Characterizing the range of the complex Monge−Ampére operator

In this note, we solve the complex Monge−Ampére equation for measures with a pluripolar part in the Cegrell class on compact Kähler manifolds, and require that the pluripolar part is solvable within this framework. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss the properties of the complex Monge−Ampére operator in some special cases.