Complex Monge–Ampère equations on singular spaces

We investigate the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain of a closed, connected, singular, and locally irreducible complex-analytic subvariety. We first examine the classes Ep for p>0 and establish a characterization of their images under the complex Monge–Ampère operator. This result answers a question posed by N. Q. Dieu, T. V. Long. We then turn to the weighted energy classes Eχ(Ω), consisting of negative plurisubharmonic functions with finite χ-energy, and provide a precise characterization of their images under the complex Monge–Ampère operator, where χ is a convex increasing function satisfying χ(0)=0 and χ(−∞)=−∞.