Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T00:48:17.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Isometries of Weighted Bergman Spaces

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.

In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear).

2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α(0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy

(1)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 910 - 915
Copyright
Copyright © Canadian Mathematical Society 1982

References

References&gt;

1. Coif, R. R. man and Rochberg, R., Representation theorems for holomorphic and harmonic functions, to appear.Google Scholar
2. Forelli, F., A theorem on isometries and the application of it to the isometries of Hp﹛S) for 2 < p < ∞, Can. J. Math. 25 (1973), 284–289.Google Scholar
3. Horowitz, C., Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693–710.Google Scholar
4. Kolaski, C., Isometries of Bergman spaces over bounded Runge domains, Can. J. Math. 33 (1981), 1157–1164.Google Scholar
5. Kolaski, C., Anew i00k ai a theorem of Forelli and Rudin, Indiana Univ. Math. J. 28 (1979), 495–499.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, 1969).Google Scholar
8. Rudin, W., Lp-isometries and equimeasurability, Indiana Univ. Math. J. 25 (1976), 215–228.Google Scholar
9. Rudin, W., Function theory in the unit ball of Cn (Springer-Verlag, 1980).Google Scholar
10. Schneider, R., Iosmetries of Hv(Un), Can. J. Math. 25 (1973), 92–95.Google Scholar
11. Taibleson, M. H. and Weiss, G., The molecular characterization of certain Hardy spaces, to appear.Google Scholar