Toeplitz operators on the weighted Bergman spaces

In this paper, we first study Toeplitz operators on a weighted Bergman space $L_a^1(\omega)$ where the weight $\omega$ is regular. Using the Carleson measure and $T1$-type condition, we obtain necessary and sufficient conditions of the positive measure $\mu$ such that the Toeplitz operator $T_{\mu,\omega}$ is bounded and compact. For the $L^1(\omega)$-integrable function $\varphi$, a boundedness criterion of $T_{\varphi,\omega}$ is presented by a bump condition. In addition, Toeplitz operators with positive symbols between weighted harmonic Bergman spaces induced by regular weights are discussed.