Function Theory and Operator Theory via Harmonic Analysis on the Polydisc

本提案的研究目标是对解析函数的Besov-Sobolev空间的某些函数论概念进行更深入的研究，在一个和多个复变量中。具体而言，有关问题的双线性形式的空间解析函数，电晕型问题的乘子代数空间的解析函数，函数理论的polydisc将接近。 方法的主要方法是应用调和分析的思想和工具来解决本提案中提出的问题，其目标是解决函数和算子理论领域中出现的重要和开放的问题。 本提案中概述的研究计划利用PI从多参数谐波分析和复变函数理论领域获得的知识和技术，提供一系列工具来解决提案中提出的挑战性问题。 建议的研究是基于最近的，由提案人作出的重大贡献，并侧重于关键问题连接到解析函数的磁盘和polydisc。在这个建议中研究的问题与复分析，功能理论，调和分析和运营商理论的领域有重要的联系。 在这个建议中提出的研究问题的解决方案将在复分析，函数论，调和分析和算子理论中有无数的应用。 它们不仅将为额外的数学探究开辟道路，而且还将对现实世界的思想产生重要的应用，特别是在工程和控制理论领域。博士后和研究生也将参与该项目，加强国家的科学基础设施。

The research objective of this Proposal is to conduct a deeper study of certain function theoretic concepts for Besov--Sobolev spaces of analytic functions, in one and several complex variables. Specifically, questions related to bilinear forms on spaces of analytic functions, Corona-type problems for multiplier algebras of spaces of analytic functions, and function theory on the polydisc will be approached. The main method of approach is to apply ideas and tools from harmonic analysis to address the problems raised in this proposal with the goal of resolving important and open questions arising in the areas of function and operator theory. The research program outlined in this Proposal utilizes the PI's knowledge and techniques gained from the areas of multi-parameter harmonic analysis and complex function theory to provide an array of tools by which to approach the challenging questions raised in the proposal. The proposed research is based on recent, significant contributions made by the Proposer and focuses on key questions connected to analytic functions on the disc and polydisc.The questions studied in this proposal have important connections with the areas of complex analysis, function theory, harmonic analysis and operator theory. Solutions to the research questions posed in this Proposal will have countless applications in complex analysis, function theory, harmonic analysis, and operator theory. Not only will they open the way to additional mathematical inquiry, but they will also have significant application to real-world ideas, in particular in the areas of engineering and control theory. Postdoctoral associates and graduate students will also be engaged in this project, enhancing the scientific infrastructure of the country.