  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.
