Essential norms and weak compactness of integral operators between weighted Bergman spaces

We consider Volterra-type integration operators $T_g$ between Bergman spaces induced by weights $\omega$ satisfying a doubling property. We derive estimates for the operator norms, essential and weak essential norms of $T_g: A_\omega^p \to A_\omega^q$, $0<p\leq q<\infty$. In particular, the operator $T_g: A_\omega^1\to A_\omega^1$ is weakly compact if and only if it is compact.