Composition operators that improve integrability on weighted Bergman spaces
HTML articles powered by AMS MathViewer
- by Wayne Smith and Liming Yang PDF
- Proc. Amer. Math. Soc. 126 (1998), 411-420 Request permission
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
- M. Essén, D. F. Shea, and C. S. Stanton, A value-distribution criterion for the class $L\,\textrm {log}\,L$, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 127–150 (English, with French summary). MR 812321, DOI 10.5802/aif.1030
- Charles Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), no. 1, 201–213. MR 427650
- Herbert Hunziker and Hans Jarchow, Composition operators which improve integrability, Math. Nachr. 152 (1991), 83–99. MR 1121226, DOI 10.1002/mana.19911520109
- Littlewood, J.E., On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925), 481–519.
- Daniel H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Michigan Math. J. 40 (1993), no. 2, 333–358. MR 1226835, DOI 10.1307/mmj/1029004756
- Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4
- Riedl, R., Composition operators and geometric properties of analytic functions, Thesis, Universität Zurich (1994).
- 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.
- Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
- Wayne Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2331–2348. MR 1357404, DOI 10.1090/S0002-9947-96-01647-9
- Smith, W. and Zhao, R., Composition operators mapping into the $Q_{p}$ spaces, preprint.
- Charles S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), no. 2, 459–468. MR 863538, DOI 10.2140/pjm.1986.125.459
Additional Information
- Wayne Smith
- Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
- MR Author ID: 213832
- Email: wayne@math.hawaii.edu
- Liming Yang
- Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
- Email: yang@math.hawaii.edu
- 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
- © Copyright 1998 American Mathematical Society
- 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