Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T05:39:03.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Isometries of Bergman Spaces over Bounded Runge Domains

Published online by Cambridge University Press:  20 November 2018

Clinton J. Kolaski
Clinton J. Kolaski*
Affiliation:
The University of Texas at San Antonio, San Antonio, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1.1. The isometries of the Hardy spaces Hp(0 < p < ∞, p ≠ 2) of the unit disc were determined by Forelli in [2]. Generalizations to several variables: For the polydisc the isometries of Hp onto itself were characterized by Schneider [9]. For the unit ball the case p > 2 was then done by Forelli [3]; Rudin [8] removed the restriction p > 2 by proving a theorem on equimeasurability. Finally, Koranyi and Vagi [6] noted that the methods developed by Forelli, Rudin and Schneider applied to bounded symmetric domains.

In this note it will be shown that their methods also apply to the Bergman spaces over bounded Runge domains. The isometries which are onto are completely characterized; the special cases of the ball and polydisc are particularly nice and are given separately.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1157 - 1164
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Alexander, H., Proper holomorphic mappings in C n , Indiana University Math. J. 26 (1977), 137–146.Google Scholar
2. Forelli, F., The isometries of Hp , Can. J. Math. 16 (1964), 721–728.Google Scholar
3. Forelli, F., A theorem on isometrics and the application of it to the isometnes of Hp(S) for 2 > p > ∞, Can. J. Math. 25 (1973), 284–289.+p+>+∞,+Can.+J.+Math.+25+(1973),+284–289.>Google Scholar
4. Gunning, R. and Rossi, H., Analytic functions of several complex variables (Prentice Hall, New York, 1965).Google Scholar
5. Koranyi, C., An F. and M. Riesz type theorem for the unit ball in complex N-space, Proc. Amer. Math. Soc. 61, (1976) 19–25.Google Scholar
6. Koranyi, A. and Vagi, S.. Isometries of Hp spaces of bounded symmetric domains, Can, J. Math. 28 (1976), 334–340.Google Scholar
7. Rudin, W., Function theory in polydiscs (Benjamin, New York, 1969)Google Scholar
8. Rudin, W., Lp-isometries and equimeasurability, Indiana Univ. Math. J. 25 (1976), 215–228.Google Scholar
9. Schneider, R., Isometries of Hp(Un), Can. J. Math. 25 (1973), 92–95.Google Scholar