Zeros of bounded holomorphic functions in strictly pseudoconvex domains in ℂ 2

Jim Arlebrink

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 2, page 437-458
  • ISSN: 0373-0956

Abstract

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Let D be a bounded strictly pseudoconvex domain in ℂ 2 and let X be a positive divisor of D with finite area. We prove that there exists a bounded holomorphic function f such that X is the zero set of f . This result has previously been obtained by Berndtsson in the case where D is the unit ball in ℂ 2 .

How to cite

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Arlebrink, Jim. "Zeros of bounded holomorphic functions in strictly pseudoconvex domains in ${\mathbb {C}}^2$." Annales de l'institut Fourier 43.2 (1993): 437-458. <http://eudml.org/doc/75003>.

@article{Arlebrink1993,
abstract = {Let $D$ be a bounded strictly pseudoconvex domain in $\{\Bbb C\}^2$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$. This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in $\{\Bbb C\}^2$.},
author = {Arlebrink, Jim},
journal = {Annales de l'institut Fourier},
keywords = {-equation; zero divisor; bounded strictly pseudoconvex domain; bounded holomorphic function},
language = {eng},
number = {2},
pages = {437-458},
publisher = {Association des Annales de l'Institut Fourier},
title = {Zeros of bounded holomorphic functions in strictly pseudoconvex domains in $\{\mathbb \{C\}\}^2$},
url = {http://eudml.org/doc/75003},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Arlebrink, Jim
TI - Zeros of bounded holomorphic functions in strictly pseudoconvex domains in ${\mathbb {C}}^2$
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 2
SP - 437
EP - 458
AB - Let $D$ be a bounded strictly pseudoconvex domain in ${\Bbb C}^2$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$. This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in ${\Bbb C}^2$.
LA - eng
KW - -equation; zero divisor; bounded strictly pseudoconvex domain; bounded holomorphic function
UR - http://eudml.org/doc/75003
ER -

References

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  1. [AC] M. ANDERSSON, H. CARLSSON, On Varopoulos' theorem about zero sets of Hp-functions, Bull. Sc. Math., 114 (1990), 463-484. Zbl0725.32005MR91j:32006
  2. [Ar] J. ARLEBRINK, Zeros of bounded holomorphic functions in C2, Preprint Göteborg (1989). 
  3. [Be] B. BERNDTSSON, Integral formulas for the ∂∂-equation and zeros of bounded holomorphic functions in the unit ball, Math. Ann., 249 (1980), 163-176. Zbl0414.31007MR81m:32012
  4. [BA] B. BERNDTSSON, M. ANDERSSON, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier, 32-3 (1982), 91-100. Zbl0466.32001MR84j:32003
  5. [Fo] J. E. FORNAESS, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math., 98 (1976), 529-569. Zbl0334.32020MR54 #10669
  6. [He1] G. M. HENKIN, Solutions with estimates of the H. Levy and Poincaré-Lelong equations. Constructions of functions of the Nevanlinna class with prescribed zeros in strictly pseudoconvex domains, Soviet Math. Dokl., 16 (1976), 3-13. 
  7. [He2] G. M. HENKIN, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math., Surveys, 32-3 (1977), 59-130. Zbl0382.35038MR56 #12318
  8. [KS] N. KERZMAN, G. STEIN, The Szegö kernel in terms of Cauchy-Fantappiè kernels, Duke Math. J., 45 (1978), 197-224. Zbl0387.32009MR58 #22676
  9. [Le1] P. LELONG, Fonctionnelles analytiques et fonctions entières (n variables), Presses Univ. Montréal, Montréal, 1968. Zbl0194.38801MR57 #6483
  10. [Le2] P. LELONG, Fonctions plurisousharmoniques et formes différentielles positives, Gordon and Breach, Paris-London-New York, 1968. Zbl0195.11603MR39 #4436
  11. [Sk1] H. SKODA, Diviseurs d'aire bornée dans la boule de C2: réflexions sur un article de B. Berndtsson, Sem. Lelong-Skoda 1978-79, LNM 822, Springer-Verlag, Berlin-Heidelberg-New York, 1980. Zbl0443.32002
  12. [Sk2] H. SKODA, Valeurs au bord pour les solutions de l’opérateur d " et caractérisation de zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France, 104 (1976), 225-299. Zbl0351.31007MR56 #8913
  13. [Ra] R. M. RANGE, Holomorphic functions and integral representations in several complex variables, Springer-Verlag, Berlin-Heidelberg-New-York, 1986. Zbl0591.32002MR87i:32001
  14. [Va] N. Th. VAROPOULOS, Zeros of Hp-functions in several variables, Pacific J. Math., 88 (1980), 189-246. Zbl0454.32006

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