## 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 $(dd^cg)^n=(dd^c\psi)^n+(dd^cv)^n\,.$

### MSC:

 32U15 General pluripotential theory 32W20 Complex Monge-Ampère operators
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### References:

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