Topics in Harmonic Analysis

数学分析学科中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。这个项目涉及调和分析的主题，旨在为这些学科提供有效的数学工具，并有望促进看似无关的领域的统一。特别感兴趣的是各种积分变换的研究，例如傅立叶变换和X射线变换，以及它们的一些相关的变换。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。它已被发现与解决许多科学和工程中出现的数学问题有关，并可视为更大类振荡积分算子的特例。X射线变换是一种运算符，它将其积分赋给一个函数。它与医学成像中的问题有关，可视为更大类Radon型变换和平均算子的特例。该项目的一个主要目标是扩展当前调和分析的数学工具箱，以促进对这些积分变换及其推广的更深入的理论理解。主要研究人员将在调和分析的几个项目上工作。第一个项目是关于解耦理论在平均算子和广义Radon变换的正则性问题以及相关极大函数的有界性问题上的应用。第二个项目讨论了Hardy-Sobolev型空间中Haar展开的一个新的乘子问题，其中Haar基不是无条件的。第三个项目研究了Heisenberg群上Kohn Laplace算子的谱乘子以及相应的乘子变换在勒贝格空间上的行为。证明了与Kohn Laplace算子有关的波动方程的解的新的时空估计，并利用它们来约束乘子算子。主要研究人员还将致力于其他项目，涉及Bochner-Riesz平均的几乎处处收敛，极大函数的局部改进不等式及其在稀疏控制结果中的应用，以及多线性奇异积分算子的有界性。该项目包括对研究生的指导。这一奖项反映了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。

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.