We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain Ω⊂ℂn. We show, under a mild technical assumption, that the unique solution u to this problem is Hölder continuous if the boundary data ϕ is Hölder continuous and the density f belongs to Lp(Ω) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.

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

International audienceIn this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $\varphi$ with an isolated singularity at $0$ in an open subset of ${\mathbb C}^n$. This threshold is defined as the supremum of constants $c>0$ such that $e^{-2c\varphi}$ is integrable on a neighborhood of $0$. We relate $c(\varphi)$ with the intermediate multiplicity numbers $e_j(\varphi)$, defined as the Lelong numbers of $(dd^c\varphi)^j$ at $0$ (so that in particular $e_0(\varphi)=1$). Our main result is that $c(\varphi)\ge\sum e_j(\varphi)/e_{j+1}(\varphi)$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(\varphi)\ge 1/e_1(\varphi)$ due to Skoda, as well as the lower estimate $c(\varphi)\ge n/e_n(\varphi)^{1/n}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals

In this article, we prove that if E is a complete pluripolar set in , then E ¼ {' ¼ À1} for some ' 2 F 1 (). Moreover, we study the subextension in Cegrell's class E p .