Nonlinear bounds in Hölder spaces for the Monge–Ampère equation

We demonstrate that C2,αC2,α estimates for the Monge–Ampère equation depend in a highly nonlinear way both on the CαCα norm of the right-hand side and 1/α1/α. First, we show that if a solution is strictly convex, then the C2,αC2,α norm of the solution depends polynomially on the CαCα norm of the right-hand side. Second, we show that the C2,αC2,α norm of the solution is controlled by exp⁡((C/α)log⁡(1/α))exp⁡((C/α)log⁡(1/α)) as α→0α→0. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results.