Quantitative homogenization of Monge-Ampère equations in periodic media

Let $u^{\epsilon }$ and $u$ be the convex solutions of $$\det D^2{u}^{\epsilon }=F(x,x\mathrm{/}\epsilon ),\det D^{2}u=\bar{F}(x)$$ on a bounded convex domain $\mathrm{\Omega }$, with the same Dirichlet data. Here, $F$ is uniformly positive and periodic in its second variable, and $\bar{F}(x)={\int }_{{\mathbb{T}}^n}F(x,y)dy$. For $m=0,1$ and $0<\alpha \le 1$, we prove $$\mathrm{\parallel }{u}^{\epsilon }-u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C{\epsilon }^{m+\alpha },\mathrm{\forall }0<\epsilon \le 1,$$ provided that $u\in C^{m+2,\alpha }(\bar{\mathrm{\Omega }})$ and $F\in C_x^{m,\alpha }(\bar{\mathrm{\Omega }};C_y^{0,\gamma }({\mathbb{T}}^n))$ for some $0<\gamma <1$. Both exponents are optimal in their respective regularity scales. We also establish the $O({\epsilon }^{\alpha })$ rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.