The general definition of the complex Monge-Ampère operator

Cegrell, Urban

doi:10.5802/aif.2014

The general definition of the complex Monge-Ampère operator
[Une définition générale de l'opérateur de Monge-Ampère complexe]

Cegrell, Urban ¹

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.

Résumé
Abstract

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.

Zbl | 8 citations dans Numdam

DOI : 10.5802/aif.2014

Classification : 32U15, 32W20
Keywords: complex Monge-Ampère operator, plurisubharmonic function
Mot clés : opérateur de Monge-Ampère complexe, fonction plurisousharmonique

Affiliations des auteurs :

Cegrell, Urban ¹

¹ Ume{\aa} University, Department of Mathematics, 901 87 Ume{\aa} (Suède)

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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. https://www.numdam.org/articles/10.5802/aif.2014/

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Hong, Nguyen Xuan; Trao, Nguyen Van; Thuy, Tran Van Convergence in capacity of plurisubharmonic functions with given boundary values, International Journal of Mathematics, Volume 28 (2017) no. 03, p. 1750018 | DOI:10.1142/s0129167x17500185
Di Nezza, Eleonora; Lu, Chinh H. Complex Monge–Ampère equations on quasi-projective varieties, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2017 (2017) no. 727, p. 145 | DOI:10.1515/crelle-2014-0090
Trao, Nguyen Van; Viet, Hoang; Hong, Nguyen Xuan Approximation of plurifinely plurisubharmonic functions, Journal of Mathematical Analysis and Applications, Volume 450 (2017) no. 2, p. 1062 | DOI:10.1016/j.jmaa.2017.01.028
Rashkovskii, Alexander Local geodesics for plurisubharmonic functions, Mathematische Zeitschrift, Volume 287 (2017) no. 1-2, p. 73 | DOI:10.1007/s00209-016-1817-4
Kang, Qianqian; Wan, Dongrui Potential theory for quaternionic plurisubharmonic functions, Michigan Mathematical Journal, Volume 66 (2017) no. 1 | DOI:10.1307/mmj/1488510022
Hawari, H.; Zaway, M. On the space of delta m-subharmonic functions, Analysis Mathematica, Volume 42 (2016) no. 4, p. 353 | DOI:10.1007/s10476-016-0404-6
Hung, Vu Viet A characterization of, Complex Variables and Elliptic Equations, Volume 61 (2016) no. 4, p. 448 | DOI:10.1080/17476933.2015.1094466
Dhouib, Abir; Elkhadhra, Fredj m-Potential theory associated to a positive closed current in the class ofm-sh functions, Complex Variables and Elliptic Equations, Volume 61 (2016) no. 7, p. 875 | DOI:10.1080/17476933.2015.1133615
Dinew, S.; Guedj, V.; Zeriahi, A. Open problems in pluripotential theory, Complex Variables and Elliptic Equations, Volume 61 (2016) no. 7, p. 902 | DOI:10.1080/17476933.2015.1121481
Hai, Le Mau; Van Khiem, Nguyen; Van Dung, Trieu Subextension of plurisubharmonic functions in unbounded hyperconvex domains and applications, Complex Variables and Elliptic Equations, Volume 61 (2016) no. 8, p. 1116 | DOI:10.1080/17476933.2016.1143466
Charabati, Mohamad Modulus of continuity of solutions to complex Hessian equations, International Journal of Mathematics, Volume 27 (2016) no. 01, p. 1650003 | DOI:10.1142/s0129167x16500038
Hai, Le Mau; Long, Tang Van; Dung, Trieu Van Equations of complex Monge–Ampère type for arbitrary measures and applications, International Journal of Mathematics, Volume 27 (2016) no. 04, p. 1650035 | DOI:10.1142/s0129167x1650035x
Son, Do Hoang Weak solution of parabolic complex Monge–Ampère equation II, International Journal of Mathematics, Volume 27 (2016) no. 12, p. 1650098 | DOI:10.1142/s0129167x16500981
Hai, Le Mau; Hiep, Pham Hoang An equality on the complex Monge–Ampère measures, Journal of Mathematical Analysis and Applications, Volume 444 (2016) no. 1, p. 503 | DOI:10.1016/j.jmaa.2016.06.023
Hung, Vu Viet Local Property of a Class of m-Subharmonic Functions, Vietnam Journal of Mathematics, Volume 44 (2016) no. 3, p. 603 | DOI:10.1007/s10013-015-0176-5
Hai, Le Mau; Hong, Nguyen Xuan; Dung, Trieu Van Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes, Complex Variables and Elliptic Equations, Volume 60 (2015) no. 11, p. 1580 | DOI:10.1080/17476933.2015.1036244
Hong, Nguyen Xuan Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values, Complex Variables and Elliptic Equations, Volume 60 (2015) no. 3, p. 429 | DOI:10.1080/17476933.2014.944177
Benelkourchi, Slimane Weighted Pluricomplex Energy II, International Journal of Partial Differential Equations, Volume 2015 (2015), p. 1 | DOI:10.1155/2015/947819
Åhag, Per; Cegrell, Urban; Czyż, Rafał Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge–Ampère operator, Journal of Mathematical Analysis and Applications, Volume 422 (2015) no. 2, p. 960 | DOI:10.1016/j.jmaa.2014.09.022
Åhag, Per; Cegrell, Urban; Phạm, Hoàng Hiệp Monge–Ampère measures on subvarieties, Journal of Mathematical Analysis and Applications, Volume 423 (2015) no. 1, p. 94 | DOI:10.1016/j.jmaa.2014.09.061
Hong, Nguyen Xuan The locally F-approximation property of bounded hyperconvex domains, Journal of Mathematical Analysis and Applications, Volume 428 (2015) no. 2, p. 1202 | DOI:10.1016/j.jmaa.2015.03.071
Lu, Chinh H. A variational approach to complex Hessian equations in Cn, Journal of Mathematical Analysis and Applications, Volume 431 (2015) no. 1, p. 228 | DOI:10.1016/j.jmaa.2015.05.067
Dinew, Sławomir; Lu, Chinh H. Mixed Hessian inequalities and uniqueness in the class $E (X, ω, m)$ E ( X , ω , m ), Mathematische Zeitschrift, Volume 279 (2015) no. 3-4, p. 753 | DOI:10.1007/s00209-014-1392-5
Avelin, Benny; Hed, Lisa; Persson, Håkan A Note on the Hyperconvexity of Pseudoconvex Domains Beyond Lipschitz Regularity, Potential Analysis, Volume 43 (2015) no. 3, p. 531 | DOI:10.1007/s11118-015-9486-1
Demailly, Jean-Pierre; Phạm, Hoàng Hiệp A sharp lower bound for the log canonical threshold, Acta Mathematica, Volume 212 (2014) no. 1, p. 1 | DOI:10.1007/s11511-014-0107-4
Zeriahi, Ahmed A viscosity approach to degenerate complex Monge-Ampère equations, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 22 (2014) no. 4, p. 843 | DOI:10.5802/afst.1390
Hai, Le Mau; Van Trao, Nguyen; Hong, Nguyen Xuan The complex Monge–Ampère equation in unbounded hyperconvex domains in ℂn, Complex Variables and Elliptic Equations, Volume 59 (2014) no. 12, p. 1758 | DOI:10.1080/17476933.2013.879385
Rashkovskii, Alexander Extremal cases for the log canonical threshold, Comptes Rendus. Mathématique, Volume 353 (2014) no. 1, p. 21 | DOI:10.1016/j.crma.2014.11.002
Hai, Le Mau; Hiep, Pham Hoang; Hong, Nguyen Xuan; Phu, Nguyen Van The Monge–Ampère type equation in the weighted pluricomplex energy class, International Journal of Mathematics, Volume 25 (2014) no. 05, p. 1450042 | DOI:10.1142/s0129167x14500426
Dieu, Nguyen Quang; Bang, Pham Hien; Hong, Nguyen Xuan Uniqueness properties of m-subharmonic functions in Cegrell classes, Journal of Mathematical Analysis and Applications, Volume 420 (2014) no. 1, p. 669 | DOI:10.1016/j.jmaa.2014.06.008
Phu, Nguyen Van; Hung, Vu Viet On some classes of ω-plurisubharmonic functions on compact Kähler manifolds, Acta Mathematica Vietnamica, Volume 38 (2013) no. 4, p. 617 | DOI:10.1007/s40306-013-0040-1
Hiệp, Phạm Hoàng A comparison principle for the log canonical threshold, Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, p. 441 | DOI:10.1016/j.crma.2013.06.013
Wan, Dongrui Estimates for k-Hessian operator and some applications, Czechoslovak Mathematical Journal, Volume 63 (2013) no. 2, p. 547 | DOI:10.1007/s10587-013-0037-x
Wan, Dongrui; Wang, Wei Lelong–Jensen type formula, k -Hessian boundary measure and Lelong number for k -convex functions, Journal de Mathématiques Pures et Appliquées, Volume 99 (2013) no. 6, p. 635 | DOI:10.1016/j.matpur.2012.10.002
Hai, Le Mau; Hiep, Pham Hoang; Quy, Hoang Nhat Local property of the classEχ,loc, Journal of Mathematical Analysis and Applications, Volume 402 (2013) no. 2, p. 440 | DOI:10.1016/j.jmaa.2013.01.048
Rashkovskii, Alexander Analytic approximations of plurisubharmonic singularities, Mathematische Zeitschrift, Volume 275 (2013) no. 3-4, p. 1217 | DOI:10.1007/s00209-013-1179-0
Hai, Le Mau; Hong, Nguyen Xuan Maximal q-Subharmonicity in $C^{n}$ , Vietnam Journal of Mathematics, Volume 41 (2013) no. 1, p. 1 | DOI:10.1007/s10013-013-0012-8
Cegrell, Urban Convergence in Capacity, Canadian Mathematical Bulletin, Volume 55 (2012) no. 2, p. 242 | DOI:10.4153/cmb-2011-078-6
Hai, Le Mau; Hiep, Pham Hoang; Phu, Nguyen Van Global and local definition of the Monge–Ampère operator on compact Kähler manifolds, Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, p. 153 | DOI:10.1016/j.crma.2012.01.025
MAGNÚSSON, JÓN I.; RASHKOVSKII, ALEXANDER; SIGURDSSON, RAGNAR; THOMAS, PASCAL J. LIMITS OF MULTIPOLE PLURICOMPLEX GREEN FUNCTIONS, International Journal of Mathematics, Volume 23 (2012) no. 06, p. 1250065 | DOI:10.1142/s0129167x12500656
Cegrell, U.; Kołodziej, S.; Zeriahi, A. Maximal subextensions of plurisubharmonic functions, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 20 (2011) no. S2, p. 101 | DOI:10.5802/afst.1307
Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Viscosity solutions to degenerate complex monge-ampère equations, Communications on Pure and Applied Mathematics, Volume 64 (2011) no. 8, p. 1059 | DOI:10.1002/cpa.20364
BENELKOURCHI, SLIMANE APPROXIMATION OF WEAKLY SINGULAR PLURISUBHARMONIC FUNCTIONS, International Journal of Mathematics, Volume 22 (2011) no. 07, p. 937 | DOI:10.1142/s0129167x11007094
Hai, Le Mau; Hiep, Pham Hoang Some Weighted Energy Classes of Plurisubharmonic Functions, Potential Analysis, Volume 34 (2011) no. 1, p. 43 | DOI:10.1007/s11118-010-9179-8
Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge–Ampère equations in big cohomology classes, Acta Mathematica, Volume 205 (2010) no. 2, p. 199 | DOI:10.1007/s11511-010-0054-7
HED, LISA APPROXIMATION OF NEGATIVE PLURISUBHARMONIC FUNCTIONS WITH GIVEN BOUNDARY VALUES, International Journal of Mathematics, Volume 21 (2010) no. 09, p. 1135 | DOI:10.1142/s0129167x10006410
Åhag, Per; Czyż, Rafał Modulability and duality of certain cones in pluripotential theory, Journal of Mathematical Analysis and Applications, Volume 361 (2010) no. 2, p. 302 | DOI:10.1016/j.jmaa.2009.07.013
Åhag, P.; Cegrell, U.; Kołodziej, S.; Phạm, H.H.; Zeriahi, A. Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Advances in Mathematics, Volume 222 (2009) no. 6, p. 2036 | DOI:10.1016/j.aim.2009.07.002
Åhag, Per; Cegrell, Urban; Czyż, Rafał; Phạm, Hoàng Hiệp Monge–Ampère measures on pluripolar sets, Journal de Mathématiques Pures et Appliquées, Volume 92 (2009) no. 6, p. 613 | DOI:10.1016/j.matpur.2009.06.001
Dinew, Sławomir Uniqueness in E(X,ω), Journal of Functional Analysis, Volume 256 (2009) no. 7, p. 2113 | DOI:10.1016/j.jfa.2009.01.019
Dinew, Sławomir An inequality for mixed Monge–Ampère measures, Mathematische Zeitschrift, Volume 262 (2009) no. 1, p. 1 | DOI:10.1007/s00209-008-0356-z
Xing, Yang Continuity of the Complex Monge–Ampère operator on compact Kähler manifolds, Mathematische Zeitschrift, Volume 263 (2009) no. 2, p. 331 | DOI:10.1007/s00209-008-0420-8
Benelkourchi, Slimane Weighted Pluricomplex Energy, Potential Analysis, Volume 31 (2009) no. 1, p. 1 | DOI:10.1007/s11118-009-9119-7
Hiep, Pham Hoang Pluripolar sets and the subextension in Cegrell's classes, Complex Variables and Elliptic Equations, Volume 53 (2008) no. 7, p. 675 | DOI:10.1080/17476930801966893
Coman, Dan; Guedj, Vincent; Zeriahi, Ahmed Domains of definition of Monge-Ampère operators on compact Kähler manifolds, Mathematische Zeitschrift, Volume 259 (2008) no. 2, p. 393 | DOI:10.1007/s00209-007-0233-1
Cegrell, Urban; Hed, Lisa Subextension and approximation of negative plurisubharmonic functions, Michigan Mathematical Journal, Volume 56 (2008) no. 3 | DOI:10.1307/mmj/1231770362
Xing, Yang A strong comparison principle of plurisubharmonic functions with finite pluricomplex energy, Michigan Mathematical Journal, Volume 56 (2008) no. 3 | DOI:10.1307/mmj/1231770360
Dieu, Nguyen Quang; Hiep, Pham Hoang Pluripolar Hulls and Complete Pluripolar Sets, Potential Analysis, Volume 29 (2008) no. 4, p. 409 | DOI:10.1007/s11118-008-9103-7
Hiep, Pham On the convergence in capacity on compact Kahler manifolds and its applications, Proceedings of the American Mathematical Society, Volume 136 (2008) no. 6, p. 2007 | DOI:10.1090/s0002-9939-08-09043-6
Ivarsson, Björn On the behavior of strictly plurisubharmonic functions near real hypersurfaces, Arkiv för Matematik, Volume 45 (2007) no. 1, p. 71 | DOI:10.1007/s11512-006-0029-2
Åhag, Per; Czyż, Rafał On the Cegrell classes, Mathematische Zeitschrift, Volume 256 (2007) no. 2, p. 243 | DOI:10.1007/s00209-006-0067-2
Åhag, Per A Dirichlet problem for the complex Monge-Ampíre operator in F(f), Michigan Mathematical Journal, Volume 55 (2007) no. 1 | DOI:10.1307/mmj/1177681988
Wiklund, Jonas On subextension of pluriharmonic and plurisubharmonic functions, Arkiv för Matematik, Volume 44 (2006) no. 1, p. 182 | DOI:10.1007/s11512-006-0014-9
Benelkourchi, Slimane A Note on the approximation of plurisubharmonic functions, Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, p. 647 | DOI:10.1016/j.crma.2006.03.002
Rashkovskii, A. Relative types and extremal problems for plurisubharmonic functions, International Mathematics Research Notices (2006) | DOI:10.1155/imrn/2006/76283
Cegrell, U.; Kołodziej, S. The equation of complex Monge-Ampère type and stability of solutions, Mathematische Annalen, Volume 334 (2006) no. 4, p. 713 | DOI:10.1007/s00208-005-0687-6
Cegrell, Urban Weak*- convergence of Monge-Ampère measures, Mathematische Zeitschrift, Volume 254 (2006) no. 3, p. 505 | DOI:10.1007/s00209-006-0953-7
Dieu, Nguyen Quang Approximation of plurisubharmonic funcitons on bounded domains in Cn, Michigan Mathematical Journal, Volume 54 (2006) no. 3 | DOI:10.1307/mmj/1163789922
Benelkourchi, Slimane; Jennane, Bensalem; Zeriahi, Ahmed Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates, Arkiv för Matematik, Volume 43 (2005) no. 1, p. 85 | DOI:10.1007/bf02383612
Cegrell, U.; Kolodziej, S.; Zeriahi, A. Subextension of plurisubharmonic functions with weak singularities, Mathematische Zeitschrift, Volume 250 (2005) no. 1, p. 7 | DOI:10.1007/s00209-004-0714-4
Czyz, Rafal Convergence in capacity of the Perron-Bremermann envelope, Michigan Mathematical Journal, Volume 53 (2005) no. 3 | DOI:10.1307/mmj/1133894161

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