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

TL;DR

This paper explores the relationship between slice monogenic Cauchy kernels and Fourier transforms, extending previous results to even dimensions using fractional Laplacian powers and explicit Fourier transform calculations.

Contribution

It demonstrates that the relation between Cauchy kernels and F-kernels holds in even dimensions through explicit Fourier transform computations involving the Poisson kernel.

Findings

Established the relation for even dimensions using fractional Laplacian powers.

Computed explicit Fourier transforms of the kernels as functions of the Poisson kernel.

Extended the integral representation of the FSQ-theorem to even dimensions.

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}^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 $\mathcal{F}^R_n$.