Range of the Hessian operator on some weighted m−subharmonic classes

In this paper we study the range of the complex Hessian operator on some weighted classes . For a given measure ν and a negative real function χ we prove that χ(Em,χ)⊂ L¹(Ω, dν) if and only if the measure ν can be written as the Hessian of a unique function in Em,χ. We extend also this result to an energy class with a given boundary data.