spaces of Hardy-Sobolev type, in ranges where the Haar basis is not unconditional. The third project investigates spectral multipliers for the Kohn Laplacian on the Heisenberg group and the behavior of the corresponding multiplier transformations on Lebesgue spaces. It is proposed to prove new space-time estimates for solutions of the wave equation associated with the Kohn Laplacian and to use them to bound the multiplier operators. The principal investigator will also work on other projects, related to almost everywhere convergence of Bochner-Riesz means, local improving inequalities for maximal functions with application to sparse domination results, and boundedness of multilinear singular integral operators. The project involves mentoring of graduate students.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.