Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

Abstract: We demonstrate that $C^{2,\alpha}$ estimates for the Monge-Amp`{e}re equation depend in a highly nonlinear way both on the $C^{\alpha}$ norm of the right-hand side and $1/\alpha$. First, we show that if a solution is strictly convex, then the $C^{2,\alpha}$ norm of the solution depends polynomially on the $C^{\alpha}$ norm of the right-hand side. Second, we show that the $C^{2,\alpha}$ norm of the solution is controlled by $\exp((C/\alpha)\log(1/\alpha))$ as $\alpha \to 0$. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results.