Growth spaces on circular domains: composition operators and Carleson measures

Doubtsov, Evgueni

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

Doubtsov, Evgueni ¹

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

Comptes Rendus. Mathématique, Tome 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 :

Doubtsov, Evgueni ¹

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

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     title = {Growth spaces on circular domains: composition operators and {Carleson} measures},
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     pages = {609--612},
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Doubtsov, Evgueni. Growth spaces on circular domains: composition operators and Carleson measures. Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 609-612. doi : 10.1016/j.crma.2009.04.003. http://archive.numdam.org/articles/10.1016/j.crma.2009.04.003/

Bibliographie
Cité par

[1] Aleksandrov, A.B. Proper holomorphic mappings from the ball to the polydisk, Dokl. Akad. Nauk SSSR, Volume 286 (1986) no. 1, pp. 11-15 (in Russian); English transl.: Soviet Math. Dokl., 33, 1, 1986, pp. 1-5

[2] Blasco, O.; Lindström, M.; Taskinen, J. Bloch-to-BMOA compositions in several complex variables, Complex Var. Theory Appl., Volume 50 (2005) no. 14, pp. 1061-1080

[3] Carleson, L. An interpolation problem for bounded analytic functions, Amer. J. Math., Volume 80 (1958), pp. 921-930

[4] Cowen, C.C.; MacCluer, B.D. Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995

[5] Girela, D.; Peláez, J.Á.; Pérez-González, F.; Rättyä, J. Carleson measures for the Bloch space, Integral Equations Operator Theory, Volume 61 (2008) no. 4, pp. 511-547

[6] Kot, P. Homogeneous polynomials on strictly convex domains, Proc. Amer. Math. Soc., Volume 135 (2007) no. 12, pp. 3895-3903

[7] Ramey, W.; Ullrich, D. Bounded mean oscillation of Bloch pull-backs, Math. Ann., Volume 291 (1991) no. 4, pp. 591-606

[8] Rudin, W. New Constructions of Functions Holomorphic in the Unit Ball of $C^{n}$ , CBMS Regional Conference Series in Mathematics, vol. 63, 1986 (Published for the Conference Board of the Mathematical Sciences, Washington, DC)

[9] Ryll, J.; Wojtaszczyk, P. On homogeneous polynomials on a complex ball, Trans. Amer. Math. Soc., Volume 276 (1983) no. 1, pp. 107-116

[10] Ullrich, D.C. A Bloch function in the ball with no radial limits, Bull. London Math. Soc., Volume 20 (1988) no. 4, pp. 337-341

[11] Wojtaszczyk, P. On highly nonintegrable functions and homogeneous polynomials, Ann. Polon. Math., Volume 65 (1997) no. 3, pp. 245-251

Cité par Sources :