#### Vol. 276, No. 2, 2015

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The complex Monge–Ampère equation on some compact Hermitian manifolds

### Jianchun Chu

Vol. 276 (2015), No. 2, 369–386
##### Abstract

We consider the complex Monge–Ampère equation on compact manifolds when the background metric is a Hermitian metric (in complex dimension 2) or a Hermitian metric satisfying an additional condition (in higher dimensions). We prove that the Laplacian estimate holds when $F$ is in ${W}^{\mathfrak{1},{q}_{\mathfrak{0}}}$ for any ${q}_{\mathfrak{0}}>\mathfrak{2}n$. As an application, we show that, up to scaling, there exists a unique classical solution in ${W}^{\mathfrak{3},{q}_{\mathfrak{0}}}$ for the complex Monge–Ampère equation when $F$ is in ${W}^{\mathfrak{1},{q}_{\mathfrak{0}}}$.

##### Keywords
complex Monge–Ampère equation, compact Hermitian manifold
##### Mathematical Subject Classification 2010
Primary: 35J96, 53C55