Applications of Harmonic Analysis to Function Theory and Operator Theory

调和分析在函数论和算子理论中的应用

• 批准号: 1560955
• 负责人: Brett Wick
• 金额: $ 18万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-17 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1560955&HistoricalAwards=false
• 关键词: Applications; Harmonic; Analysis; Function; Theory

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.

该研究项目将使用谐波分析的工具和技术对功能理论和运营商理论中的基本问题进行研究。 由于首席调查员及其合作者的最新进展，该项目将解决现在对探索的重要问题。 提出的这些问题的解决将在功能理论操作者理论中找到应用，并产生大型分析社区可以采用的新工具和技术。 首席调查员将为研究生和博士后研究员提供建议，将他们包括在拟议的研究项目中，并提供指导，以帮助他们过渡到下一阶段的职业。 通过参与会议并将研究发布给ARXIV预印式服务器的大规模传播结果。该项目将将主要研究者的最新结果与功能理论和运营商理论的动机相结合，以研究与模型空间的两重Hilbert Transform和属性有关的问题。 首先要探索的研究方向是首席研究者的结果，其疑问是关于Toeplitz运营商产品的有限性和可逆性性能的问题。 特别是，要研究的问题旨在更好地理解副群的组成，并确定其界限的必要条件和足够条件。 与希尔伯特变革的两次重量不平等的联系提出了要调查的相关问题。 解决提出的问题将为Hilbert Transform的最新表征提供更多的洞察力，用于Hilbert Transform和Toeplitz Operators在Hardy空间上的相关特性。 基于主要研究者的最新结果及其与Carleson对模型空间的描述的联系，另一个研究方向将被追求。 获得模型空间的Riesz基础表征的开放问题导致与Carleson量度相关的模型空间量度有关，以及它们与Cauchy和Hilbert Transforcs的两重量不等式的关系。 调查的其他方向连接到模型空间上的双线性形式和换向器。