Sobolev regularity for Monge-Amp\`ere 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 $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1+\kappa }$ for some $\kappa >0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp`ere equation with right hand side bounded away from zero and infinity.