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

Abstract

The Bergman projection $P_{\alpha }$, induced by a standard radial weight, is bounded 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 be emended via the boundedness of the operator $P_{\alpha }:BM{O}_2(\Delta )\to \mathcal{B}$, where $BM{O}_2(\Delta )$ is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection $P_{\omega }$ and the space $BM{O}_{\omega ,p}(\Delta )$ of functions of bounded mean oscillation induced by $1<p<\infty$ and a radial weight $\omega \in \mathcal{M}$. Here $\mathcal{M}$ 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 $P_{\omega }:BM{O}_{\omega ,p}(\Delta )\to \mathcal{B}$ 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 radial weights when $p\ne 2$.