Applications of Harmonic Analysis to Riesz Transforms and Commutators beyond the Classical Settings

谐波分析在经典设置之外的 Riesz 变换和换向器中的应用

基本信息

批准号：
1800057
负责人：
Brett Wick
金额：
$ 23万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800057&HistoricalAwards=false
关键词：
Applications Harmonic Analysis Riesz Transforms

项目摘要

The mathematical discipline of analysis has been fundamental in understanding physical phenomena in the natural sciences and engineering. The behavior of a function (size, smoothness, quantitative information) that are solutions to differential equations are important. In understanding a function it is frequently useful to have "simpler" building blocks to work with. Particular mathematical tools that have proved extremely useful in addressing questions of these types and providing a framework to analyze these simple building blocks lie within the realm of harmonic analysis. A main goal of this project is to provide a deeper understanding of some of the simple building blocks that arise in important function spaces connected to function theory and partial differential equations by using and advancing the tools of harmonic analysis.This project outlines a research program combining recent results with motivation from function theory and operator theory to study questions related to the boundedness of commutators associated to Riesz transforms arising from differential operators and understanding the boundedness of the Riesz transform on a manifold with ends. The research direction couples the past work by the principal investigator with questions about boundedness of commutators with Riesz transforms associated to differential operators. In particular, the problems discussed are aimed at obtaining a better understanding of the differential operators and geometry where one can characterize appropriate BMO spaces in terms of commutators with Riesz transforms; equivalently demonstrate that the appropriate Hardy space possesses a weak factorization. A second research direction provides a holomorphic functional calculus on a manifold with ends and studies the open question of obtaining the boundedness of the Riesz transform using ideas from non-homogeneous harmonic analysis and the techniques developed by the principal investigator. Graduate students with whom the principal investigator works will be included in these and related projects, and will receive advising and career mentoring.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.

分析的数学学科对于理解自然科学和工程学中的物理现象至关重要。作为微分方程解决方案的函数（大小，平滑度，定量信息）的行为很重要。在理解功能时，拥有“更简单”的构建块通常是有用的。事实证明，特定的数学工具在解决这些类型的问题上非常有用，并提供了一个框架来分析这些简单的构建块，位于谐波分析领域。该项目的一个主要目标是通过使用和推进谐波分析的工具，对与功能理论和部分微分方程相关的重要功能空间和部分差分方程的一些简单构建块提供更深入的了解。该项目概述了一个研究计划，概述了一个研究计划，概述了与函数理论和操作者与与界限的界限相关的功能理论和操作者的动力理论的动机，并理解了与Riessiase and Indressies and Inderiase and Inderiase and Inderiase and Inderiase and Inderiase and Inderiase Arive and Insperies and Issiase Arivesiase Arivesiase的界定问题。带有末端的歧管。研究方向将主要研究人员的过去工作与有关RIESZ的界限有关的问题与差异操作员相关。特别是，所讨论的问题旨在更好地了解差异操作员和几何形状，在这种情况下，人们可以根据Riesz Transforms的换向器来表征适当的BMO空间；等效地表明，适当的耐寒空间具有弱分解。第二个研究方向在末端的流动性上提供了一个全体形态功能演算，并研究了使用非均匀谐波分析和主要研究者开发的技术来获得Riesz变换的界限的开放问题。与主要调查员作品一起纳入这些项目和相关项目的研究生，并将获得咨询和职业指导。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估审查标准来评估的。