Growth spaces on circular domains: composition operators and Carleson measures

Evgueni Doubtsov

doi:10.1016/j.crma.2009.04.003

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]

Présenté par : Jean-Pierre Demailly

Evgueni Doubtsov ¹

¹ St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 609-612.

Résumé
Abstract

Soit Ω un domaine circulaire, strictement convexe et borné dans $C^{n}$ dont le bord est de classe $C^{2}$ . Nous désignons par $H o l (Ω)$ l'espace des fonctions holomorphes dans Ω. Soient $g \in H o l (Ω)$ et $φ : Ω \to Ω$ une transformation holomorphe. Posons $C_{φ}^{g} f = g \cdot (f ○ φ)$ pour $f \in H o l (Ω)$ . Nous caractérisons les fonctions g et φ pour lesquelles $C_{φ}^{g}$ est un opérateur borné ou compact de l'espace à croissance $A^{- \log} (Ω)$ ou de $A^{- β} (Ω)$ , $β > 0$ , dans l'espace de Bergman à poids $A_{α}^{p} (Ω)$ , $0 < p < \infty$ , $α > - 1$ . Nous caractérisons aussi les mesures positives μ sur Ω telles que $A^{- β} (Ω) \subset L^{q} (Ω, μ)$ et les mesures positives μ telles que $A^{- \log} (Ω) \subset L^{q} (Ω, μ)$ pour $0 < q < \infty$ et $β > 0$ .

Let $Ω \subset C^{n}$ be a bounded, circular and strictly convex domain with the boundary of class $C^{2}$ . Denote by $H o l (Ω)$ the space of all holomorphic functions in Ω. Given $g \in H o l (Ω)$ and a holomorphic mapping $φ : Ω \to Ω$ , put $C_{φ}^{g} f = g \cdot (f ○ φ)$ for $f \in H o l (Ω)$ . We characterize those g and φ for which $C_{φ}^{g}$ is a bounded or compact operator from the growth space $A^{- \log} (Ω)$ or $A^{- β} (Ω)$ , $β > 0$ , to the weighted Bergman space $A_{α}^{p} (Ω)$ , $0 < p < \infty$ , $α > - 1$ . Also, given $0 < q < \infty$ and $β > 0$ , we describe those positive measures μ on Ω for which $A^{- β} (Ω) \subset L^{q} (Ω, μ)$ and those μ for which $A^{- \log} (Ω) \subset L^{q} (Ω, μ)$ .

Reçu le : 2008-11-07
Accepté le : 2009-03-20
Publié le : 2009-04-18

DOI : 10.1016/j.crma.2009.04.003

Affiliations des auteurs :

Evgueni Doubtsov ¹

¹ St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

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     author = {Evgueni Doubtsov},
     title = {Growth spaces on circular domains: composition operators and {Carleson} measures},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {609--612},
     publisher = {Elsevier},
     volume = {347},
     number = {11-12},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.003},
     language = {en},
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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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