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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence domains of holomorphic functions of restricted growth


Authors: M. Jarnicki and P. Pflug
Journal: Trans. Amer. Math. Soc. 304 (1987), 385-404
MSC: Primary 32D05; Secondary 32A07, 32D10
MathSciNet review: 906821
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Abstract: The paper presents a large class of domains $ G$ of holomorphy in $ {{\mathbf{C}}^n}$ such that, for any $ N > 0$, there exists a nonextendable holomorphic function $ f$ on $ G$ with $ \vert f\vert\delta _G^N$ bounded where $ {\delta _G}(z): = \min ({(1 + \vert z{\vert^2})^{ - 1 / 2}},\,\operatorname{dist} (z,\,\partial G))$. Any fat Reinhardt domain of holomorphy belongs to this class.

On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.


References [Enhancements On Off] (What's this?)

  • [1] O. U. Balashova, Description of $ 2$-circled $ {H^\infty }$-domains of holomorphy, Some Problems of Multidimensional Complex Analysis, Akad. Nauk SSSR Sibirsk Otdel. Inst. Fiz., Krasnoyarsk, 1980, pp. 221-224. (Russian)
  • [2] N. Bourbaki, Topologie générale, Hermann, Paris, 1961.
  • [3] T. W. Gamelin, Peak points for algebras on circled sets, Math. Ann. 238 (1978), 131-139. MR 0508099 (58:22663)
  • [4] J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699. MR 0276456 (43:2203)
  • [5] R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 0180696 (31:4927)
  • [6] R. Narasimhan, On holomorphic functions of polynomial growth in a bounded domain, Ann. Scuola Norm. Sup. Pisa 21 (1967), 161-166. MR 0217332 (36:423)
  • [7] M. Jarnicki and P. Pflug, Non-extendable holomorphic functions of bounded growth in Reinhardt domains, Ann. Polon. Math. 46 (1985), 129-140. MR 841818 (87j:32038)
  • [8] M. Jarnicki and P. Tworzewski, A note on holomorphic functions with polynomial growth, Osnabrücker Schriften Math. 67 (1983).
  • [9] P. Pflug, Über polynomiale Funktionen auf Holomorphiegebieten, Math. Z. 139 (1974), 133-139. MR 0355102 (50:7579)
  • [10] -, Quadratintegrale holomorphe Funktionen und die Serre-Vermutung, Math. Ann. 216 (1975), 285-288. MR 0382717 (52:3599)
  • [11] -, About the Carathéodory completeness of all Reinhardt domains, Functional Analysis, Holomorphy and Approximation Theory II, North-Holland Math. Studies 86, North-Holland, Amsterdam, 1984, pp. 331-337. MR 771335 (86b:32026)
  • [12] N. Sibony, Prolongement des fonctions holomorphes bornées et métrique de Carathéodory, Invent. Math. 29 (1975), 205-230. MR 0385164 (52:6029)
  • [13] J. Siciak, Extremal plurisubharmonic functions and capacities in $ {{\mathbf{C}}^n}$, Sophia Kokyuroku Math. 14 (1982).
  • [14] -, Circled domains of holomorphy of type $ {H^\infty }$, Bull. Soc. Sci. Lett. Łodź 1 (1984/85), 1-20.
  • [15] -, Balanced domains of holomorphy of type $ {H^\infty }$, Mat. Vesnik 37 (1985), 134-144. MR 791577 (87b:32022)
  • [16] P. Tworzewski and T. Winiarski, Analytic sets with proper projections, J. Reine Angew. Math. 337 (1982), 68-76. MR 676042 (84f:32006)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0906821-9
PII: S 0002-9947(1987)0906821-9
Article copyright: © Copyright 1987 American Mathematical Society