  We consider the complex Monge-Amp\`ere equation on a compact K\"ahler
manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In
particular we prove that estimate of $\t\phi$ 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-Amp\`ere equation when $F$ is in $W^{1, p_0}$.
