This research originates from recent results by M. Goldman and F. Otto concerning regularity of optimal transport maps for the quadratic cost. We consider cost functions having the form c(x, y) = h(x − y), where h is positively homogeneous of degree p ≥ 2 and h ∈ C2(Rn). A mapping T : Rn → Rn is c-monotone if c(Tx,x) + c(Ty,y) ≤ c(Tx,y) + c(Ty,x). Using Green’s representation formulas, if T is c-monotone, we prove local L∞- estimates of Tx−x in terms of Lp-averages of Tx−x. From this we deduce estimates for the interpolating maps between T and Id, and when T is optimal, L∞-estimates of T −1x − x. As a consequence of the technique, we also obtain a.e. differentiability of monotone maps. This is joint work with Annamaria Montanari.