arXiv:1304.1750v2 [math.CA] 4 Sep 2013 SARASON CONJECTURE ON THE BERGMAN SPACE ALEXANDRU ALEMAN, SANDRA POTT, AND MARIA CARMEN REGUERA 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. 1. Introduction Let dA denote Lebesgue area measure on the unit disc D, normalized so that the measure of D equals 1. The Bergman space A 2 (D) is the closed subspace of analytic functions in the Hilbert space L 2 (D, dA). Likewise, the Hardy space H 2 (T) is the closed subspace of L 2 (T) consisting of analytic functions. The Bergman projection P B , given by P B f (z)=  D f (ζ ) (1 − ζz) 2 dA(ζ ), is the orthogonal projection from L 2 (D, dA) onto A 2 (D), while the Riesz projection P R denotes the orthogonal projection from L 2 (T) to H 2 (T). For each function f ∈ L 2 (D) we have the densely defined Bergman space Toeplitz operator T f on A 2 (D), given by T f u = P B fu. In the same way, given f ∈ L 2 (T), the Hardy space Toeplitz operator T f on H 2 is given by T f v = P R fv, where u and v are suitable elements in A 2 and H 2 , respectively. For analytic f , it is easy to see that both the Bergman space Toeplitz operator T f and the Hardy space Toeplitz operator T f are bounded, if and only if f is a bounded function on D. In this paper, we shall study the question as to which pairs of functions f,g ∈ A 2 (D) give rise to a bounded Toeplitz product operator T f T ∗ g : A 2 (D) → A 2 (D). 2010 Mathematics Subject Classification. Primary: Primary: 47B38, 30H20 Secondary: 42C40, 42A61,42A50 . Key words and phrases. Bergman spaces, Toeplitz products, two-weight inequalities, Bekoll´ e weights. Supported partially by the Crafoord Foundation and by Lund University, Mathematics in the Faculty of Science, with a postdoctoral research grant. 1