  The subject of this paper is the estimation of a probability measure on
${\mathbb R}^d$ from data observed with an additive noise, under the
Wasserstein metric of order $p$ (with $p\geq 1$). We assume that the
distribution of the errors is known and belongs to a class of supersmooth
distributions, and we give optimal rates of convergence for the Wasserstein
metric of order $p$. In particular, we show how to use the existing lower
bounds for the estimation of the cumulative distribution function in dimension
one to find lower bounds for the Wasserstein deconvolution in any dimension.
