Applications of Harmonic Analysis to Function Theory and Operator Theory

本研究项目将利用调和分析的工具和技术对函数论和算子论中的基本问题进行研究。由于主要研究人员和他的合作者最近取得的进展，该项目将解决目前有待探索的重要问题。这些问题的解决将在函数论算子理论中找到应用，并产生可被更大的分析社区采用的新工具和技术。首席研究员将为研究生和博士后研究员提供建议，将他们纳入拟议的研究项目，并提供指导，以帮助他们过渡到职业生涯的下一个阶段。结果的广泛传播将通过参加会议和将研究结果张贴到arxiv预印本服务器上进行。这个项目将结合主要研究人员的最新结果与函数论和算子理论的动机来研究与双权Hilbert变换和模型空间的性质有关的问题。第一个要探索的研究方向是将主要研究者的结果与关于Toeplitz算子乘积的有界性和可逆性的问题结合起来。特别是，要研究的问题旨在更好地了解副产物的组成，并确定它们有界的充要条件。与希尔伯特变换的双权不等有关的联系提出了需要研究的相关问题。所提出的问题的解决将使我们更深入地了解最近关于Hilbert变换的双权不等式的刻画以及Hardy空间上Toeplitz算子的相关性质。根据首席研究员的最新结果及其与模型空间Carleson测度的描述的联系，将继续进行另一个研究方向。得到模型空间的Riesz基的特征的公开问题导致了与模型空间的逆Carleson测度有关的问题，以及它们与Cauchy变换和Hilbert变换的双权不等式的关系。其他研究方向与模型空间上的双线性形式和交换子有关。

This research project will conduct a study of fundamental questions in function theory and operator theory using the tools and techniques of harmonic analysis. The project will address important questions now open to exploration because of recent advances made by the principal investigator and his collaborators. Resolution of these problems raised will find applications in function theoretic operator theory and yield new tools and techniques that can be adopted by the larger analysis community. The principal investigator will advise graduate students and postdoctoral fellows, include them in the proposed research projects, and provide mentoring, in order to assist them in transitioning to the next stage of their careers. Broad dissemination of the results will take place by participation in conferences and posting of the research to the arxiv preprint server.This project will combine recent results of the principal investigator with motivation from function theory and operator theory to study questions related to the two-weight Hilbert transform and properties of model spaces. The first research direction to be explored couples the results of the principal investigator with questions about boundedness and invertibility properties of products of Toeplitz operators. In particular, the problems to be studied are aimed at obtaining a better understanding of the composition of paraproducts and determining necessary and sufficient conditions for their boundedness. Connections to the two-weight inequality for the Hilbert transform suggest related problems to investigate. Resolving the proposed problems will provide more insight into the recent characterization of the two-weight inequality for the Hilbert transform and related properties for Toeplitz operators on the Hardy space. An additional research direction, based upon the principal investigator's recent results and their connection to the description of the Carleson measure for model spaces, will be pursued. The open question of obtaining a characterization of the Riesz bases for model spaces leads to problems related to reverse Carleson measures for the model spaces, and their relation to two-weight inequalities for the Cauchy and Hilbert transforms. Additional directions of investigation connect to bilinear forms and commutators on model spaces.