Comptes Rendus
Complex Analysis
Growth spaces on circular domains: composition operators and Carleson measures
[Espaces à croissance sur les domaines circulaires : opérateurs de composition et mesures de Carleson]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 609-612.

Soit Ω un domaine circulaire, strictement convexe et borné dans Cn dont le bord est de classe C2. Nous désignons par Hol(Ω) l'espace des fonctions holomorphes dans Ω. Soient gHol(Ω) et φ:ΩΩ une transformation holomorphe. Posons Cφgf=g(fφ) pour fHol(Ω). Nous caractérisons les fonctions g et φ pour lesquelles Cφg est un opérateur borné ou compact de l'espace à croissance Alog(Ω) ou de Aβ(Ω), β>0, dans l'espace de Bergman à poids Aαp(Ω), 0<p<, α>1. Nous caractérisons aussi les mesures positives μ sur Ω telles que Aβ(Ω)Lq(Ω,μ) et les mesures positives μ telles que Alog(Ω)Lq(Ω,μ) pour 0<q< et β>0.

Let ΩCn be a bounded, circular and strictly convex domain with the boundary of class C2. Denote by Hol(Ω) the space of all holomorphic functions in Ω. Given gHol(Ω) and a holomorphic mapping φ:ΩΩ, put Cφgf=g(fφ) for fHol(Ω). We characterize those g and φ for which Cφg is a bounded or compact operator from the growth space Alog(Ω) or Aβ(Ω), β>0, to the weighted Bergman space Aαp(Ω), 0<p<, α>1. Also, given 0<q< and β>0, we describe those positive measures μ on Ω for which Aβ(Ω)Lq(Ω,μ) and those μ for which Alog(Ω)Lq(Ω,μ).

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Accepté le :
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DOI : 10.1016/j.crma.2009.04.003

Evgueni Doubtsov 1

1 St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
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Evgueni Doubtsov. Growth spaces on circular domains: composition operators and Carleson measures. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 609-612. doi : 10.1016/j.crma.2009.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.003/

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