The general definition of the complex Monge-Ampère operator. (English) Zbl 1065.32020

Denote by PSH\((\Omega)\) the plurisubharmonic functions on a hyperconvex domain \(\Omega\) and by PSH\(^-(\Omega)\) the subclass of negative functions. The author considers global approximation of negative plurisubharmonic functions by decreasing sequences of negative plurisubharmonic functions that are continuous on \(\overline\Omega\), equal to zero on \(\partial\Omega\) and with Monge-Ampère mass. The elements of this class of functions serves as “test functions”.
The author proves that global approximation is possible in PSH\(^-(\Omega)\). Furthermore, the author discusses a general definition of the complex Monge-Ampère operator. This is done by introducing a class \(\mathcal E\) = \({\mathcal E}(\Omega)\) of plurisubharmonic functions which consists of all functions that are locally equal to decreasing limits of test functions. The Monge-Ampère operator can be extended to \(\mathcal E\), and this is the most general definition if we require the operator to be continuous under decreasing limits. Consider a class \(\mathcal K\) (=\({\mathcal K}(\Omega)\)) \(\subset \text{PSH}^-(\Omega)\) such that:
(1) If \(u\in\mathcal K\), \(v\in PSH^-(\Omega)\) then \(\max(u,v)\in\mathcal K\).
(2) \(u\in\mathcal K\), \(\varphi_j \in PSH^-(\Omega)\cap L^\infty_{\text{loc}}\), \(\varphi_j\searrow u\), \(j\to +\infty\), then \(((dd^c\varphi_j)^n)^\infty_{j=1}\) is weak*-convergent.
The author proves that \(\mathcal E\) is the largest class for which (1) and (2) holds true. Thus, \(\mathcal E\) is optimal in this sense.
In the remaining part of the paper, the author studies the Monge-Ampère operator using this general definition and solves a Dirichlet problem.
Let \({\mathcal E}_0(\Omega)\) be the convex cone of bounded plurisubharmonic functions \(\varphi\) with \(\lim_{z\to\xi}\varphi(z)=0\), for all \(\xi\in\partial\Omega\) and \(\int_\Omega(dd^c\varphi)^n<+\infty\). We denote by \({\mathcal F}(\Omega)\) the subclass of functions \(u\) in \(\mathcal E\) such that there exists a decreasing sequence \(u_j\in{\mathcal E}_0(\Omega)\) such that \(u_j\searrow u\) on \(\Omega\) and \(\sup_j\int_\Omega(dd^cu_j)^n<+\infty\). The author considers the Dirichlet problem in \(\mathcal F\). Suppose \(\psi\in{\mathcal E}_0\), \(v\in{\mathcal F}\) where \((dd^cv)^n\) is carried by a pluripolar set. Then there is a \(g\in{\mathcal F}\) with \[ (dd^cg)^n=(dd^c\psi)^n+(dd^cv)^n\,. \]


32U15 General pluripotential theory
32W20 Complex Monge-Ampère operators
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[1] Survey of pluripotential theory. several complex variables, Proceedings of the Mittag-Leffler Inst. (1987-88), 38, 48-95, (1994), Princeton University Press · Zbl 0786.31001
[2] The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math, 37, 1-44, (1976) · Zbl 0315.31007
[3] A new capacity for plurisubharmonic functions, Acta Math, 149, 1-40, (1982) · Zbl 0547.32012
[4] Estimates for the complex Monge-Ampère operator, Bull. Pol. Acad. Sci. Math, 41, 151-157, (1993) · Zbl 0795.32003
[5] The complex Monge-Ampère operator in hyperconvex domains, Annali della Scuola Normale Superiore di Pisa, 23, 4, 721-747, (1996) · Zbl 0878.31003
[6] Potentials in pluripotential theory, Ann. de la Fac. Sci. de Toulouse (6), 8, 3, 439-469, (1999) · Zbl 0961.31005
[7] Pluricomplex energy, Acta Mathematica, 180, 2, 187-217, (1998) · Zbl 0926.32042
[8] Explicit calculation of a Monge-Ampère measure, Actes des rencontres d’analyse complexe (Université de Poitiers, 25-28 mars 1999), 39-42, (2000), Poitiers: Atlantique · Zbl 1036.32023
[9] Convergence in capacity, (2001)
[10] Exhaustion functions for hyperconvex domains, (2001)
[11] The Dirichlet problem for the complex Monge-Ampère operator: Perron classes and rotation invariant measures, Michigan. Math. J, 41, 563-569, (1994) · Zbl 0820.31005
[12] Integration by parts for currents and applications to the relative capacity and Lelong numbers, Mathematica, 39(62), 1, 45-57, (1997) · Zbl 0914.32003
[13] Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z, 194, 519-564, (1987) · Zbl 0595.32006
[14] Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut, Math. Ann, 257, 171-184, (1981) · Zbl 0451.32012
[15] The complex Monge-Ampère equation, Acta Mathematica, 180, 69-117, (1998) · Zbl 0913.35043
[16] Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J, 52, 157-197, (1985) · Zbl 0578.32023
[17] Extremal plurisubharmonic functions and capacities in \(\Bbb C^n,\) Sophia Kokyuroko in Mathematics, (1982) · Zbl 0579.32025
[18] Continuity of envelopes of plurisubharmonic functions, J. Math. Mech, 18, 143-148, (1968) · Zbl 0159.16002
[19] Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat, 39, 181-200, (2001) · Zbl 1021.32014
[20] Complex Monge-Ampère equations with a countable number of singular points, Indiana Univ. Math. J, 48, 749-765, (1999) · Zbl 0934.32027
[21] Pluricomplex Green functions and the Dirichlet problem for the complex Monge-Ampère operator, Michigan Math. J, 44, 579-596, (1997) · Zbl 0899.31007
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