The inverse Fueter mapping theorem

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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,

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

Received January 2010 Revised November 2010 Published April 2011

Abstract
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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 and Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165

References:

[1]	F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982.
[2]	F. Brackx, H. De Schepper and F. Sommen, A theoretical framework for wavelet analysis in a Hermitean Clifford setting, Commun. Pure Appl. Anal., 6 (2007), 549-567. doi: 10.3934/cpaa.2007.6.549.
[3]	P. Cerejeiras, M. Ferreira, U. Kahler and F. Sommen, Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis, Commun. Pure Appl. Anal., 6 (2007), 619-641. doi: 10.3934/cpaa.2007.6.619.
[4]	F. Colombo, G. Gentili, I. Sabadini and D. C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math., 222 (2009), 1793-1808. doi: 10.1016/j.aim.2009.06.015.
[5]	F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, in "Hypercomplex Analysis," Trends in Mathematics, Birkhäuser, 2009, 101-114.
[6]	F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19 (2009), 601-627. doi: 10.1007/s12220-009-9075-x.
[7]	F. Colombo and I. Sabadini, The Cauchy formula with $s$-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl., 373 (2011), 655-679. doi: 10.1016/j.jmaa.2010.08.016.
[8]	F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcal{F}$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066. doi: 10.1002/mma.1315.
[9]	F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra," Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004.
[10]	F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), 2255-2274. doi: 10.1016/j.jfa.2007.12.008.
[11]	F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385-403. doi: 10.1007/s11856-009-0055-4.
[12]	F. Colombo, I. Sabadini and D. C. Struppa, An extension theorem for slice monogenic functions and some of its consequences, Israel J. Math., 177 (2010), 369-389. doi: 10.1007/s11856-010-0051-8.
[13]	F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math., 645 (2010), 85-104. doi: 10.1515/CRELLE.2010.060.
[14]	F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics Vol. 289, Birkhäuser Basel, 2011.
[15]	A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions, J. Math. Anal. Appl., 179 (1993), 610-629. doi: 10.1006/jmaa.1993.1372.
[16]	C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32 (1965), 139-148. doi: 10.1215/S0012-7094-65-03212-6.
[17]	R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.
[18]	G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279-301.
[19]	R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), 1662-1691.
[20]	J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26, 1991.
[21]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Accademic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.
[22]	K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.
[23]	H. Hochstadt, "The Functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.
[24]	K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.
[25]	D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent, 2008.
[26]	D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922. doi: 10.1080/17476930600667650.
[27]	T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.
[28]	T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.
[29]	M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.
[30]	F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902.

show all references

References:

[1]	F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982.
[2]	F. Brackx, H. De Schepper and F. Sommen, A theoretical framework for wavelet analysis in a Hermitean Clifford setting, Commun. Pure Appl. Anal., 6 (2007), 549-567. doi: 10.3934/cpaa.2007.6.549.
[3]	P. Cerejeiras, M. Ferreira, U. Kahler and F. Sommen, Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis, Commun. Pure Appl. Anal., 6 (2007), 619-641. doi: 10.3934/cpaa.2007.6.619.
[4]	F. Colombo, G. Gentili, I. Sabadini and D. C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math., 222 (2009), 1793-1808. doi: 10.1016/j.aim.2009.06.015.
[5]	F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, in "Hypercomplex Analysis," Trends in Mathematics, Birkhäuser, 2009, 101-114.
[6]	F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19 (2009), 601-627. doi: 10.1007/s12220-009-9075-x.
[7]	F. Colombo and I. Sabadini, The Cauchy formula with $s$-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl., 373 (2011), 655-679. doi: 10.1016/j.jmaa.2010.08.016.
[8]	F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcal{F}$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066. doi: 10.1002/mma.1315.
[9]	F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra," Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004.
[10]	F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), 2255-2274. doi: 10.1016/j.jfa.2007.12.008.
[11]	F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385-403. doi: 10.1007/s11856-009-0055-4.
[12]	F. Colombo, I. Sabadini and D. C. Struppa, An extension theorem for slice monogenic functions and some of its consequences, Israel J. Math., 177 (2010), 369-389. doi: 10.1007/s11856-010-0051-8.
[13]	F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math., 645 (2010), 85-104. doi: 10.1515/CRELLE.2010.060.
[14]	F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics Vol. 289, Birkhäuser Basel, 2011.
[15]	A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions, J. Math. Anal. Appl., 179 (1993), 610-629. doi: 10.1006/jmaa.1993.1372.
[16]	C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32 (1965), 139-148. doi: 10.1215/S0012-7094-65-03212-6.
[17]	R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.
[18]	G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279-301.
[19]	R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), 1662-1691.
[20]	J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26, 1991.
[21]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Accademic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.
[22]	K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.
[23]	H. Hochstadt, "The Functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.
[24]	K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.
[25]	D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent, 2008.
[26]	D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922. doi: 10.1080/17476930600667650.
[27]	T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.
[28]	T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.
[29]	M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.
[30]	F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902.

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