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The domain of definition of the complex Monge-Ampère operator. (English) Zbl 1102.32018
The complex Monge-Ampère operator, which associates to a $$C^2$$ function the determinant of its complex Hessian, has been defined, as a positive measure, for locally bounded plurisubharmonic functions by E. Bedford and B. A. Taylor [Invent. Math. 37, 1–44 (1976; Zbl 0315.31007)]. J.-P. Demailly, [Math. Z. 194, 519–564 (1987; Zbl 0595.32006)] extended the definition to plurisubharmonic functions locally bounded outside a compact set.
In the present paper, the author characterizes those plurisubharmonic functions for which the complex Monge-Ampère operator can be well defined in terms of the boundedness of the Monge-Ampère masses of any smooth decreasing regularizing sequence. Namely, let $$u$$ be a negative plurisubharmonic function defined on some open set of $$\mathbb C^n$$. Let $$\{u_j\}$$ be a decreasing sequence of smooth plurisubharmonic functions which pointwise tends to $$u$$. Then the complex Monge-Ampère operator $$(dd^c u)^n$$ can be defined as a regular Borel measure such that $$(dd^c u_j)$$ tends (in the weak$$^\star$$ topology) to $$(dd^c u)^n$$ if and only if the sequence $$\{(dd^c u_j)\}$$ is locally weakly bounded.
Other more technical characterizations are also provided. Also, the author shows that if $$u$$ belongs to the domain of definition of the complex Monge-Ampère operator, $$v$$ is a plurisubharmonic function such that $$u\leq v$$ outside a compact set, then also $$v$$ is in the domain of definition of the complex Monge-Ampère operator.

##### MSC:
 32W20 Complex Monge-Ampère operators 31B99 Higher-dimensional potential theory
##### Keywords:
plurisubharmonic functions
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