Sarason Conjecture on the Bergman Space

Abstract

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.

Two weight inequality for Bergman projection

“…This is not only due to the mathematical difficulties the question raises, but also to its numerous applications in operator theory. Recently, the bounded projections P 0 : L 2 |g| −2 → L 2 |f | 2 were characterized on the way to disprove the Sarason conjecture on the Toeplitz product operator T f T ⋆ g : A 2 → A 2 , induced by analytic symbols f and g [2]. However, the most commonly known results concerning the two weight inequality (1.1) have been obtained when the inducing weight ω is standard [5,6].…”

Bergman projection induced by kernel with integral representation

“…The two-weight inequality P + (f ) L p u ≤ C f L p v was recently characterized in terms of testing conditions on the indicators of Carleson squares [2]. The last of our main results offers a generalization of this result to the class of radial weights with kernels of the form (1.5).…”

“…To prove the right hand inequality, let z, ζ ∈ D. Let J = J(z, ζ) ⊂ T such that z, ζ ∈ S(J) and |J| ≍ |1 − ζz|, see [2] for details. There exist β ∈ {0, 1/2} and K ∈ D β such that J ⊂ K and |K| ≤ 4|J|.…”

On products of some Toeplitz operators on polyanalytic Fock spaces

“…Actually, the previous condition is only necessary and the conjecture fails for both the Hardy space and Bergman space of the unit disk. Counterexamples were given in [8] and [2]. However, in the context of classical Fock spaces, Cho, Park and Zhu in [6] show that the Sarason's conjecture is true.…”