Plurisubharmonic functions with weak singularities

Abstract

We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of \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.Comment: 15 pages, dedicated to Christer Kiselman on the occasion of his retiremen