On the Cegrell classes (Q2385007)

In [Acta Math. 180, 187--217 (1998; Zbl 0926.32042)], \textit{U. Cegrell} defined new classes of plurisubharmonic functions on a hyperconvex domain \(\Omega \subset\mathbb C^n\), to study the solutions of the complex Monge-Ampère equation on \(\Omega\). In particular, \(\mathcal E_0\) consists of negative and bounded plurisubharmonic functions \(u\) on \(\Omega\), with finite Monge-Ampère mass and such that \(\lim_{z\in \xi} u(z)=0\), for any \(\xi\in \partial \Omega\). Any Cegrell class \(\mathcal K\) can be generalized as follows: given a continuous function \(f\colon \partial\Omega\to\mathbb R\), let \(U(0,f)\) be the Perron-Bremermann envelope on \(\Omega\) such that \(\lim_{z\in \xi} U(0,f)(z)=f(\xi)\), for any \(\xi\in \partial \Omega \). Then \(u\in \mathcal K(f)\) if there exists \(\varphi\in \mathcal K\) such that \(U(0,f)\geq u\geq\varphi + U(0,f)\). In particular \(\mathcal K=\mathcal K(0)\). In the paper under review, the authors show that it is possible to define \(\mathcal K(f)\) in terms of decreasing sequences of functions in \(\mathcal E_0(f)\), with some extra properties depending on \(\mathcal K\). As an application, they prove an existence result for the Dirichlet problem for certain singular measures.