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The complex Monge-Ampere equation on compact Kaehler manifolds

We consider the complex Monge-Ampère equation on a compact Kähler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of $\tϕ$ and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical solution in $W^{3, p_0}$ for the complex Monge-Ampère equation when $F$ is in $W^{1, p_0}$.