*Published Paper*

**Inserted:** 11 sep 2007

**Journal:** Proc. Amer. Math. Soc.

**Volume:** 135

**Number:** 11

**Pages:** 3525-3535

**Year:** 2007

**Abstract:**

Let 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 $\nu$ which realizes the Monge-Kantorovich distance $W_p(\mu,\nu)$. In this paper, we establish an $L^\infty$ bound for the displacement map $T_px-x$ which depends only on $p$, on the shape of $\Omega$ and on the essential infimum of the density $f$.

**Keywords:**
optimal mass transport, p-Wasserstein distance, p>1, $L^\infty$ estimate

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