Abstract
Denote by L the class of plurisubharmonic functions f on ₵n such that
where af 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.
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.
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.
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 ey(Ve) is a capacity was proved by the same author in: Capacities associated to the Siciak extremal function, Manuscript. Cracow. 1986.
The relationship between γ and Γ has also been studied by J. Siciak, On logarithmic capacities and pluripolar sets in ₵n. Manuscript, October 1986.
Using Corollary 6.7 in E. Bedford and B.A. Taylor, Plurisubharmonic functions with logarithmic singularities, Manuscript 1987, one can prove that ey(Ve) is an outer regular capacity.
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.
Extremal plurisubharmonic functions, orthogonal polynomials and the Bernstein-Walsh theorem for analytic functions of several complex variables. Ann. Polon. Math. 33 (1976).
Nguyen Thanh Van and Ahmed Zeriahi, Familles de polynômes presque partout bornées. Bull. Sc. Math. 2c Série 107 (1983).
W. Plesniak and W. Pawłucki, Markov’s inequality and C∞ functions on sets with polynomial cusps. Math. Ann. 275 (1986), 467–480.
A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds. Russian Math. Surveys 36 (1981).
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© 1988 Springer Fachmedien Wiesbaden
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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
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