Sobolev Regularity for Monge-Ampère Type Equations

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W 2,1+κ loc for some κ > 0. This generalizes some recents results [10, 11, 24] concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.