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

分析的数学学科一直是理解自然科学和工程中的物理现象的基础。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。在理解函数时，使用“更简单”的构建块通常是有用的。事实证明，在解决这类问题并提供分析这些简单构件的框架方面非常有用的特殊数学工具属于调和分析领域。这个项目的一个主要目标是通过使用和发展调和分析的工具，对与函数论和偏微分方程有关的重要函数空间中出现的一些简单构件有更深入的理解。这个项目概述了一个研究方案，将最近的结果与函数论和算子理论的动机相结合，研究与由微分算子产生的Riesz变换的交换子的有界性有关的问题，并理解Riesz变换在有端流形上的有界性。研究方向结合了主要研究者过去的工作和关于交换子的有界性的问题，以及与微分算子相关的Riesz变换。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变换的交换子来刻画适当的BMO空间；等价地证明适当的Hardy空间具有弱因式分解。第二个研究方向是在有端点的流形上建立一个全纯泛函演算，并利用非齐次调和分析的思想和主要研究者发展的技巧研究Riesz变换的有界性这一公开问题。与首席研究员合作的研究生将被包括在这些项目和相关项目中，并将得到建议和职业指导。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.