Interior $W^{2,p}$ estimate for small perturbations to the complex Monge-Ampere equation

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 $L^{\mathrm{\infty }}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\text{eps}\le MA(u)\le 1+\text{eps}$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\text{eps}>0$ 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.