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 W1,q0 for any q0 > 2n. As an application, we show that, up to scaling, there exists a unique classical solution in W3,q0 for the complex Monge-Ampère equation when F is in W1,q0.
- Compact hermitian manifold
- Complex monge-ampère equation
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