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On Lelong-Bremermann Lemma

Authors:	Aydin Aytuna Vyacheslav Zakharyuta

Institution:	FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey ; FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey

Abstract:	The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma: Let be a continuous plurisubharmonic function on a Stein manifold $\Omega$ of dimension Then there exists an integer $m\leq 2n+1$ , natural numbers $p_{s}$ , and analytic mappings $G_{s}=\left( g_{j}^{\left( s\right) }\right) :\Omega \rightarrow \mathbb{C}^{m}, s=1,2,...,$ such that the sequence of functions $\displaystyle u_{s}\left( z\right) =\frac{1}{p_{s}}\max \left( \ln \left\vert g_{j}^{\left( s\right) }\left( z\right) \right\vert :\text{ }j=1,\ldots ,m\right)$ converges to uniformly on each compact subset of $\Omega$ . In the case when $\Omega$ is a domain in the complex plane, it is shown that one can take in the theorem above (Section 3); on the other hand, for -circular plurisubharmonic functions in $\mathbb{C}^{n}$ the statement of this theorem is true with (Section 4). The last section contains some remarks and open questions.

Keywords:	Plurisubharmonic functions Lelong-Bremermann Lemma

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