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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.

分析的数学学科是理解自然科学和工程中物理现象的基础。作为微分方程解的函数的行为（大小，光滑度，定量信息）很重要。在理解一个函数时，使用“更简单”的构建块通常是有用的。特定的数学工具，已被证明是非常有用的，在解决这些类型的问题，并提供一个框架来分析这些简单的积木位于谐波分析的领域。本项目的主要目标是通过使用和改进调和分析的工具，提供对与函数论和偏微分方程相关的重要函数空间中出现的一些简单构建块的更深入的理解。本项目概述了一个研究计划，该研究计划将最近的结果与函数论和算子论的动机相结合，以研究与Riesz相关的算子有界性问题从微分算子产生的变换和理解Riesz变换在有端流形上的有界性。该研究方向将主要研究者过去的工作与微分算子的Riesz变换相关的有界性问题结合起来。特别是，所讨论的问题是为了获得更好的理解微分算子和几何，其中一个可以表征适当的BMO空间的Riesz变换，等价地证明，适当的哈代空间具有弱因子分解。第二个研究方向提供了一个全纯函数演算的流形结束和研究的开放问题获得有界的Riesz变换使用的想法从非齐次谐波分析和技术开发的主要研究者。与主要研究者一起工作的研究生将被包括在这些和相关项目中，并将获得建议和职业指导。该奖项反映了NSF的法定使命，并被认为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来支持。