  The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure
to extend holomorphic functions of one complex variable to slice monogenic
functions and to monogenic functions. Using the Cauchy formula of slice
monogenic functions the FSQ-theorem admits an integral representation for $n$
odd. In this paper we show that the relation $
\Delta_{n+1}^{(n-1)/2}S_L^{-1}=\mathcal{F}^L_n $
  between the slice monogenic Cauchy kernel $S_L^{-1}$ and the F-kernel
$\mathcal{F}^L_n$, that appear in the integral form of the FSQ-theorem for $n$
odd, holds also in the case we consider the fractional powers of the Laplace
operator $\Delta_{n+1}$ in dimension $n+1$, i.e., for $n$ even. Moreover, this
relation is proven computing explicitly Fourier transform of the kernels
$S_L^{-1}$ and $\mathcal{F}^L_n$ as functions of the Poisson kernel. Similar
results hold for the right kernels $S_R^{-1}$ and of $\mathcal{F}^R_n$.
