Problems in Function Theory and Operator Theory

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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.