On infinitely divisible distributions with polynomially decaying   characteristic functions
  We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $\phi$ decays polynomially. Under a mild regularity condition this polynomial decay is equivalent to $1/\phi$ being a Fourier multiplier on Besov spaces. 
Mathias Trabs
math.PR math.FA math.ST stat.TH