A sharp constant for the Bergman projection

For the Bergman projection operator $P$ we prove that $\|P\|_{L^1(B,dλ)\rightarrow B_1}= \frac {(2n+1)!}{n!}.$ Here $λ$ stands for the invariant metric in the unit ball $B$ of $\mathbf{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

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