Averaging, spectral multipliers, sparse domination and subelliptic operators

平均、谱乘数、稀疏支配和次椭圆算子

• 批准号: 2054220
• 负责人: Andreas Seeger
• 金额: $ 27.6万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054220&HistoricalAwards=false
• 关键词: Averaging; spectral; multipliers; sparse; domination

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.

这个项目的重点是谐波分析，一个广阔的领域内的数学学科的分析。谐波分析涉及将函数或信号表示为基本波的叠加，以及研究和推广傅里叶级数和傅里叶变换的概念。它的方法在理解自然科学和工程中的现象方面有着广泛的应用。调和分析为这些学科提供了有效的数学工具，并有助于统一看似无关的领域。 一个主要的目标是扩大目前的数学工具箱谐波分析，有助于更深入的理论理解，最终将有利于应用。指导研究生的研究是该项目的重要教育组成部分。首席研究员将从事谐波分析的几个项目。该项目的第一部分是关于精确的正则性的某些平均曲线和相关的极大算子在勒贝格空间的有界性。模型案例往往是卷积运算符，但最终的目标是理解更一般的非卷积平均运算符。该项目的第二部分旨在约束最大函数与部分，可能分形，膨胀集，并了解各种概念的尺寸是如何相关的勒贝格空间有界性等最大运营商。该项目的第三部分涉及海森堡群上的乘子变换（某个次椭圆算子的函数）。通过傅立叶变换，它们被连接到波算子，并且在海森堡群上的波传播的精细结构在乘子问题中起着重要作用。该项目的第四部分是关于稀疏支配不平等;一个是寻求扩大理论的范围，以涵盖端点的情况。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

• DOI: 10.1016/j.jfa.2022.109775
• 发表时间: 2021-12
• 期刊: ArXiv
• 作者: 'Oscar Dom'inguez;A. Seeger;B. Street;Jean Van Schaftingen;Po-Lam Yung

On the Korányi spherical maximal function on Heisenberg groups

关于海森堡群上的 Koranyi 球极大函数

• DOI: 10.1007/s00208-022-02533-2
• 发表时间: 2022
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Srivastava, Rajula

Sobolev spaces revisited

• DOI: 10.4171/rlm/976
• 发表时间: 2022-02
• 期刊: Rendiconti Lincei - Matematica e Applicazioni
• 作者: H. Brezis;A. Seeger;Jean Van Schaftingen;Po-Lam Yung

Haar frame chacterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts

Besov-Sobolev 空间的 Haar 框架表征及其二元对应空间的最佳嵌入

• 发表时间: 2023
• 期刊: Journal of fourier analysis applications
• 作者: Garrigós, Gustavo;Seeger, Andreas;Ullrich, Tino.
• 通讯作者: Ullrich, Tino.

Lebesgue space estimates for spherical maximal functions on Heisenberg groups

海森堡群上球面极大函数的勒贝格空间估计

• 发表时间: 2022
• 期刊: International mathematics research notices
• 影响因子: 1
• 作者: Roos, Joris;Seeger, Andreas;Srivastava, Rajula
• 通讯作者: Srivastava, Rajula