$W^{2,p}$-estimates for the linearized Monge-Ampère equation

Let $\Omega \subseteq \mathbb{R}^n$ be a strictly convex domain and let $\phi \in C^2(\Omega)$ be a convex function such that $\lambda \leq$   det$D^2\phi \leq\Lambda$ in $\Omega$. The linearized Monge-Ampère equation is

$L_{\Phi}u=\textrm{trace}(\Phi D^2u)=f,$

where $\Phi = ($det$D^2\phi)(D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove that there exist $p>0$ and $C>0$ depending only on $n,\lambda,\Lambda$, and $\textrm{dist}(\Omega^\prime,\Omega)$ such that

$\Vert D^2u\Vert _{L^p(\Omega^\prime)}\leq C(\Vert u\Vert _{L^\infty(\Omega)}+\Vert f\Vert _{L^n(\Omega)})$

for all solutions $u\in C^2(\Omega)$ to the equation $L_{\Phi}u=f$.