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

The Bergman projection $$P_\alpha $$P α , induced by a standard radial weight, is bounded and onto from $$L^\infty $$L ∞ to the Bloch space $$\mathcal {B}$$B . However, $$P_\alpha : L^\infty \rightarrow \mathcal {B}$$P α : L ∞ → B is not a projection. This fact can be emended via the boundedness of the operator $$P_\alpha :\textrm{BMO}_2(\Delta )\rightarrow \mathcal {B}$$P α : BMO 2 ( Δ ) → B , where $$\textrm{BMO}_2(\Delta )$$BMO 2 ( Δ ) is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection $$P_\omega $$P ω and the space $$\textrm{BMO}_{\omega ,p}(\Delta )$$BMO ω , p ( Δ ) of functions of bounded mean oscillation induced by $$1<p<\infty $$1 < p < ∞ and a radial weight $$\omega \in \mathcal {M}$$ω ∈ M . Here $$\mathcal {M}$$M is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that $$P_\omega :\textrm{BMO}_{\omega ,p}(\Delta )\rightarrow \mathcal {B}$$P ω : BMO ω , p ( Δ ) → B is bounded. They coincide with the weights for which $$P_\omega : L^\infty \rightarrow \mathcal {B}$$P ω : L ∞ → B is bounded and onto. This result seems to be new even for the standard radial weights when $$p\ne 2$$p ≠ 2 .