Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball

Abstract

We prove a sufficient condition for products of Toeplitz operators T f Tḡ, where f, g are square integrable holomorphic functions in the unit ball in C n , to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators H f H * g is also given.

On the weighted estimate of the Bergman projection

“…For the usual Bergman spaces, α > −1, there have been many works on the problem. It has been proved in [18] that condition (4.3) is necessary but the authors did not manage to prove whether the same condition is sufficient or not (see also [8], [10], [14], [16], [17] for related discussions and other domains). It is only two years ago that Aleman, Pott and Reguera exhibited in [1] an example of f, g ∈ A 2 (D) = A 2 0 (D) such that (4.3) holds but the product T f T g is not bounded on A 2 (D).…”

Products of Toeplitz and Hankel operators on the Bergman space in the polydisk

“…Next, Miao in [4] gave an interesting way to transfer Theorem 1 and Theorem 2 to the space A p α , 1 < p < ∞, α > −1, of the unit ball. Recently, Michalska and Sobolewski [5] improved a sufficient condition on boundedness of T f Tḡ on A p α . A similar problem concerns the products of the Hankel operators H f H * g .…”

Sarason Conjecture on the Bergman Space

“…Sufficient conditions close to Sarason's condition 1.2 for the boundedness of Toeplitz products in the style of the so-called bump conditions can be found in [45] and in [31].…”