A new $L^\infty$ estimate in optimal mass transport

Let $\Omega$ be a bounded Lipschitz regular open subset of $\mathbb{R}^d$ and let $\mu,\nu$ be two probablity measures on $\overline{\Omega}$. It is well known that if $\mu=f\, dx$ is absolutely continuous, then there exists, for every $p>1$, a unique transport map $T_p$ pushing forward $\mu$ on $\nu$ 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 $\Omega$ and on the essential infimum of the density $f$.