Applications of Harmonic Analysis to Function Theory and Operator Theory

本研究课题将利用调和分析的工具和技术对函数论和算子论的基本问题进行研究。 由于首席研究员及其合作者最近取得的进展，该项目将解决现在开放探索的重要问题。 这些问题的解决将在函数论算子理论中找到应用，并产生新的工具和技术，可以通过更大的分析社区。 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阶段。 通过参加会议和将研究成果发布到arxiv预印本服务器上，广泛传播研究成果。本项目将联合收割机主要研究者的最新成果与函数论和算子论的动机相结合，研究与双权希尔伯特变换和模型空间性质有关的问题。 第一个要探索的研究方向是将主要研究者的结果与Toeplitz算子乘积的有界性和可逆性问题结合起来。 特别是，要研究的问题是为了获得更好的理解的组成paraproducts和确定其有界性的充分必要条件。 与希尔伯特变换的双权不等式的联系表明了需要研究的相关问题。 解决提出的问题将提供更多的洞察到最近的两个重量不等式的Hilbert变换和相关性质的Toeplitz算子的哈代空间上的特征。 一个额外的研究方向，根据主要研究者的最新成果和他们的连接到描述模型空间的Carleson措施，将被追求。 获得模型空间的Riesz基的特征的公开问题导致与模型空间的反向Carleson测度相关的问题，以及它们与Cauchy和Hilbert变换的双权不等式的关系。 其他的研究方向与模型空间上的双线性形式和双线性算子有关。

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.