A BMO TYPE CHARACTERIZATION OF WEIGHTED LIPSCHITZ FUNCTIONS IN TERMS OF THE BEREZIN TRANSFORM

Title & Authors
A BMO TYPE CHARACTERIZATION OF WEIGHTED LIPSCHITZ FUNCTIONS IN TERMS OF THE BEREZIN TRANSFORM
Cho, Hong-Rae; Seo, Yeoung-Tae;

Abstract
The Berezin transform is the analogue of the Poisson transform in the Bergman spaces. Dyakonov characterize the holomorphic weighted Lipschitz function in the unit disk in terms of the Possion integral. In this paper, we characterize the harmonic weighted Lispchitz function in terms of the Berezin transform instead of the Poisson integral.
Keywords
weighted Lipschitz space of harmonic functions;Berezin transform;a regular modulus of continuity;
Language
English
Cited by
References
1.
P. Ahern, M. Flores and W. Rudin, An invariant volume-mean value Property, J. Funct. Anal. 111 (1993), 380-397

2.
Bekolle, Berger, Coburn, and Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Func. Anal. 93 (1990), 310-350

3.
F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR-Izv.6 (1972), 1117-1151

4.
F. A. Berezin, The relation between covariant and contravariant symbols of operators on classical complex symmetric spaces, Soviet Math. Dokl. 19 (1978), 786-789

5.
S. Bloom and G. S. De Souza, Atomic decomposition of generalized Lipschitz spaces, Illinois J. Math. 33 (1989), no. 2, 181-209

6.
S. Bloom and G. S. De Souza, Weighted Lipschitz spaces and their analytic characterizations, Constr. Approx, 10 (1994), no. 3, 339-376

7.
B. R. Choe, H. Koo, and H. Yi, Derivatives of Harmonic Bergman and Bloch functions on the ball, J. Math. Anal. Appl. 260 (2001), 100-123

8.
K. M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta. Math. 178 (1997), 143-167

9.
K. M. Dyakonov, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math. 187 (2004), 146-172

10.
K. M. Dyakonov, Strong Hardy-Littlewood theorems for analytic functions and mappings of finite distortion, Math. Z. 249 (2005), no. 3, 597-611

11.
H. Hedenrnanlrn, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Springer-Verlag, 2000

12.
M. Pavlovic, On Dyakonov's paper 'Equivalent norms on Lipschitz-type spaces of holomorphic functions', Acta. Math. 183 (1999), 141-143

13.
M. Pavlovic, Introduction to Function Spaces on the Disk, Posebna izdanja 20, Math-ematicki Institut SANU, Beograd 2004

14.
W. Rudin, Function theory in the unit ball of \$\mathbb{C}^n\$, Springer-Verlag Press, New York, 1980

15.
K. Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (1987), 617-646

16.
K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990