The Monge-Ampère type equation in the weighted pluricomplex energy class (Q2874717)

Letl \(\chi\) be a continuous increasing function which is negative on the negative half-axis, and let \(\mu\) be a positive, finite Radon measure in a hyperconvex domain \(\Omega\) in \(\mathbb C ^n\). Assume that \(\mu\) is bounded above by \(-\chi (v)(dd^c v)^n\) for some plurisubharmonic function \(v\) which belongs to the weighted energy class \(\mathcal E _{\chi } (\Omega )\) as defined by \textit{S. Benelkourchi} et al. [in: Complex analysis and digital geometry. Proceedings from the Kiselmanfest, Uppsala, Sweden, 2006. Uppsala: Univ. Uppsala. 57--74 (2009; Zbl 1200.32021)]. Then it is shown that the equation \(-\chi (u)(dd^c u)^n = \mu\) has a solution in \(\mathcal E _{\chi } (\Omega )\).NEWLINENEWLINE\textit{U. Cegrell} [Ann. Pol. Math. 94, No. 2, 131--147 (2008; Zbl 1166.32017)] introduced the family \(\mathcal N (\Omega ) \) of those functions in the maximal domain of definition of the Monge-Ampère operator which have, ``essentially'', zero boundary data. In the second main result of the paper the authors solve the equation \((dd^c u)^n = \mu\) in the class \(\mathcal N (\Omega ) \) provided that the measure \(\mu\) admits local subsolutions and for some bounded \(h \in \mathcal N (\Omega ) \) the integral \(\int _{\Omega }-h\, d\mu\) is finite.