Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

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

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