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

Abstract

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 ${\Delta }_{n+1}^{(n-1)/2}{S}_L^{-1}={\mathcal{F}}_n^L$ between the slice monogenic Cauchy kernel $S_L^{-1}$ and the F-kernel ${\mathcal{F}}_n^L$, 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 ${\Delta }_{n+1}$ 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 ${\mathcal{F}}_n^L$ as functions of the Poisson kernel. Similar results hold for the right kernels $S_R^{-1}$ and of ${\mathcal{F}}_n^R$.