Deconvolution for some singular density errors with respect to the $\mathbb{L}^1$ loss

We present a versatile and model-based procedure for estimating a density in a deconvolution setting where the error density is assumed to be singular enough. We assess the quality of our estimator by establishing non-asymptotic risk bounds for the $\mathbb{L}^1$ loss. We specify them under various constraints on the target. We investigate cases where the density is multimodal, (piecewise) concave/convex, belongs to a (possibly inhomogeneous) Besov space or a particular parametric model. Moreover, our estimation procedure is robust with respect to model misspecification