Norm of the Bergman projection

This paper deals with the the norm of the weighted Bergman projection operator $P_{\alpha }:L^{\mathrm{\infty }}\to \mathcal{B}$ where $\alpha >-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space ${\mathbf{C}}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.