Zeros of bounded holomorphic functions in strictly pseudoconvex domains in 2
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 437-458

Let D be a bounded strictly pseudoconvex domain in 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 2 .

Soit D un domaine strictement pseudoconvexe borné dans 2 , et soit X un diviseur positif de D d’aire finie. On montre l’existence d’une fonction bornée f dont X est l’ensemble des zéros de f. Ceci généralise un résultat de B. Berndtsson dans le cas où D est la boule unité de 2 .

@article{AIF_1993__43_2_437_0,
     author = {Arlebrink, Jim},
     title = {Zeros of bounded holomorphic functions in strictly pseudoconvex domains in ${\mathbb {C}}^2$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {437-458},
     doi = {10.5802/aif.1339},
     zbl = {0782.32013},
     mrnumber = {94f:32021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_2_437_0}
}
Arlebrink, Jim. Zeros of bounded holomorphic functions in strictly pseudoconvex domains in ${\mathbb {C}}^2$. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 437-458. doi : 10.5802/aif.1339. http://www.numdam.org/item/AIF_1993__43_2_437_0/

[AC] M. Andersson, H. Carlsson, On Varopoulos' theorem about zero sets of Hp-functions, Bull. Sc. Math., 114 (1990), 463-484. | MR 91j:32006 | Zbl 0725.32005

[Ar] J. Arlebrink, Zeros of bounded holomorphic functions in C2, Preprint Göteborg (1989).

[Be] B. Berndtsson, Integral formulas for the ∂∂-equation and zeros of bounded holomorphic functions in the unit ball, Math. Ann., 249 (1980), 163-176. | MR 81m:32012 | Zbl 0414.31007

[BA] B. Berndtsson, M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier, 32-3 (1982), 91-100. | Numdam | MR 84j:32003 | Zbl 0466.32001

[Fo] J. E. Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math., 98 (1976), 529-569. | MR 54 #10669 | Zbl 0334.32020

[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.

[He2] G. M. Henkin, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math., Surveys, 32-3 (1977), 59-130. | MR 56 #12318 | Zbl 0382.35038

[KS] N. Kerzman, G. Stein, The Szegö kernel in terms of Cauchy-Fantappiè kernels, Duke Math. J., 45 (1978), 197-224. | MR 58 #22676 | Zbl 0387.32009

[Le1] P. Lelong, Fonctionnelles analytiques et fonctions entières (n variables), Presses Univ. Montréal, Montréal, 1968. | MR 57 #6483 | Zbl 0194.38801

[Le2] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Gordon and Breach, Paris-London-New York, 1968. | MR 39 #4436 | Zbl 0195.11603

[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. | Zbl 0443.32002

[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. | Numdam | MR 56 #8913 | Zbl 0351.31007

[Ra] R. M. Range, Holomorphic functions and integral representations in several complex variables, Springer-Verlag, Berlin-Heidelberg-New-York, 1986. | MR 87i:32001 | Zbl 0591.32002

[Va] N. Th. Varopoulos, Zeros of Hp-functions in several variables, Pacific J. Math., 88 (1980), 189-246. | Zbl 0454.32006