Nathan Wagner
Brown University
Title
Boundedness and compactness of Bergman projection commutators in two-weight setting
Abstract
The Bergman projection is a fundamental operator in complex analysis with connections to singular integral theory, and it is of interest to study the commutator operator of the Bergman projection with multiplication by a measurable function b. In particular, we study the boundedness and compactness of the Bergman projection commutators in two weighted settings via weighted BMO (bounded mean oscillation) and VMO (vanishing mean oscillation) spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. Complete characterizations of two weight boundedness and compactness are obtained in the analytic case, which parallel results of S. Bloom for the Hilbert transform. Our work initiates a study of the commutators acting on complex function spaces with different symbols. In this talk, we will discuss our main results, as well as the principal ideas of the proofs. This talk is based on joint work with Bingyang Hu and Ji Li.
