Applications of Harmonic Analysis to Function Theory and Operator Theory

本研究项目将运用调和分析的工具和技术，对函数理论和算子理论中的基本问题进行研究。由于首席研究员和他的合作者最近取得的进展，该项目将解决一些重要的问题，这些问题现在还有待探索。这些问题的解决将在函数理论算子理论中找到应用，并产生新的工具和技术，可以被更大的分析社区采用。首席研究员将为研究生和博士后提供建议，将他们纳入拟议的研究项目，并提供指导，以帮助他们过渡到他们的职业生涯的下一个阶段。将通过参加会议和将研究成果发布到arxiv预印本服务器来广泛传播结果。该项目将结合主要研究者的最新成果，从函数理论和算子理论的动机，研究与双权希尔伯特变换和模型空间的性质有关的问题。要探索的第一个研究方向是将主要研究者的结果与Toeplitz算子积的有界性和可逆性问题耦合在一起。特别要研究的问题是为了更好地了解副积的组成和确定其有界性的充分必要条件。与希尔伯特变换的二权不等式的联系提出了相关的研究问题。解决所提出的问题将为最近Hilbert变换的双权不等式的表征和Hardy空间上Toeplitz算子的相关性质提供更多的见解。一个额外的研究方向，基于首席研究员最近的结果及其与模型空间的Carleson测度描述的联系，将被追求。获得模型空间的Riesz基的特征的开放问题导致了与模型空间的反向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。附加的研究方向与模型空间上的双线性形式和换向子有关。

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.