  This paper deals with the estimation of a probability measure on the real
line from data observed with an additive noise. We are interested in rates of
convergence for the Wasserstein metric of order $p\geq 1$. The distribution of
the errors is assumed to be known and to belong to a class of supersmooth or
ordinary smooth distributions. We obtain in the univariate situation an
improved upper bound in the ordinary smooth case and less restrictive
conditions for the existing bound in the supersmooth one. In the ordinary
smooth case, a lower bound is also provided, and numerical experiments
illustrating the rates of convergence are presented.
