The inverse Fueter mapping theorem

Fabrizio Colombo; Irene Sabadini; Frank Sommen

doi:10.3934/cpaa.2011.10.1165

Article Contents

2011, Volume 10, Issue 4: 1165-1181. Doi: 10.3934/cpaa.2011.10.1165

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The inverse Fueter mapping theorem

1.
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano
2.
Politecnico di Milano, Dipartimento di Matematica, Via Bonardi, 9, 20133 Milano
3.
Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent

Received: December 31, 2009

Revised: October 31, 2010

Published: June 2011

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Abstract

In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function $f$ of the form $f=\alpha+\underline{\omega}\beta$ (where $\alpha$, $\beta$ satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function $\bar{f}=A+\underline{\omega}B$ (where $A,B$ satisfy the Vekua's system) given by $\bar{f}(x)=\Delta^{\frac{n-1}{2}}f(x)$ where $\Delta$ is the Laplace operator in dimension $n+1$. In this paper we solve the inverse problem: given an axially monogenic function $\bar{f}$ determine a slice monogenic function $f$ (called Fueter's primitive of $\bar{f}$ such that $\bar{f}=\Delta^{\frac{n-1}{2}}f(x)$. We prove an integral representation theorem for $f$ in terms of $\bar{f}$ which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution $f$ of the equation $\Delta^{\frac{n-1}{2}}f(x)=\bar{f} (x)$ in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.

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Mathematics Subject Classification: Primary: 30G35.

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