The Poisson kernel and the Fourier transform of the slice monogenic Cauchy kernels

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 Δn+1^(n−1)/2 SL^−1=Fn^L between the slice monogenic Cauchy kernel SL^−1 and the F-kernel Fn^L, 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 Δn+1 in dimension n+1, i.e., for n even. Moreover, this relation is proven computing explicitly the Fourier transform of the kernels SL^−1 and Fn^L as functions of the Poisson kernel. Similar results hold for the right kernels SR^−1 and of Fn^R.