  We study the non-parametric estimation of an unknown density f with support
on R+ based on an i.i.d. sample with multiplicative measurement errors. The
proposed fully data driven procedure is based on the estimation of the Mellin
transform of the density f , a regularisation of the inverse of the Mellin
transform by a spectral cut-off and a data-driven model selection in order to
deal with the upcoming bias-variance trade-off. We introduce and discuss
further Mellin-Sobolev spaces which characterize the regularity of the unknown
density f through the decay of its Mellin transform. Additionally, we show
minimax-optimality over Mellin-Sobolev spaces of the data-driven density
estimator and hence its adaptivity.
