Computational Methods and Function Theory 1 (2001), No. 1, 179--192
Copyright Heldermann Verlag 2001
Sirkka-Liisa Eriksson-Bique
Sirkka-Liisa.Eriksson-Bique@joensuu.fi, Department of Mathematics, University of Joensuu, P. O. Box 111, 80101 Joensuu, Finland.
Heinz Leutwiler
leutwil@mi.uni-erlangen.de, University of Erlangen-Nürnberg, Department of Mathematics, Bismarckstrasse 1 1/2, D-91054 Erlangen, Germany.
Let $\mathbb{H}$ be the algebra of quaternions and $\mathcal{C}\ell_{0,3}$ the Clifford algebra generated by $e_1$, $e_2$, $e_3$, subject to the conditions ${e_1}^2={e_2}^2={e_3}^2=-1$ and $e_i e_k = -e_k e_i$ $(i,k=1,2,3:\, i\neq k)$. Embedding $\mathbb{H}$ in $\mathcal{C}\ell_{0,3}$, any element in $\mathcal{C}\ell_{0,3}$ may be written in the form $q_0 + q_1 e_3$, where $q_0,q_1\in\mathbb{H}$. We thus write $\mathbb{H}_2$ rather than $\mathcal{C}\ell_{0,3}$. Let $Q$ denote the projection operator given by $Q(q_0 + q_1 e_3) =q_1$. We are interested in the modified Dirac operator in $\mathbb{R}^4$, defined by $$ Mf(x) = Df(x) +\frac{2(Qf)^{\prime}(x)}{x_3}, $$ where $x=\sum_{i=0}^{3} x_i e_i,$ $D=\sum_{i=0}^{3} e_i \partial/\partial x_i$ and $^\prime$ designates the main involution in $\mathbb{H}_2$. An infinitely differentiable function $f\colon \Omega\rightarrow\mathbb{H}_2$ is called hyperholomorphic in an open set $\Omega\subset\mathbb{R}^4$, if $Mf(x)=0$, for all $x\in\Omega$ with $x_3\neq0$. The second author noticed in \cite{le1} that the power function $x\rightarrow x^m$ is hyperholomorphic. \ A function $f\colon \Omega\rightarrow\mathbb{H}_2$ is called hyperbolic harmonic if $M\overline{M}f=0$, where $$ \overline{M}f(x) = \overline{D}f(x) - \frac{2(Qf)^{\prime}(x)}{x_3}. $$ Note that a real-valued function $f$ is hyperbolic harmonic if and only if it satisfies the Laplace-Beltrami equation $x_3\triangle f-2\partial f/\partial x_3=0$, associated with the hyperbolic metric on $\mathbb{R}_+^4$. We show that $f$ is hyperholomorphic if and only if both $f$ and $xf$ are hyperbolic harmonic. Hyperholomorphic functions are holomorphic Cliffordian functions, a concept introduced by G. Laville and~I.~Ramadanoff in \cite{laville}. Our main result states that, locally, holomorphic Cliffordian functions can always be written in terms of four hyperholomorphic functions. At last we give a representation theorem.
Keywords: Hyperholomorphic functions, hyperbolic metrics, monogenic, hypermonogenic, quaternions, generalized function theory.
MSC 2000: Primary 30G35; Secondary 30A05, 30F45.
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