## 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 $$h \in C (\mathbb R \times \Omega)$$.

### MSC:

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

Zbl 1343.32030
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