Chen, ShibingFigalli, Alessio2020-12-202020-12-200022-1236http://hdl.handle.net/1885/218301We 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 costapplication/pdfen-AU201710.1016/j.jfa.2017.02.0252020-11-23