Fueter’s Theorem for Monogenic Functions in Biaxial Symmetric Domains

Abstract

Fueter’s theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions in \(\mathbb R^{m+1}\) when m is odd. It states that \(\Delta _{m+1}^{k+\frac{m-1}{2}}\bigl [\bigl (u(x_0,\vert \underline{x}\vert )+\frac{\underline{x}}{\vert \underline{x}\vert }\,v(x_0,\vert \underline{x}\vert )\bigr )P_k(\underline{x})\bigr ]\) is monogenic if \(u+iv\) is holomorphic and \(P_k(\underline{x})\) is a homogeneous monogenic polynomial in \(\mathbb R^m\). Eelbode et al. (AIP Conf Proc 1479:340–343, 2012) proved that this statement is still valid if the monogenicity condition on \(P_k(\underline{x})\) is dropped. To obtain this result, the authors used representation theory methods but their result also follows from a direct calculus we established in our paper Peña Peña and Sommen (J Math Anal Appl 365:29–35, 2010). In this paper we generalize the result from Eelbode et al. (2012) to the case of monogenic functions in biaxially symmetric domains. In order to achieve this goal we first generalize Peña Peña and Sommen (2010) to the biaxial case and then derive the main result from that.

Full text

Results Math 72 (2017), 1747–1758
c
2017 Springer International Publishing AG
1422-6383/17/041747-12
published online August 8, 2017
DOI 10.1007/s00025-017-0732-2 Results in Mathematics
Fueter’s Theorem for Monogenic Functions
in Biaxial Symmetric Domains
Dixan Pe˜na Pe˜na, Irene Sabadini, and Franciscus Sommen
Abstract. Fueter’s theorem discloses a remarkable connection existing be-
tween holomorphic functions and monogenic functions in Rm+1 when m
is odd. It states that Δk+m−1
2
m+1 u(x0,|x|)+ x
|x|v(x0,|x|)Pk(x)is mono-
genic if u+iv is holomorphic and Pk(x) is a homogeneous monogenic
polynomial in Rm. Eelbode et al. (AIP Conf Proc 1479:340–343, 2012)
proved that this statement is still valid if the monogenicity condition on
Pk(x) is dropped. To obtain this result, the authors used representation
theory methods but their result also follows from a direct calculus we
established in our paper Pe˜na Pe˜na and Sommen (J Math Anal Appl
365:29–35, 2010). In this paper we generalize the result from Eelbode
et al. (2012) to the case of monogenic functions in biaxially symmetric
domains. In order to achieve this goal we first generalize Pe˜na Pe˜na and
Sommen (2010) to the biaxial case and then derive the main result from
that.
Mathematics Subject Classification. 30G35, 31A05.
Keywords. Clifford monogenic functions, Fueter’s theorem, Fischer
decomposition.
1. Introduction
Let Rmbe the real Clifford algebra generated by the standard basis {e1,...,e
m}
of the Euclidean space Rm(see [2,17]). The multiplication in this associative al-
gebra is determined by the relations: e2
j=−1, ejek+ekej=0,1≤j=k≤m.
Any Clifford number a∈Rmmay thus be written as
a=
A
aAeA,a
A∈R,
where the basis elements eA=ej1...e
jkare defined for every subset A=
{j1,...,j
k}of {1,...,m}with j1<···<j
k(for A=∅one puts e∅= 1).
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