Problems in Function Theory and Operator Theory

本课题涉及函数理论、算子理论和交换谐波分析中的几个相关问题的研究。一个主要目标是更好地理解函数理论和算子理论在狄利克雷空间和相关势空间中的相互作用。主要工具之一将是首席研究员（与Arcozzi和Sawyer共同）最近对Dirichlet空间上Carleson测度的描述结果。然而，在经典环境中可用的其他基本分析工具还不能用于这些空间，项目的一部分是开发这些工具。另外两个较新的、相对未开发的地区也将被研究。第一个是从Lacey和Thiele关于双线性希尔伯特变换的结果发展而来的函数理论分支。二是材料BMO与材料Muckenhoupt权值类的关系问题。本质上，关于这个关系我们唯一知道的就是它和标量的情况有本质的不同。这项工作将促进对全纯函数空间的函数论算子理论和欧几里得调和分析中出现的算子的理解。它将发展在这些领域具有普遍适用性的新观点和技术。从本质上讲，这些观点和技术关注的是信息的数学分析和综合。这些分析和综合的问题在数学中普遍存在，并且往往是数学在科学和工程中新用途发展的核心。

This project involves research on several related problems in function theoretic operator theory and in commutative harmonic analysis. One main goal is to understand better the interaction of function theory and operator theory in the context of the Dirichlet space and related potential spaces. One of the major tools will be the recent results of the Principal Investigator (jointly with Arcozzi and Sawyer) characterizing the Carleson measures on the Dirichlet space. However other basic analytical tools that are available in classical contexts are not yet available for these spaces, and part of the project is to develop those tools. Two newer, and still relatively unexplored, areas will also be studied. The first is the function theoretic ramifications of the body of work that has grown from the results of Lacey and Thiele on the bilinear Hilbert transform. The second is the question of the relationship of matricial BMO to the class of matricial Muckenhoupt weights. Essentially the only thing currently known about that relationship is that it is fundamentally different than in the scalar case.This work will advance the understanding of the function theoretic operator theory on spaces of holomorphic functions and of operators arising in Euclidean harmonic analysis. It will develop new viewpoints and techniques that will have general applicability in those areas. At their hearts, such viewpoints and techniques focus on the mathematical analysis and synthesis of information. Such questions of analysis and synthesis pervade mathematics and are often central to the development of new uses of mathematics in science and engineering.