The general definition of the complex Monge-Ampère operator
[Une définition générale de l'opérateur de Monge-Ampère complexe]
Cegrell, Urban1
1 Ume{\aa} University, Department of Mathematics, 901 87 Ume{\aa} (Suède)
Annales de l'Institut Fourier, Tome 54 (2004) no. 1, pp. 159-179.

On définit et étudie le domaine de définition de l'opérateur de Monge-Ampère complexe. Ce domaine est le plus général possible si on impose que l'opérateur soit continu pour les limites décroissantes. Ce domaine est donné à l'aide d'approximation par certaines fonctions plurisousharmoniques jouant le rôle de "fonctions test". On démontre des estimations, on étudie un théorème de décomposition pour les mesures positives et on résout le problème de Dirichlet.

We define and study the domain of definition for the complex Monge-Ampère operator. This domain is the most general if we require the operator to be continuous under decreasing limits. The domain is given in terms of approximation by certain " test"-plurisubharmonic functions. We prove estimates, study of decomposition theorem for positive measures and solve a Dirichlet problem.

DOI : 10.5802/aif.2014
Classification : 32U15, 32W20
Keywords: complex Monge-Ampère operator, plurisubharmonic function
Mot clés : opérateur de Monge-Ampère complexe, fonction plurisousharmonique
Cegrell, Urban 1

1 Ume{\aa} University, Department of Mathematics, 901 87 Ume{\aa} (Suède) 
@article{AIF_2004__54_1_159_0,
     author = {Cegrell, Urban},
     title = {The general definition of the complex {Monge-Amp\`ere} operator},
     journal = {Annales de l'Institut Fourier},
     pages = {159--179},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {1},
     year = {2004},
     doi = {10.5802/aif.2014},
     zbl = {1065.32020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2014/}
}
TY  - JOUR
AU  - Cegrell, Urban
TI  - The general definition of the complex Monge-Ampère operator
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 159
EP  - 179
VL  - 54
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2014/
DO  - 10.5802/aif.2014
LA  - en
ID  - AIF_2004__54_1_159_0
ER  - 
%0 Journal Article
%A Cegrell, Urban
%T The general definition of the complex Monge-Ampère operator
%J Annales de l'Institut Fourier
%D 2004
%P 159-179
%V 54
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2014/
%R 10.5802/aif.2014
%G en
%F AIF_2004__54_1_159_0
Cegrell, Urban. The general definition of the complex Monge-Ampère operator. Annales de l'Institut Fourier, Tome 54 (2004) no. 1, pp. 159-179. doi : 10.5802/aif.2014. http://archive.numdam.org/articles/10.5802/aif.2014/

[1] E. Bedford; John Erik Fornaess Survey of pluripotential theory. Several complex variables, Proceedings of the Mittag-Leffler Inst. (1987-88) (Mathematical Notes), Volume 38 (1994), pp. 48-95 | MR | Zbl

[2] E. Bedford; B.A. Taylor The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math, Volume 37 (1976), pp. 1-44 | MR | Zbl

[3] E. Bedford; B.A. Taylor A new capacity for plurisubharmonic functions, Acta Math, Volume 149 (1982), pp. 1-40 | MR | Zbl

[4] Z. Blocki Estimates for the complex Monge-Ampère operator, Bull. Pol. Acad. Sci. Math, Volume 41 (1993), pp. 151-157 | MR | Zbl

[5] Z. Blocki The complex Monge-Ampère operator in hyperconvex domains, Annali della Scuola Normale Superiore di Pisa, Volume 23 (1996) no. 4, pp. 721-747 | Numdam | MR | Zbl

[6] M. Carlehed Potentials in pluripotential theory, Ann. de la Fac. Sci. de Toulouse (6), Volume 8 (1999) no. 3, pp. 439-469 | Numdam | MR | Zbl

[7] U. Cegrell Pluricomplex energy, Acta Mathematica, Volume 180 (1998) no. 2, pp. 187-217 | MR | Zbl

[8] U. Cegrell; Gilles Raby and Frédéric Symesak Explicit calculation of a Monge-Ampère measure, Actes des rencontres d'analyse complexe (Université de Poitiers, 25-28 mars 1999) (2000), pp. 39-42 | MR | Zbl

[9] U. Cegrell Convergence in capacity (2001) (Isaac Newton Institute for Mathematical Sciences, Preprint Series NI01046-NPD, Cambridge)

[10] U. Cegrell Exhaustion functions for hyperconvex domains (2001) (Research reports, No 10, Mid Sweden University)

[11] U. Cegrell; S. Kolodziej The Dirichlet problem for the complex Monge-Ampère operator: Perron classes and rotation invariant measures, Michigan. Math. J, Volume 41 (1994), pp. 563-569 | MR | Zbl

[12] D. Coman Integration by parts for currents and applications to the relative capacity and Lelong numbers, Mathematica, Volume 39(62) (1997) no. 1, pp. 45-57 | MR | Zbl

[13] J.-P. Demailly Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z, Volume 194 (1987), pp. 519-564 | MR | Zbl

[14] N. Kerzman; J.-P. Rosay Fonctions plurisousharmoniques d'exhaustion bornées et domaines taut, Math. Ann, Volume 257 (1981), pp. 171-184 | MR | Zbl

[15] S. Kolodziej The complex Monge-Ampère equation, Acta Mathematica, Volume 180 (1998), pp. 69-117 | MR | Zbl

[16] N. Sibony Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J, Volume 52 (1985), pp. 157-197 | MR | Zbl

[17] J. Siciak Extremal plurisubharmonic functions and capacities in ℂ n , Sophia Kokyuroko in Mathematics (1982) | Zbl

[18] J.B. Walsh Continuity of envelopes of plurisubharmonic functions, J. Math. Mech, Volume 18 (1968), pp. 143-148 | MR | Zbl

[19] F. Wikström Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat, Volume 39 (2001), pp. 181-200 | MR | Zbl

[20] Y. Xing Complex Monge-Ampère equations with a countable number of singular points, Indiana Univ. Math. J, Volume 48 (1999), pp. 749-765 | MR | Zbl

[21] A. Zeriahi Pluricomplex Green functions and the Dirichlet problem for the Complex Monge-Ampère operator, Michigan Math. J, Volume 44 (1997), pp. 579-596 | MR | Zbl

Cité par Sources :