Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An extension theorem for separately holomorphic functions with pluripolar singularities

Authors: Marek Jarnicki and Peter Pflug
Journal: Trans. Amer. Math. Soc. 355 (2003), 1251-1267
MSC (2000): Primary 32D15, 32D10
Published electronically: November 5, 2002
MathSciNet review: 1938756
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $D_j\subset\mathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put

\begin{displaymath}X:=\bigcup_{j=1}^N A_1\times\dots\times A_{j-1}\times D_j\tim... ...thbb{C} ^{n_1}\times\dots\times\mathbb{C} ^{n_N}=\mathbb{C} ^n.\end{displaymath}

Let $U\subset\mathbb{C} ^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,\dots,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times\dots\times A_{j-1}) \times(A_{j+1}\times\dots\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb{C} ^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,\dots,\Sigma_N$ are pluripolar. Put
\begin{multline*}X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times\dots\times A_{j-... ...imes(A_{j+1}\times\dots\times A_N):\\ (z',z'')\notin\Sigma_j\}. \end{multline*}
Then there exists a relatively closed pluripolar subset $\widehat{M}\subset\widehat X$ of the ``envelope of holomorphy'' $\widehat{X}\subset\mathbb{C} ^n$ of $X$ such that:

$\bullet$ $\widehat M\cap X'\subset M$,

$\bullet$ for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\widehat f$ holomorphic on $\widehat X\setminus\widehat M$ with $\widehat f=f$ on $X'\setminus M$, and

$\bullet$ $\widehat M$ is singular with respect to the family of all functions $\widehat f$.

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

  • [1] O. Alehyane and A. Zeriahi, Une nouvelle version du théorème d'extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques, Ann. Polon. Math. 76 (2001), 245-278. MR 2002f:32018
  • [2] D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag, New York, 2001. MR 2001m:31001
  • [3] E. M. Chirka, The extension of pluripolar singularity sets, Proc. Steklov Inst. Math. 200 (1993), 369-373. MR 93a:32015
  • [4] E. M. Chirka and A. Sadullaev, On continuation of functions with polar singularities, Math. USSR-Sb. 60 (1988), 377-384. MR 89a:32014
  • [5] M. Jarnicki and P. Pflug, Extension of Holomorphic Functions, de Gruyter Expositions in Mathematics 34, Walter de Gruyter, 2000. MR 2001k:32017
  • [6] M. Jarnicki and P. Pflug, Cross theorem, Ann. Polon. Math. 77 (2001), 295-302. MR 2002i:32009
  • [7] M. Jarnicki and P. Pflug, An extension theorem for separately holomorphic functions with singularities, IMUJ Preprint 2001/27 (2001).
  • [8] M. Jarnicki and P. Pflug, An extension theorem for separately holomorphic functions with analytic singularities, Ann. Polon. Math., to appear.
  • [9] M. V. Kazaryan, On holomorphic continuation of functions with pluripolar singularities, Dokl. Akad. Nauk Arm. SSR 87 (1988), 106-110 (in Russian). MR 90g:32016
  • [10] Nguyen Thanh Van, Separate analyticity and related subjects, Vietnam J. Math. 25 (1997), 81-90. MR 99m:32002
  • [11] Nguyen Thanh Van and J. Siciak, Fonctions plurisousharmoniques extrémales et systèmes doublement orthogonaux de fonctions analytiques, Bull. Sci. Math. 115 (1991), 235-244. MR 92a:32018
  • [12] Nguyen Thanh Van and A. Zeriahi, Une extension du théorème de Hartogs sur les fonctions séparément analytiques in Analyse Complexe Multivariables, Récents Dévelopements, A. Meril (éd.), EditEl, Rende 1991, 183-194. MR 94e:32004
  • [13] Nguyen Thanh Van and A. Zeriahi, Systèmes doublement othogonaux de fonctions holomorphes et applications, Banach Center Publ. 31 (1995), 281-297. MR 96g:32004
  • [14] O. Öktem, Extension of separately analytic functions and applications to range characterization of exponential Radon transform, Ann. Polon. Math. 70 (1998), 195-213. MR 2000c:32037
  • [15] O. Öktem, Extending separately analytic functions in $\mathbb C^{n+m}$with singularities in Extension of separately analytic functions and applications to mathematical tomography (Thesis), Dep. Math. Stockholm Univ. 1999.
  • [16] B. Shiffman, On separate analyticity and Hartogs theorem, Indiana Univ. Math. J. 38 (1989), 943-957. MR 91a:32018
  • [17] J. Siciak, Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $\mathbb C^n$, Ann. Polon. Math. 22 (1969-1970), 147-171. MR 40:5893
  • [18] J. Siciak, Extremal plurisubharmonic functions in $\mathbb C^N$, Ann. Polon. Math. 39 (1981), 175-211. MR 83e:32018
  • [19] J. Siciak, Holomorphic functions with singularities on algebraic sets, Univ. Iag. Acta Math. No. 39 (2001), 9-16.
  • [20] V. P. Zahariuta, Separately analytic functions, generalizations of Hartogs theorem, and envelopes of holomorphy, Math. USSR-Sb. 30 (1976), 51-67. MR 54:13128

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32D15, 32D10

Retrieve articles in all journals with MSC (2000): 32D15, 32D10

Additional Information

Marek Jarnicki
Affiliation: Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland

Peter Pflug
Affiliation: Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany

Received by editor(s): February 12, 2002
Received by editor(s) in revised form: June 3, 2002
Published electronically: November 5, 2002
Additional Notes: The first author was supported in part by KBN grant no. 5 P03A 033 21.
Both authors were supported in part by the Niedersächsisches Ministerium für Wissenschaft und Kultur, Az. 15.3 – 50 113(55) PL
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society