Treffer: BMO in the Bergman metric on bounded symmetric domains

The results of this paper have been announced in \textit{C. A. Berger, L. A. Coburn}, and \textit{K. H. Zhu}, Bull. Am. Math. Soc., New Ser. 17, 133-136 (1987; Zbl 0621.32014). Let \(\Omega{}\) be a bounded symmetric domain in \(\mathbb{C}^ n\). Consider \(L^ 2\) and \(H^ 2\) defined with respect to the Lebesgue measure. For \(f \in{} L^ 2\), let \(M_ f, M_ fg=fg,\) and let \(P\) be the Bergman projection of \(L^ 2\) onto \(H^ 2\). \(H^ 2\) has a reproducing kernel and associated to it there is a Bergman metric in \(\Omega{}\). Using this metric one can define ''bounded mean oscillation'' and ''vanishing mean oscillation at the boundary of \(\Omega{}\)''. For \(f\in{} L^ 2\), let \([M_ f,P]= M_ f P-P M_ f\) denote the commutator of \(M_ f\) and \(P\). The authors prove the following results: a) \([M_ f,P]\) is bounded if and only if \(f\) is of bounded mean oscillation in \(\Omega{}\); b) \([M_ f,P]\) is compact if and only if \(f\) is of vanishing mean oscillation at the boundary of \(\Omega\). The paper concludes by an interesting conjecture concerning properties of strictly pseudo-convex domains.