A sharp constant for the Bergman projection

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For the Bergman projection operator P we prove that Pₗ₁₍B,dλ₎→ B₁= (2n+1)!/n!. Here λ stands for the invariant metric in the unit ball B of Cⁿ, and B₁ 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.