Problems in Function Theory and Operator Theory

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The applicant proposes to work on two sets of problems. The first is a continuation of the applicant's collaboration with Arcozzi (Bologna), Sawyer (Hamilton), and Wick (Atlanta). This group is working on specific problems in the operator theoretic function theory of spaces of holomorphic functions which are subspaces of potential spaces. The classical Dirichlet space is a fundamental example; the Drury-Arveson-Hardy space is perhaps the most important example. The questions are similar to the questions classically considered for subspaces of Lebesgue spaces; questions about interpolation, zero sets, multiplier algebras, coronas, etc. However the techniques required are quite different, involving capacity theory and involving use of discrete models for both the space of holomorphic functions and the containing potential space. The work on a second set of problems is work in collaboration with Xiang Tang (St. Louis) trying to understand the various roles of the Rankin-Cohen brackets. The brackets are constant coefficient bidifferential operators whose coefficients are combinatorial numbers. These operators arise naturally in a wide range of settings. Some of the structural reasons for their ubiquity are now understood, but the occurrence of the brackets in certain contexts of particular interest to the applicant is poorly understood. A goal of the proposed research is to improve that situation.In the 1970s and 1980s there were profound mathematical advances at the interface of commutative harmonic analysis and function theory. The research was driven by a desire to see how several very productive, but seemingly very different, viewpoints could be used together to get a deeper understanding of transformation rules for waveforms. The successful unification of the viewpoints allowed resolution of long standing mathematical questions. The improved insight also led to fundamental innovations in data analysis and signal processing; using wavelets for image compression was an early success and by now the descendents of these ideas are a standard part of the toolkit of many computer scientist and electrical engineers. The theoretical tools and insights developed during that time also brought the questions in this proposal within reach. The specific questions being considered are of direct mathematical interest. The analytical tools being developed to work on the questions will, again, broaden and deepen the understanding of how waveforms can be analyzed and manipulated. The questions about the Rankin-Cohen brackets are of a different sort. When very complicated and very elegant expressions arise in several seemingly unrelated contexts there is, for many mathematicians, a compelling aesthetic imperative to find the underlying reason. There is a great tradition of working on such questions and although the answers are often mundane, sometimes they are quite profound.