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空间在一个或多个复变量下的某些函数理论概念进行更深入的研究。具体而言，将讨论解析函数空间上的双线性形式、解析函数空间上乘数代数的冕型问题以及多盘上的函数理论。方法的主要方法是应用谐波分析的思想和工具来解决本提案中提出的问题，目的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具有挑战性的问题。提议的研究是基于最近，由提案人作出的重大贡献，并重点关注与椎间盘和多椎间盘分析功能相关的关键问题。本文所研究的问题与复分析、函数理论、调和分析和算符理论等领域有着重要的联系。本提案所提出的研究问题的解决方案将在复分析、函数理论、调和分析和算子理论中有无数的应用。它们不仅将为进一步的数学研究开辟道路，而且还将对现实世界的想法有重要的应用，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.