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 Δn+1(n−1)/2SL−1=FnL between the slice monogenic Cauchy kernel SL−1 and the F-kernel FnL, 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 FnL as functions of the Poisson kernel. Similar results hold for the right kernels SR−1 and of FnR.