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.
The research objective of this Proposal was to conduct a deeper study of certain function theoretic concepts for spaces of analytic functions, in one and several complex variables. The main method of approach was 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 Scientific Outcomes of this proposal are the following, many in collaboration with junior and early career mathematicians. The following is a brief synopsis of the mathematical results obtained, which have been submitted or accepted for publication. The PI studied various extensions of well known interpolation results in spaces of analytic functions. The PI was able to show that if the Corona data depends smoothly on a parameter, the solutions of the corresponding Bezout equations can be chosen to have the same smoothness in the parameter. The PI has been exploring the question of computing the essential norm of an operator on the weighted Bergman space on the unit ball. We have obtained a complete characterization of when an operator is compact in terms of the vanishing of the Berezin transform and membership in the Toeplitz algebra. This problem was also settled for the Bergman space of the polydisc as well. Several alternate and streamlined proofs of results of the PI related to compactness of operators on the Bergman space of the ball and polydisc were also found. The PI obtained a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc and applied this to the Completion Problem (an analogue of the Corona Problem). This analogue of the Completion Problem settles a related question raised in DMS #0955432. Finally, the PI revisited well-known results for thin sequences and their connections with interpolation and bases in model spaces. This Proposal also provided the opportunity for collaboration with numerous other mathematicians. Training of future generations of mathematicians was also supported by this project. Graduate students were supported with funds related to this proposal, and were able to participate in research conferences. Postdoctoral Fellows were also incorporated into the research program associated with this project.