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 C_T-capacity, where T is a closed positive current and C_T is the capacity associated to T. This derived from some esimate on the growth of the C_T-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 C_T (K) = ∫_K T ∧ (dd^cv)^p, where K is a compact subset. Finally, we give some conditions on the capacity C_T that guarantees the weak convergence u_kT_k → uT, for positive closed currents T, T_k and psh functions u_k u.