$$L^\infty $$-estimates in optimal transport for non quadratic costs

For cost functions $$c(x,y)=h(x-y)$$ , with $$h\in C^2\left( {{\mathbb {R}}}^n\setminus \{0\}\right) \cap C^1\left( {R}}}^n\right) $$ homogeneous of degree $$p>1$$ we show $$L^\infty -estimates $$Tx-x$$ on balls, where T is an h-monotone map. Estimates for the interpolating mappings $$T_t=t(T-I)+I$$ are deduced from this.