The complex Monge-Ampére equation on the complement of a divisor

Abstract

We consider the complex Monge-Ampére equation on complete Kähler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some \emph{weighted} $W^{1,p_0}$ space for $p_0>2n$, the Monge-Ampére equation has a classical $W^{3,p_0}$ solution.