Differentiability of monotone maps related to non quadratic costs

The cost functions considered are c(x,y)=h(x−y), with h∈C2Rn, homogeneous of degree p≥2, with positive definite Hessian in the unit sphere. We consider monotone maps T with respect to that cost and establish local scale invariant L∞-estimates of T minus affine functions, which are applied to obtain differentiability properties of T a.e. It is also shown that these maps are related to maps of bounded deformation, and further differentiability and Hölder continuity properties are derived.