The Fueter primitive of biaxially monogenic functions

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March 2014, 13(2): 657-672. doi: 10.3934/cpaa.2014.13.657

The Fueter primitive of biaxially monogenic functions

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
3.	Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent

Received February 2013 Revised July 2013 Published October 2013

Abstract
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In the recent papers by F. Colombo, I. Sabadini, F. Sommen, "The inverse Fueter mapping theorem", Commun. Pure Appl. Anal., 10 (2011), 1165--1181, and "The inverse Fueter mapping theorem in integral form using spherical monogenics", Israel J. Math., 194 (2013), 485--505, the authors have started a systematic study of the inverse Fueter mapping theorem. In this paper we show that the inversion theorem holds for the case of biaxially monogenic functions. Here there are several additional difficulties with respect to the cases already treated. However, we are still able to prove an integral version of the inverse Fueter mapping theorem. The kernels appearing in the integral representation formula have an explicit representation that can be computed depending on the dimension of the Euclidean space in which the problem is considered.

Keywords: Inverse Fueter mapping theorem, biaxially monogenic functions, integral representation formula..

Mathematics Subject Classification: Primary: 30G3.

Citation: Fabrizio Colombo, Irene Sabadini, Frank Sommen. The Fueter primitive of biaxially monogenic functions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 657-672. doi: 10.3934/cpaa.2014.13.657

References:

[1]	F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982.
[2]	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.
[3]	F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem, Commun. Pure Appl. Anal., 10 (2011), 1165-1181. doi: 10.3934/cpaa.2011.10.1165.
[4]	F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel J. Math., 194 (2013), 485-505. doi: 10.1007/s11856-012-0090-4.
[5]	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.
[6]	F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics, Vol. 289, Birkhäuser, 2011.
[7]	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.
[8]	R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.
[9]	R. Fueter, Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta\Delta u = 0$ mit vier reellen Variablen, Comm. Math. Helv., 7 (1934), 307-330.
[10]	J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26 (1991).
[11]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Academic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.
[12]	K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.
[13]	H. Hochstadt, "The functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.
[14]	G. Jank and F. Sommen, Clifford analysis, biaxial symmetry and pseudoanalytic functions, Compl. Var. Theory Appl., 13 (1990), 195-212.
[15]	K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.
[16]	D. Pena-Pena, "Cauchy-Kowalevski Extensions, Fueter's Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent (2008).
[17]	D. Pena-Pena, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922. doi: 10.1080/17476930600667650.
[18]	T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.
[19]	T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.
[20]	T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann., 310 (1998), 601-630.
[21]	T. Qian and F. Sommen, Deriving harmonic functions in higher dimensional spaces, Zeit. Anal. Anwen., 2 (2003), 1-12.
[22]	M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.
[23]	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. 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.
[3]	F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem, Commun. Pure Appl. Anal., 10 (2011), 1165-1181. doi: 10.3934/cpaa.2011.10.1165.
[4]	F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel J. Math., 194 (2013), 485-505. doi: 10.1007/s11856-012-0090-4.
[5]	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.
[6]	F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics, Vol. 289, Birkhäuser, 2011.
[7]	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.
[8]	R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.
[9]	R. Fueter, Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta\Delta u = 0$ mit vier reellen Variablen, Comm. Math. Helv., 7 (1934), 307-330.
[10]	J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26 (1991).
[11]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Academic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.
[12]	K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.
[13]	H. Hochstadt, "The functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.
[14]	G. Jank and F. Sommen, Clifford analysis, biaxial symmetry and pseudoanalytic functions, Compl. Var. Theory Appl., 13 (1990), 195-212.
[15]	K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.
[16]	D. Pena-Pena, "Cauchy-Kowalevski Extensions, Fueter's Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent (2008).
[17]	D. Pena-Pena, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922. doi: 10.1080/17476930600667650.
[18]	T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.
[19]	T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.
[20]	T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann., 310 (1998), 601-630.
[21]	T. Qian and F. Sommen, Deriving harmonic functions in higher dimensional spaces, Zeit. Anal. Anwen., 2 (2003), 1-12.
[22]	M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.
[23]	F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902.

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