Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Lelong-Bremermann Lemma


Authors: Aydin Aytuna and Vyacheslav Zakharyuta
Journal: Proc. Amer. Math. Soc. 136 (2008), 1733-1742
MSC (2000): Primary 32U05; Secondary 31C10
Published electronically: January 17, 2008
MathSciNet review: 2373603
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma:

Let $ u$ be a continuous plurisubharmonic function on a Stein manifold $ \Omega $ of dimension $ n.$ 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 $ u$ 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 $ m=2$ in the theorem above (Section 3); on the other hand, for $ n$-circular plurisubharmonic functions in $ \mathbb{C}^{n}$ the statement of this theorem is true with $ m=n+1$ (Section 4). The last section contains some remarks and open questions.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32U05, 31C10

Retrieve articles in all journals with MSC (2000): 32U05, 31C10


Additional Information

Aydin Aytuna
Affiliation: FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey
Email: aytuna@sabanciuniv.edu

Vyacheslav Zakharyuta
Affiliation: FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey
Email: zaha@sabanciuniv.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09166-1
PII: S 0002-9939(08)09166-1
Keywords: Plurisubharmonic functions, Lelong-Bremermann Lemma
Received by editor(s): October 24, 2006
Received by editor(s) in revised form: March 7, 2007
Published electronically: January 17, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.