On the mean oscillation of the Hessian of solutions to the Monge–Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge–Ampère equation detD2u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D2u∈VMO.