Interior a priori estimates for the Monge-Ampere equation
In this paper we prove the strict convexity, the interior C1,α, C2,α and W2,p estimates for convex solutions to the Monge-Amp`ere type equation. For the strict convexity and C1,α estimate, we assume that the inhomogeneous term f satisfies a doubling condition. For the C2,α and W2,p estimates, we assume that f is H¨older continuous or continuous. These estimates are mainly due to Caffarelli. We also give a brief discussion on the regularity for more general Monge-Amp`ere type equations arising in optimal transportation.