The Global Extremal Function

Cegrell, Urban

doi:10.1007/978-3-663-14203-4_9

Urban Cegrell²

Part of the book series: Aspects of Mathematics / Aspekte der Mathematik ((ASMA,volume E 14))

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Abstract

Denote by L the class of plurisubharmonic functions f on ₵ⁿ such that

$$f\left( z \right) \leqslant {a_f} + {\log ^ + }\left| z \right|,z \in {\not \subset ^n}$$

where a_f is a constant (depending on f).

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Remarks and references

A proof of Proposition VIII:1 is in J. Siciak, Extremal plurisubharmonic functions in ₵ⁿ, Proceedings of the first Finnish-Polish Summerschool in Complex Analysis in Podlesice, 1977, pg. 123-124.
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Theorem VIII:1 is due to N. Levenberg, Monge-Ampère Measures Associated to Extremal Plurisubharmonic Functions in ₵ⁿ. Trans. Am. Math. Soc. 289 (1985), 333–343.
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Lemma VIII:1 is proved by B.A. Taylor, An estimate for an extremal plurisubharmonic function. Séminare d’Analyse P. belong, Dolbeault-H. Skoda, 1981/1983. Springer Lecture Notes in Mathematics 1028. This paper also contains a somewhat weaker version of Corollary VIII:1.
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In S. Kołodziej, The logarithmic capacity in ₵ⁿ (To appear in Ann. Pol. Math.), it was proved that C(E)=e is a capacity in Choquets sense. That e^y(Ve) is a capacity was proved by the same author in: Capacities associated to the Siciak extremal function, Manuscript. Cracow. 1986.
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The relationship between γ and Γ has also been studied by J. Siciak, On logarithmic capacities and pluripolar sets in ₵ⁿ. Manuscript, October 1986.
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Using Corollary 6.7 in E. Bedford and B.A. Taylor, Plurisubharmonic functions with logarithmic singularities, Manuscript 1987, one can prove that e^y(Ve) is an outer regular capacity.
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V.P. Zaharjuta has studied capacities and extremal plurisubharmonic functions in connection with transfinite diameter and the Bernstein-Walsh theorem: Transfinite diameter ĉebychêv constants and capacity for compact in ₵ⁿ. Math. USSR Sbornik, Vol. 25 (1975), No. 3.
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Extremal plurisubharmonic functions, orthogonal polynomials and the Bernstein-Walsh theorem for analytic functions of several complex variables. Ann. Polon. Math. 33 (1976).
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Nguyen Thanh Van and Ahmed Zeriahi, Familles de polynômes presque partout bornées. Bull. Sc. Math. 2c Série 107 (1983).
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W. Plesniak and W. Pawłucki, Markov’s inequality and C∞ functions on sets with polynomial cusps. Math. Ann. 275 (1986), 467–480.
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A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds. Russian Math. Surveys 36 (1981).
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Authors and Affiliations

Department of Mathematics, University of Umeå, Sweden
Prof. Dr. Urban Cegrell

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Prof. Dr. Urban Cegrell
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Cegrell, U. (1988). The Global Extremal Function. In: Capacities in Complex Analysis. Aspects of Mathematics / Aspekte der Mathematik, vol E 14. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14203-4_9

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DOI: https://doi.org/10.1007/978-3-663-14203-4_9
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-06335-1
Online ISBN: 978-3-663-14203-4
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