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The Bergman projection $P_\alpha$, induced by a standard radial weight, isbounded and onto from $L^\infty$ to the Bloch space $\mathcal{B}$. However,$P_\alpha: L^\infty\to \mathcal{B}$ is not a projection. This fact can beemended via the boundedness of the operator$P_\alpha:BMO_2(\Delta)\to\mathcal{B}$, where $BMO_2(\Delta)$ is the space offunctions of bounded mean oscillation in the Bergman metric.  We consider the Bergman projection $P_\omega$ and the space$BMO_{\omega,p}(\Delta)$ of functions of bounded mean oscillation induced by$1<p<\infty$ and a radial weight $\omega\in\mathcal{M}$. Here $\mathcal{M}$ isa wide class of radial weights defined by means of moments of the weight, andit contains the standard and the exponential-type weights. We describe theweights such that $P_\omega:BMO_{\omega,p}(\Delta)\to\mathcal{B}$ is bounded.They coincide with the weights for which $P_\omega: L^\infty \to \mathcal{B}$is bounded and onto. This result seems to be new even for the standard radialweights when $p\ne2$.

Bergman projection and BMO in hyperbolic metric -- improvement of classical result