Estimates for Monge–Ampère operators acting on positive plurisubharmonic currents

Let Ω∼ be an open subset of CN=Cn×CmCN=Cn×Cm, let T be a positive plurisubharmonic (psh: meaning ddcT ⩾ 0) current of bidegree (k, k  ) on Ω∼ and let U be the Lelong-Skoda potential current associated to the d-closed positive current ddcT  . We denote (z,t)∈Cn×Cm(z,t)∈Cn×Cm and consider φ: (z,t) ↦ φ(z) a C2 positive semi-exhaustive plurisubharmonic (psh: meaning ddcφ ⩾ 0) function on Ω∼ such that log φ is also plurisubharmonic on the open set {φ > 0}. For p∈Np∈N such that 1 ⩽ p ⩽ n − k, we generalize some properties of the current U ∧ (ddcω)p where ω = log φ, known when ω(z) = log ∣z∣ (see [12]). Finally we want to define the current T ∧ (ddcω)p and as an application, we prove a version of the chern-Levine-Nirenberg for a positive or negative psh current which is defined out side a pluripolar set A⊂Ω∼.