Boundedness of the Bergman type operators on mixed norm spaces

Author:
Yongmin Liu

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2363-2367

MSC (2000):
Primary 47B38; Secondary 32A30, 46E15

DOI:
https://doi.org/10.1090/S0002-9939-02-06332-3

Published electronically:
January 23, 2002

MathSciNet review:
1897461

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Conditions sufficient for boundedness of the Bergman type operators on certain mixed norm spaces of functions on the unit ball of are given, and this is used to solve Gleason's problem for the mixed norm spaces .

**1.**G.B. Ren and J.H. Shi,*Bergman type operator on mixed norm spaces with applications*, Chin. Ann. of Math. 18(B)(1997), 265-276. MR**98k:32006****2.**A. Brown and C. Pearcy,*Introduction to operator theory I: elements of functional analysis*, GTM 55, Springer-Verlag, New York, Berlin, Heidelberg. MR**58:23463****3.**F. Forelli and W. Rudin,*Projections on spaces of holomorphic functions on balls*, Indiana Univ. Math. J., 24(6)(1974): 593-602. MR**50:10332****4.**W. Rudin,*Function theory in the unit ball of ,*Springer-Verlag, New York, 1980. MR**82i:32002****5.**B.R. Choe,*Projection, weighted Bergman spaces, and the Bloch space,*Proc. Amer. Math. Soc., 108(1)(1990): 127-136. MR**90h:32009****6.**J. Xiao,*Compactness for Toeplitz and Hankel operators on weighted Bergman spaces in ball in*, Science in China, 23A(8)(1993): 811-818. MR**95b:47030****7.**A.L. Shields and D.L. Williams,*Bounded projections, duality and multipliers in spaces of analytic functions,*Trans. Amer. Math. Soc., 162(1971): 287-302. MR**44:790****8.**S. Gadbois,*Mixed-norm generalization of Bergman space, and duality,*Proc. Amer. Math. Soc., 104(4)(1988): 1171-1180.MR**89m:46041****9.**G.H. Hardy and J.E. Littlewood,*Some property of fractional integrals II*Math Z, 34(1932): 403-439.**10.**K.H. Zhu,*The Bergman spaces, the Bloch spaces and Gleason's problem*, Trans. Amer. Soc. 309(1)(1988): 253-268.MR**89j:46025****11.**J.M. Ortega,*The Gleason problem in Bergman-Sobolev spaces,*Complex Variables, 20(1992): 157-170. MR**95d:32011****12.**G.B. Ren and J.H. Shi,*Forelli-Rudin type theorem on pluriharmonic Bergman spaces with small exponent*, Science in China, 29A(10)(1999): 909-913.**13.**G.B. Ren and J.H. Shi,*Gleason's problem in weighted Bergman space egg domains*, Science in China, 41A(3)(1998): 225-231.MR**99f:32038**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B38,
32A30,
46E15

Retrieve articles in all journals with MSC (2000): 47B38, 32A30, 46E15

Additional Information

**Yongmin Liu**

Affiliation:
Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, People’s Republic of China

Email:
minliu@263.net

DOI:
https://doi.org/10.1090/S0002-9939-02-06332-3

Keywords:
Bergman type operator,
normal function,
boundedness,
H\"older inequality,
Gleason's problem

Received by editor(s):
November 14, 2000

Received by editor(s) in revised form:
March 19, 2001

Published electronically:
January 23, 2002

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2002
American Mathematical Society