A New$L^{\infty}$Estimate in Optimal Mass Transport

Let Ω be a bounded Lipschitz regular open subset of ${\Bbb R}^{d}$ and let μ, ν be two probability measures on $\overline{\Omega}$ . It is well known that if μ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map $T_{p}$ pushing forward μ on ν and which realizes the Monge-Kantorovich distance $W_{p}(\mu,\nu)$ . In this paper, we establish an $L^{\infty}$ bound for the displacement map $T_{p}x-x$ which depends only on p, on the shape of Ω and on the essential infimum of the density f.