Complex Analysis
Growth spaces on circular domains: composition operators and Carleson measures
Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 609-612.

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).

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.04.003
Doubtsov, Evgueni 1

1 St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
@article{CRMATH_2009__347_11-12_609_0,
     author = {Doubtsov, Evgueni},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.003/}
}
TY  - JOUR
AU  - Doubtsov, Evgueni
TI  - Growth spaces on circular domains: composition operators and Carleson measures
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 609
EP  - 612
VL  - 347
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.04.003/
DO  - 10.1016/j.crma.2009.04.003
LA  - en
ID  - CRMATH_2009__347_11-12_609_0
ER  - 
%0 Journal Article
%A Doubtsov, Evgueni
%T Growth spaces on circular domains: composition operators and Carleson measures
%J Comptes Rendus. Mathématique
%D 2009
%P 609-612
%V 347
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.04.003/
%R 10.1016/j.crma.2009.04.003
%G en
%F CRMATH_2009__347_11-12_609_0
Doubtsov, Evgueni. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.04.003/

[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 Cn, 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

Cited by Sources: