BMO in the Bergman metric on bounded symmetric domains

AbstractFor bounded symmetric domains Ω in Cn, a notion of “bounded mean oscillation” in terms of the Bergman metric is introduced. It is shown that for ƒ in L2(Ω, dv), ƒ is in BMO(Ω) if and only if the densely-defined operator [Mƒ, P] ≡ MƒP − PMƒ on L2(Ω, dv) is bounded (here, Mƒ is “multiplication by ƒ” and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mƒ, P] is provided by functions of “vanishing mean oscillation at the boundary of Ω”