Note on a general complex Monge-Ampère equation on pseudoconvex domains of infinite type. (English) Zbl 1368.32025

Recently, L. K. Ha and T. V. Khanh [Math. Res. Lett. 22, No. 2, 467–484 (2015; Zbl 1343.32030)] proved the regularity of the solution to the following Dirichlet problem for the complex Monge-Ampère operator: \[ \begin{cases} (dd^cu)^n h dV \quad & \text{in } \quad \Omega, \\ u \phi \quad & \text{on } \quad \partial \Omega,\end{cases} \] here \(\Omega\) is a bounded pseudoconvex domain of \(\mathbb{C}^n\) with the \(f\)-property.
Recall that if \(f : [1 , +\infty[ \to [1 , +\infty[ \) is a smooth increasing function with \(f(t)t^{-1}\) decreasing, a pseudoconvex domain \(\Omega \subset \mathbb{C}^n\) has the \(f\)-property if there exist a neighborhood \(U\) of \(\partial \Omega\) and a family of \(C^2\)-smooth plurisubharmonic functions \((\phi_\delta) : U \to [-1, 0]\) such that \[ dd^c \phi_\delta \geq C_1 f(\delta ^{-1})^2 \quad \text{and}\quad |D\phi_\delta| \leq C_2 \delta ^{-1}, \quad \forall z \in U\cap \{z\in \Omega ; -\delta <r(z) <0\}, \] where \(r\) is a \(C^1\)-defining function of \(\Omega.\)
The purpose of the paper is to generalize the previous result to the case when \(h \in C (\mathbb R \times \Omega)\).


32W20 Complex Monge-Ampère operators
32T15 Strongly pseudoconvex domains
32T25 Finite-type domains
32U05 Plurisubharmonic functions and generalizations


Zbl 1343.32030
Full Text: DOI Euclid