Monge-Amp\`ere operators on compact K\"ahler manifolds

We study the complex Monge-Amp` ere operator on compact K"ahler manifolds. We give a complete description of its range on the set of $\omega -$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this compact setting results of U.Cegrell from the local pluripoltential theory. We give some applications to complex dynamics and to the existence of K"ahler-Einstein metrics on singular manifolds.