Toeplitz operators on weighted Bergman spaces induced by a class of radial weights

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 $T_{\mu,\omega}:L^p_a(\omega)\to L_a^1(\omega)$ is bounded and compact for $0<p\leq 1$. 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 \cite{zhu1989}.