Partial regularity for optimal transport with p-cost away from fixed points

We consider maps $T$ solving the optimal transport problem with a cost $c(x-y)$ modeled on the $p$-cost. For Hölder continuous marginals, we prove a $C^{1,\alpha }$-partial regularity result for $T$in the set $\{\mathrm{\mid }T(x)-x\mathrm{\mid }>0\}$.