Multipliers of BMO in the Bergman metric with applications to Toeplitz operators. (English) Zbl 0705.47025

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.


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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