In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.
Liu, J. & Trudinger, N. (2010). On Pogorelov estimates for Monge-Ampere type equations. Discrete and Continuous Dynamical Systems Series A, 28 (3), 1121-1135.