On infinitely divisible distributions with polynomially decaying characteristic functions

We provide necessary and sufficient conditions on the characteristic triplet of an infinitely divisible distribution under which its characteristic function $\phi$ decays with polynomial rate. As the main result we show under a mild regularity condition on the L\'evy measure that $1/\phi$ is a Fourier multiplier on Besov spaces if and only if $\phi$ decays polynomially. The resulting mapping properties of the deconvolution operator are important for the statistical analysis of the deconvolution model and related models.