BMO in the Bergman metric on bounded symmetric domains

Abstract

For bounded symmetric domains Ω in C n , a notion of “bounded mean oscillation” in terms of the Bergman metric is introduced. It is shown that for ƒ in L 2 (Ω, dv) , ƒ is in BMO(Ω) if and only if the densely-defined operator [M ƒ , P] ≡ M ƒ P − PM ƒ on L 2 (Ω, dv) is bounded (here, M ƒ is “multiplication by ƒ” and P is the Bergman projection with range the Bergman subspace H 2 (Ω, dv) = L a 2 (Ω, dv) of holomorphic functions in L 2 (Ω, dv)) . An analogous characterization of compactness for [ M ƒ , P ] is provided by functions of “vanishing mean oscillation at the boundary of Ω”.