Plurisubharmonic functions with weak singularities

We study the complex Monge-Amp`ere operator in bounded hyperconvex domains of ${\text{C}}^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp`ere energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Amp`ere capacity of sublevel sets and solve associated complex Monge-Amp`ere equations.