  Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on
$B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$
with the following property: for any continuous, plurisubharmonic function $u$
in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in
W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only
on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$
estimates for real Monge-Ampere to the complex version.
