Sharp optimality for density deconvolution with dominating bias

We consider estimation of the common probability density f of i.i.d. random variables Xᵢ that are observed with an additive i.i.d. noise. We assume that the unknown density f belongs to a class A of densities whose characteristic function is described by the exponent exp(-α|u|ʳ) as |u|→ ∞, where $α>0$, $r>0$. The noise density is supposed to be known and such that its characteristic function decays as exp(-β|u|ˢ), as |u| → ∞, where $β>0$, $s>0$. Assuming that $r