Toeplitz Operators on Weighted Bergman Spaces Induced by a Class of Radial Weights

Abstract

Suppose that \(\omega \) is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and T1-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure \(\mu \) such that the Toeplitz operator \(T_{\mu ,\omega }:L^p_a(\omega )\rightarrow L_a^1(\omega )\) is bounded and compact for \(0<p\le 1\). In addition, we obtain a bump condition for the bounded Toeplitz operators with \(L^1(\omega )\) symbol on \(L^1_a(\omega )\). This generalizes a result of Zhu in (J Funct Anal 87(1):31-50, 1989).