Abstract

Let Ω be a bounded Lipschitz regular open subset of ℝd and let μ, ν be two probablity measures on Ω̄. It is well known that if μ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map Tp pushing forward μ on ν and which realizes the Monge-Kantorovich distance Wp(μ, ν). In this paper, we establish an L∞ bound for the displacement map Tp x - x which depends only on p, on the shape of Ω and on the essential infimum of the density f. © 2007 American Mathematical Society.