Lelong-Demailly Numbers in Terms of Capacity and Weak Convergence for Closed Positive Currents

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic (psh for short) function. These estimates enable us to give another proof of the Demailly's comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT (K) = ∫K T ∧ (ddcv)p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk u.