No abstract

Proof. Since uEC1, it follows from Lemma 3.4 that vjEC1, VjEN, and there is a decreasing sequence uj EC0 with lim uj --u Then f and sup ] -uj (ddCuj) n = ~ < +oc. J J max(uj, Vk)(ddCmax(uj, Vk)) n >//uj(ddCuj) n >1 -a, Vj, k E N, so it is enough to prove that lim Juj(ddCuj)n=/u(ddCu) n. j~+cr We have for k>~j, f -uj(ddCuj)n <~ J-uj(dd~uk) ~ = f~ -uj(dd~uk)n+/ -uj(dd~uk) n j >~--e J uj <-e SO sup j(d+( u~+U(O,-f)))~ = ~ < +~.

In this article we solve the complex Monge-Ampère equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko lodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampère measure, then it is a complex Monge-Ampère measure.2000 Mathematics Subject Classification. Primary 32W20; Secondary 32U15.

International audienc

We prove that in a family of plurisubharmonic functions with Monge-Ampère measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampère type equation. Subject Classification (1991): 32U05, 32U40 Mathematics