The Bergman projection with operator weights

Abstract

We characterize boundedness of the Bergman projection on operator-weighted $L^p$ spaces over a separable Hilbert space, for $1<p<\infty$. Under the directional integrability conditions, the criterion is uniform boundedness of averages over Carleson squares in the unit disc. We obtain norm estimates independent of dimension. The proof uses a separated kernel expansion and a layer decomposition of the tent tree. No reverse H\"older assumption is needed. We also give a finer matrix estimate with a dimension-dependent constant. Boundedness is unchanged by compactly supported changes of the weight that preserve the integrability conditions.