Minimax rates for Wasserstein deconvolution of regular distributions with known ordinary smooth errors

We study the problem of univariate distribution deconvolution under Wasserstein metrics, with known and ordinary smooth error distributions. We fill a gap present in the literature extending the result on the upper bound rates under Wasserstein metrics for a recently proposed isotone distribution function estimator to the case when the mixing distribution has some level of regularity either in a Holder or in a Sobolev scale. The found rates turn out to be minimax-optimal (up to a logarithmic factor) over the full scale of values of the smoothness index β>0 of the Fourier transform of the error distribution under the 1-Wasserstein distance, while, for Wasserstein metrics of order p>1, these rates are known to be minimax-optimal only for β≤1/2.