Weighted composition operators on growth spaces

Denote by $\mathcal Hol(B_n)$ the space of all holomorphic functions in the unit ball $B_n$ of $\mathbb C^n$, $n\ge1$. Given $g\in\mathcal Hol(B_m)$ and a holomorphic mapping $\varphi\colon B_m\to B_n$, put $C^g_\varphi f=g\cdot(f\circ\varphi)$ for $f\in\mathcal Hol(B_n)$. We characterize those $g$ and $\varphi$ for which $C^g_\varphi$ is a bounded (or compact) operator from the growth space $\mathscr A^{-\log}(B_n)$ or $\mathscr A^{-\beta}(B_n)$, $\beta>0$, to the weighted Bergman space $A^p_\alpha(B_m)$, $0<p<\infty$, $\alpha>-1$. We obtain some generalizations of these results and study related integral operators.