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July 2011, 10(4): 1165-1181. doi: 10.3934/cpaa.2011.10.1165
The inverse Fueter mapping theorem
Fabrizio Colombo 1, , Irene Sabadini 2, and Frank Sommen 3,
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano
Politecnico di Milano, Dipartimento di Matematica, Via Bonardi, 9, 20133 Milano, Italy
Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent, Belgium
Received January 2010 Revised November 2010 Published April 2011
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.
Keywords: slice monogenic functions, axially monogenic function, Fueter mapping theorem in integral form, inverse Fueter mapping theorem in integral form., Cauchy-Riemann equations, Vekua's system, Fueter's primitive.
Mathematics Subject Classification: Primary: 30G3.
Citation: Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
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Fabrizio Colombo Irene Sabadini Frank Sommen