  We consider the complex Monge-Amp\`{e}re equation on compact manifolds when
the background metric is a Hermitian metric (in complex dimension two) or a
kind of Hermitian metric (in higher dimensions). We prove that the Laplacian
estimate holds when $F$ is in $W^{1,q_{0}}$ for any $q_{0}>2n$. As an
application, we show that, up to scaling, there exists a unique classical
solution in $W^{3,q_{0}}$ for the complex Monge-Amp\`{e}re equation when $F$ is
in $W^{1,q_{0}}$.
