As a Corollary to the main result of the paper, we give a new proof of the inequalitywhere T is either the Hilbert transform ( Amer J Math 129 (5):1355-1375, 2007), a Riesz transform (Proc Amer Math Soc 136(4):1237-1249, 2008), or the Beurling operator (Duke Math J 112(2):281-305, 2002). The weight w is non-negative, and the linear growth in the A 2 characteristic on the right is sharp. Prior proofs relied strongly on Haar shift operators (CR Acad Sci Paris Sér I Math 330(6):455-460, 2000) and Bellman function techniques. The new proof uses Haar shifts, and then uses an elegant 'two weight T 1 theorem' of Nazarov-Treil-Volberg (Math Res Lett 15(3):583-597, 2008) to immediately identify relevant Carleson measure estimates, which are in turn verified using an appropriate corona decomposition of the weight w.

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 ∞ and give sharp estimates for the one-weighted Bergman projection.

We disprove the following a priori estimate for the Hilbert transform H and the Hardy Littlewood maximal operator M :This is a sequel to paper [5] by the first author, which shows the existence of a Haar multiplier operator for which the inequality holds.

For 1 < p < ∞, weight w ∈ A p and any L 2 -bounded Calderón-Zygmund

We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Békollé constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Pérez.2010 Mathematics Subject Classification. Primary: 47B38, 30H20 Secondary: 42C40, 42A61,42A50 .