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

Zhu, Kehe

J. Funct. Anal. 87, No. 1, 31-50 (1989).

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.

Reviewer: N.K.Karapetyants (Rostov-na-Donu)

MSC:

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

Keywords:

multipliers of BMO and VMO in the Bergman metric; bounded symmetric domain; Bursin transform; multipliers of the Bloch space; Toeplitz operators

References:

[1]	Berger, C.A; Coburn, L.A; Zhu, K.H, Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus, Amer. J. math., 110, 921-953, (1988) · Zbl 0657.32001
[2]	Berger, C.A; Coburn, L.A; Zhu, K.H, BMO on the Bergman spaces of the classical domains, Bull. amer. math. soc., 17, 133-136, (1987) · Zbl 0621.32014
[3]	{\scD. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu}, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal., in press. · Zbl 0765.32005
[4]	Coefman, R.R; Rochberg, R, Representation theorems for holomorphic and harmonic functions in Lp, Astérisque, 77, 11-66, (1980) · Zbl 0472.46040
[5]	Janson, S, On functions with conditions on the Mean oscillation, Ark. mat., 14, 189-196, (1976) · Zbl 0341.43005
[6]	Janson, S; Peetre, J; Semmes, S, On the action of Hankel and Toeplitz operators on some function spaces, Duke math. J., 51, 937-958, (1984) · Zbl 0579.47022
[7]	Rudin, W, Function theory in the unit ball of \(C\)^{n}, (1980), Springer New York
[8]	Stegenga, D, Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded Mean oscillation, Amer. J. math., 98, 573-589, (1976) · Zbl 0335.47018
[9]	Timoney, R, Bloch functions in several complex variables, I, Bull. London math. soc., 12, 241-267, (1980) · Zbl 0416.32010
[10]	Zhu, K.H, Duality and Hankel operators on the Bergman spaces of bounded symmetric domains, J. funct. anal., 81, 260-278, (1988) · Zbl 0669.47019