Averaging, spectral multipliers, sparse domination and subelliptic operators

该项目侧重于谐波分析，这是数学分析学科中的一个广阔领域。谐波分析关注的是函数或信号作为基本波的叠加的表示，以及傅里叶级数和傅里叶变换概念的研究和推广。它的方法在理解自然科学和工程现象方面得到了广泛的应用。谐波分析为这些学科提供了有效的数学工具，并有助于将看似不相关的领域统一起来。主要目标是扩展谐波分析中当前的数学工具箱，以促进对最终有利于应用的更深层次的理论理解。研究生的研究指导是该项目重要的教育组成部分。首席研究员将从事谐波分析方面的几个项目。项目的第一部分讨论了某些曲线上的平均的精确正则性和相关的极大算子在Lebesgue空间中的有界性。模型的情况往往是卷积算子，但最终的目标是了解更一般的非卷积平均算子。项目的第二部分试图用部分的，可能是分形的，扩张集来约束极大函数，并理解不同的维度概念是如何与这些极大算子的勒贝格空间有界性相关的。项目的第三部分处理海森堡群上的乘子变换（某个亚椭圆算子的函数）。通过傅里叶变换将它们与波算子连接起来，波在海森堡群上传播的精细结构在乘法器问题中起着重要的作用。这个项目的第四部分是关于稀疏支配不平等；有人试图扩大理论的范围，以涵盖端点情况。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 focuses on harmonic analysis, a vast area within the mathematical discipline of analysis. Harmonic analysis is concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms. Its methods have found wide applications in understanding phenomena in the natural sciences and engineering. Harmonic analysis provides efficient mathematical tools for these disciplines and contributes to the unification of seemingly unrelated areas. A main objective is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding that will ultimately be beneficial for applications. The mentoring of graduate students in research is an important educational component of the project.The principal investigator will work on several projects in harmonic analysis. The first part of the project is concerned with the precise regularity properties of certain averages over curves and the boundedness of associated maximal operators in Lebesgue spaces. The model cases tend to be convolution operators but eventually the goal is to understand more general non-convolution averaging operators. A second part of the project seeks to bound maximal functions with partial, possibly fractal, dilation sets, and to understand how various notions of dimensions are relevant for the Lebesgue space boundedness properties of such maximal operators. A third part of the project deals with multiplier transformations on the Heisenberg group (functions of a certain subelliptic operator). Via a Fourier transform they are connected to wave operators, and the fine structure of wave propagation on the Heisenberg group plays an important role in the multiplier problem. A fourth part of the project is about sparse domination inequalities; one seeks to expand the scope of the theory to cover endpoint situations.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.