The domain of definition of the complex Monge-Ampère operator.

*(English)*Zbl 1102.32018The 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.

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.

Reviewer: Filippo Bracci (Roma)