Problems in Function Theory and Operator Theory

本项目涉及函数论、算子论和交换调和分析中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的潜在空间的背景下的相互作用。 主要工具之一将是最近的结果，主要研究者（与Arcozzi和索耶）表征的Carleson措施的狄利克雷空间。然而，在古典环境中可用的其他基本分析工具尚未用于这些空间，该项目的一部分是开发这些工具。两个较新的，仍然相对未开发的领域也将进行研究。首先是函数理论的分支机构的身体的工作，已经增长的结果莱西和Thiele的双线性希尔伯特变换。第二个问题是矩阵BMO与矩阵Muckenhoupt权类的关系。从本质上讲，唯一的事情，目前已知的关系是，它是根本不同的比在标量cases.This工作将推进理解的功能理论算子理论空间的全纯函数和运营商所产生的欧几里德调和分析。它将发展在这些领域具有普遍适用性的新观点和技术。在他们的心脏，这样的观点和技术集中在数学分析和综合信息。这种分析和综合的问题遍及数学，并且常常是数学在科学和工程中新用途的发展的中心。

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.