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

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、平滑度、定量信息）是重要的。在理解一个函数时，使用“更简单”的构建块通常是有用的。在处理这些类型的问题和提供分析这些简单构建块的框架方面，已被证明非常有用的特定数学工具属于谐波分析领域。该项目的主要目标是通过使用和推进调和分析工具，对与函数理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提供更深入的理解。本项目概述了一项研究计划，结合函数理论和算子理论的最新成果，研究微分算子引起的Riesz变换相关的换向子的有界性问题，并理解有端流形上Riesz变换的有界性。该研究方向将主要研究者过去的工作与与微分算子相关的Riesz变换的对易子的有界性问题耦合在一起。特别地，所讨论的问题旨在更好地理解微分算子和几何，其中人们可以用Riesz变换的换向子来表征适当的BMO空间；等价地证明了适当的Hardy空间具有弱分解。第二个研究方向是在有端流形上的全纯泛函演算，并利用非齐次调和分析的思想和主要研究者开发的技术，研究获得Riesz变换的有界性的开放性问题。与首席研究员一起工作的研究生将包括在这些和相关的项目中，并将获得建议和职业指导。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。

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.