Differentiability of monotone maps related to non quadratic costs

Abstract

The cost functions considered are $c(x,y)=h(x-y)$, with $h\in C^2\left(R^n\right)$, homogeneous of degree $p\ge 2$, with positive definite Hessian in the unit sphere. We consider monotone maps $T$ with respect to that cost and establish local scale invariant $L^{\infty }$-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.