Abstract
Nous donnons une caractérisation des domaines D⊂X pour lesquels la fonction extrémale relative ω*(⋅,E,D) a la propriété de stabilité pour tout E⊂D, i.e. lim k→∞ω*(⋅,E,D k )=ω*(⋅,E,D), ∀E⊂D. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim k→∞ω*(⋅,E k ,D)=ω*(⋅,E,D).
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Références
Bedford, E.: 'The operator (dd c)n on complex spaces', in Séminaire d'analyse Lelong-Skoda, Lectures Notes in Math. 919, 1981, pp. 294-324.
Bedford, E. and Taylor, B.A.: 'A new capacity for plurisubharmonic functions', Acta. Math. 149 (1982), 1-40.
Cegrell, U.: Capacities in Complex Analysis, Aspects of Mathematics, Vieweg, Viesbaden, 1988.
Cegrell, U. and Poletsky, E.A.: 'A counterexample to the “fine” problem in pluripotential theory', Proc. Amer. Math. Soc. 123(12) (1995), 3677-3679.
Edigarian, A. and Poletsky, A.E.: 'Product property of the relative extremal function', Bull. Polish Acad. Sci. Math. 45 (1997), 331-335.
Hayman, W.K. and Kennedy, P.B.: Subharmonic Functions I, London Math. Soc. Student Texts 28, Academic Press, London, 1976.
Josefson, B.: 'On the the equivalence between locally and globally polar sets for plurisubharmonic functions in ℂn', Ark. Mat. 16 (1978), 109-115.
Klimek, M.: Pluripotential Theory, Oxford Sciences Publications, London, 1991.
Levenberg, N. and Poletsky, E.A.: 'Pluripolar hulls', Michigan Math. J. 46 (1999), 151-162.
Nguyen, T.V. and Zeriahi, A.: 'Une extension du théorème de Hartogs sur les fonctions séparément analytiques', in A. Meril (ed.), Analyse Complexe Multivariable: Recents Developpements, 1988, pp. 183-194.
Nguyen, T.V. and Zeriahi, A.: 'Systèmes doublement orthogonaux de fonctions holomorphes et applications', in Topics in Complex Analysis 31, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1995, pp. 281-297.
Sadullaev, A.: 'Plurisubharmonic measures and capacities on complex manifolds', Russian Math. Surveys 36 (1981), 61-119.
Sadullaev, A.: 'Plurisubharmonic functions, several complex variables II, Geometric function theory', Encyclopaedia Math. Sci. 8 (1989), 59-106.
Siciak, J.: 'Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ℂn', Ann. Polon. Math. 22 (1970), 145-171.
Siciak, J.: 'Extremal plurisubharmonic functions in ℂn', Ann. Polon. Math. 34 (1981), 175-211.
Wiegerinck, J.: 'Graphs of holomorphic functions with isolated singularities are complete pluripolar', Préprint, 1999.
Wiegerinck, J.: 'The pluripolar hull of w = e1/z', Ark. Mat. (1999).
Zahariuta, V.P.: 'Fonctions plurisousharmoniques extrémales: échelles hilbertiennes et isomorphismes d'espaces de fonctions analytiques de plusieurs variables, I, II', Teor. Funcii Funktsional. Anal. i Prilozhen. 19, 21 (1974), 133-157, 65-83 (en russe).
Zahariuta, V.P.: 'Separately holomorphic functions, generalizations of Hartogs theorem and envelopes of holomorphy', Math. USSR Sb. 30 (1976), 51-76.
Zeriahi, A.: 'Ensembles pluripolaires exceptionnelles pour la croissance partielle des fonctions holomorphes', Ann. Polon. Math. 50 (1989), 81-91.
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Alehyane, O., Hecart, JM. Propriété de stabilité de la fonction extrémale relative. Potential Analysis 21, 363–373 (2004). https://doi.org/10.1023/B:POTA.0000034326.53466.fe
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DOI: https://doi.org/10.1023/B:POTA.0000034326.53466.fe