Partial W2,p regularity for optimal transport maps

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class W2,ploc for any p ≥ 1 outside of a closed singular set of measure zero. We also establish global W2,p estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost