  In this paper, we prove second derivative estimates together with classical
solvability for the Dirichlet problem of certain Monge-Ampere type equations
under sharp hypotheses. In particular we assume that the matrix function in the
augmented Hessian is regular in the sense used by Trudinger and Wang in their
study of global regularity in optimal transportation [28] as well as the
existence of a smooth subsolution. The latter hypothesis replaces a barrier
condition also used in their work. The applications to optimal transportation
and prescribed Jacobian equations are also indicated.
