Dyadic harmonic analysis and weighted theory in the Bergman space

Purpose and goal

The purpose of this project is the study of two fundamental interdisciplinary questions in complex analysis and operator theory using tools from dyadic harmonic analysis. The first goal is the development of a theory for the B_\infty class of weights. This goal has been completely fulfilled. The second goal is to characterize a two weight inequality for the Bergman projection. An application of the latter was recently used to resolve a famous conjecture in operator theory. Partial progress has been made towards fulfilling this goal, when restricting the class of weights to B_\infty ones

Results and expected effects

The harmonic analysis tools that have been used to solve the B_infty question have provided a new and effective way of approaching problems in complex analysis and operator theory. These techniques are expected to provide substantial benefit to a large community of mathematicians working on the above areas. The two weight question addressed during the project is related to fine properties of Toeplitz and Hankel operators, that have been studied over the last 100 years and whose use in control theory is well-established. Our contribution to its understanding is at early stages.

Approach and implementation

A multidisciplinary approach has been implemented to solve questions at the interface of two different areas of mathematics. Techniques from harmonic analysis, area of expertise corresponding to the project leader, have been used to solve problems in complex analysis and operator theory, areas of strength of the Swedish organization. The three members of the group, two professors and a early career researcher, have exchanged ideas on a weekly basis. This has resulted in a complete understanding of the class of B_\infty weights, the central question in the proposal.