that the estimate depends on $\inf f$, but is independent of $\sup f$.

By Lemmas \[l202\], \[l203\] and the Caffarelli-Gutiérrez theory [@CG], we have the following Hölder and Sobolev space estimates.

\[t201\] Let $u\in C^4(\Omega)$ be a locally uniformly convex solution of .

-   Assume $f\in L^\infty(\Omega)$. Then we have the estimate $$\label{e213}
       \|u\|_{W^{4,p}(\Omega')}\leq C,$$ for any $p>1$ and $\Omega'\Subset\Omega$, where the constant $C$ depends on $n,p,\sup_\Omega|f|, {\mbox{dist}}(\Omega',\partial\Omega)$, and the modulus of convexity of $u$.

-   Assume $f\in C^\alpha(\Omega)$ for some $\alpha\in(0,1)$. Then $$\label{e214}
       \|u\|_{C^{4,\alpha}(\Omega')}\leq C,$$ where $C$ depends on $n,\alpha,\|f\|_{C^\alpha(\Omega)}, {\mbox{dist}}(\Omega',\partial\Omega)$, and the modulus of convexity of $u$.

Therefore, to prove the regularity of the maximizer $u_0$ in Lemma \[l201\], it suffices to prove, in view of Lemmas \[l202\], \[l203\] and Theorem \[t201\], that (a) the maximizer $u_0$ can be approximated by smooth solutions to equation and (b) it is strictly convex. We will prove (a) and (b) in Sections 3 and 4, respectively.

Rotations in $\mathbb{R}^{n+1}$
-------------------------------

In order to establish the estimate of the modulus of convexity, we need to treat convex functions as graphs in $\mathbb{R}^{n+1}$, and rotate the graphs in $\mathbb{R}^{n+1}$. When $\alpha=1/(n+2)$, the affine maximal surface equation is invariant under uni-modular transformations in $\mathbb{R}^{n+1}$. 