Sarason Conjecture on the Bergman space

TL;DR

This paper disproves the Sarason Conjecture for Bergman spaces, characterizes bounded Toeplitz products using test functions, and provides sharp estimates for weighted Bergman projections.

Contribution

It offers a counterexample to the Sarason Conjecture and introduces a dyadic model approach for characterizing Toeplitz products on Bergman spaces.

Findings

Counterexample to Sarason Conjecture for Bergman space

Characterization of bounded Toeplitz products via test functions

Sharp estimates for one-weighted Bergman projection

Abstract

We provide a counterexample to the Sarason Conjecture for the Bergman space and present a characterisation of bounded Toeplitz products on the Bergman space in terms of test functions by means of a dyadic model approach. We also present some results about two-weighted estimates for the Bergman projection. Finally, we introduce the class $B_{\infty}$ and give sharp estimates for the one-weighted Bergman projection.