Multipliers of BMO in the Bergman metric with applications to Toeplitz operators (Q917881)

The author considers multipliers of BMO and VMO in the Bergman metric. The main result is as follows: Theorem A. For any bounded symmetric domain \(\Omega\) and \(f\in L^{\infty}(\Omega,dv)\) the following conditions are all equivalent: (1) f multiples BMO, i.e. fBMO\(\subset BMO;\) (2) f(0,z)[\(| \tilde f|^ 2(z)-| \tilde f(z)|^ 2]^{1/2}\) is bounded in \(\Omega\) ; (3) \(\beta\) (0,z)[\(| \hat f_ r|^ 2(z)-| \hat f_ r(z)|^ 2]^{1/2}\) is bounded in \(\Omega\) for all \(r>0;\) (4) \(\beta\) (0,z)[\(| \hat f_ r|^ 2(z)-| \hat f_ r(z)|^ 2]^{1/2}\) is bounded in \(\Omega\) for some \(r>0.\) Here \(\beta\) (\(\cdot,\cdot)\) is a Bergman distance, \(\tilde f\) is a Bursin transform and \(\hat f_ r(z)=| E(z,r)|^{- 1}\int_{E(z,r)}f(\omega)dv(\omega)\). In application he gives a characterization for the multipliers of the Bloch space and to Toeplitz operators.