On Lelong-Bremermann Lemma
HTML articles powered by AMS MathViewer
- by Aydin Aytuna and Vyacheslav Zakharyuta PDF
- Proc. Amer. Math. Soc. 136 (2008), 1733-1742 Request permission
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 \begin{equation*} 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 ) \end{equation*} 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
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. MR 1805196, DOI 10.1007/978-1-4757-8137-3
- David H. Armitage and Stephen J. Gardiner, Classical potential theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. MR 1801253, DOI 10.1007/978-1-4471-0233-5
- Bloom, T., Levenberg, N., and Lyubarskii, Yu., A Hilbert Lemniscate Theorem in $\mathbb {C}^{2}$, preprint
- Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-AmpĂšre equation, Invent. Math. 37 (1976), no. 1, 1â44. MR 445006, DOI 10.1007/BF01418826
- H. J. Bremermann, On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions, Math. Ann. 131 (1956), 76â86. MR 77644, DOI 10.1007/BF01354666
- Jean-Pierre Demailly, On the Ohsawa-Takegoshi-Manivel $L^2$ extension theorem, Complex analysis and geometry (Paris, 1997) Progr. Math., vol. 188, BirkhĂ€user, Basel, 2000, pp. 47â82 (English, with English and French summaries). MR 1782659
- Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR 1052649
- T. W. Gamelin and N. Sibony, Subharmonicity for uniform algebras, J. Functional Analysis 35 (1980), no. 1, 64â108. MR 560218, DOI 10.1016/0022-1236(80)90081-6
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
- Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, BirkhÀuser Boston, Inc., Boston, MA, 1994. MR 1301332
- Pierre Lelong, Sur quelques problĂšmes de la thĂ©orie des fonctions de deux variables complexes, Ann. Sci. Ăcole Norm. Sup. (3) 58 (1941), 83â177 (French). MR 0013421
- StĂ©phanie Nivoche, Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the $\epsilon$-entropy, Invent. Math. 158 (2004), no. 2, 413â450. MR 2096799, DOI 10.1007/s00222-004-0372-5
- Evgeny A. Poletsky, Approximation of plurisubharmonic functions by multipole Green functions, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1579â1591. MR 1946406, DOI 10.1090/S0002-9947-02-03215-4
- Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
- A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Uspekhi Mat. Nauk 36 (1981), no. 4(220), 53â105, 247 (Russian). MR 629683
- Nessim Sibony, Prolongement des fonctions holomorphes bornĂ©es et mĂ©trique de CarathĂ©odory, Invent. Math. 29 (1975), no. 3, 205â230 (French). MR 385164, DOI 10.1007/BF01389850
- V. Zahariuta, Spaces of analytic functions and complex potential theory, Linear Topol. Spaces Complex Anal. 1 (1994), 74â146. MR 1323360
- V. Zahariuta, On approximation by special analytic polyhedral pairs, Proceedings of Conference on Complex Analysis (Bielsko-BiaĆa, 2001), 2003, pp. 243â256. MR 1972851, DOI 10.4064/ap80-0-22
Additional Information
- Aydin Aytuna
- Affiliation: FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey
- MR Author ID: 28620
- Email: aytuna@sabanciuniv.edu
- Vyacheslav Zakharyuta
- Affiliation: FENS, Sabanci University, 34956 Tuzla/Istanbul, Turkey
- Email: zaha@sabanciuniv.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1733-1742
- MSC (2000): Primary 32U05; Secondary 31C10
- DOI: https://doi.org/10.1090/S0002-9939-08-09166-1
- MathSciNet review: 2373603