Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Composition operators that improve integrability on weighted Bergman spaces


Authors: Wayne Smith and Liming Yang
Journal: Proc. Amer. Math. Soc. 126 (1998), 411-420
MSC (1991): Primary 47B38; Secondary 30D55, 46E15
DOI: https://doi.org/10.1090/S0002-9939-98-04206-3
MathSciNet review: 1443167
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Composition operators between weighted Bergman spaces with a smaller exponent in the target space are studied. An integrability condition on a generalized Nevanlinna counting function of the inducing map is shown to characterize both compactness and boundedness of such an operator. Composition operators mapping into the Hardy spaces are included by making particular choices for the weights.


References [Enhancements On Off] (What's this?)

  • [ESS] Essén, M., Shea, D.F. and Stanton, C.S., A value-distribution criterion for the class $L\log L$, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), 127-150. MR 87e:30041
  • [H] Horowitz, C., Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), 201-213. MR 55:681
  • [HJ] Hunziker, H. and Jarchow, H., Composition operators which improve integrability, Math. Nachr. 152 (1991), 83-91. MR 93d:47061
  • [Li] Littlewood, J.E., On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925), 481-519.
  • [Lu] Luecking, D. H., Embedding theorems for spaces of analytic functions via Khinchine's inequality, Michigan Math. J. 40 (1993), 333-358. MR 94e:46046
  • [MS] MacCluer, B.D. and Shapiro, J.H., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878-906. MR 87h:47048
  • [Ri] Riedl, R., Composition operators and geometric properties of analytic functions, Thesis, Universität Zurich (1994).
  • [Ro] Rockberg, R., Decomposition theorems for Bergman spaces and their applications, in Operators and Function Theory (S.C. Power, editor), D. Reidel, Dordrecht, 1985, pp. 225-277.
  • [Sh] Shapiro, J.H., The essential norm of a composition operator, Annals of Math. 127 (1987), 375-404. MR 88c:47058
  • [Sm] Smith, W., Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), 2331-2348. MR 96i:47056
  • [SZ] Smith, W. and Zhao, R., Composition operators mapping into the $Q_{p}$ spaces, preprint.
  • [St] Stanton, C. S., Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), 459-468. MR 88c:32002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B38, 30D55, 46E15

Retrieve articles in all journals with MSC (1991): 47B38, 30D55, 46E15


Additional Information

Wayne Smith
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: wayne@math.hawaii.edu

Liming Yang
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: yang@math.hawaii.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04206-3
Keywords: Bergman spaces, Hardy spaces, composition operators, Nevanlinna counting function
Received by editor(s): July 24, 1996
Additional Notes: The second author was partially supported by National Science Foundation grant DMS9531917 and a seed-money grant from the University of Hawaii.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society