Further Aspects of the Fueter Mapping Theorem

TL;DR

This paper explores the properties of the Fueter mapping in even and odd dimensions, showing its axial nature, surjectivity, and its action on monomials, extending previous results to even dimensions and Laurent series.

Contribution

It generalizes the Fueter and Sce theorems to even dimensions and demonstrates the equivalence of different definitions of the Fueter mapping on monomials and Laurent series.

Findings

Fueter mapping images are axial functions for even n.

The Fueter mapping is surjective on axial monogenic functions.

The action of the Fueter mapping on monomials coincides with the Fourier-based mapping, extending previous odd-dimensional results.

Abstract

In this paper we first show that for $n$ being even the images of the Fueter mapping, as monogenic functions, like for the odd $n$ cases proved through computation on the pointwise differential operator, are also of the axial type (the Axial Form Theorem). Due to a recent result of B. Dong, K. I. Kou, T. Qian, I. Sabadini (2016), we know that the Fueter mapping is surjective on the set of all left- and right-monogenic functions of the axial type in axial domains (the Surjectivity Theorem). The second part results of this paper address the action of the Fueter mapping on the monomials $f^{(l)}_0(z)=z^l, l=0, \pm 1, \pm 2,...,$ in one complex variable. In generalizing the Fueter and the Sce theorems to the even dimensions $n,$ Qian used the following mapping $\tau$ (applicable for odd dimensions as well): $$\tau (f^{(-k)}_0)=\mathcal{F}^{-1}((-2\pi |\cdot |)^{n-1}\mathcal{F}({\vec{f_0^{(-k)}}})), \ \ \ \tau (f^{(n-2+k)}_0)=I(\tau (f^{(-k)}_0)), $$ where $k$ is any positive integer, $\mathcal{F}$ is the Fourier transformation in $\mathbb{R}^{n+1},$ $I$ is the Kelvin inversion. We show that on the monomials the action of the mapping $\tau$ defined through the Kelvin inversion coincides with that of the Fueter mapping defined through the Fourier transform (the Monomial Theorem). In the mentioned 1997 paper this result is proved for the case $n$ being odd integers. The Monomial Theorem further implies that the extended mapping $\tau$ on Laurent series of real coefficients is identical with the Fueter mapping defined through pointwise differentiation or the Fourier multiplier.