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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

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 & 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.  Google Scholar

[2]

F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcalF$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066. doi: 10.1002/mma.1315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[11]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Academic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.  Google Scholar

[12]

K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.  Google Scholar

[13]

H. Hochstadt, "The functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.  Google Scholar

[14]

G. Jank and F. Sommen, Clifford analysis, biaxial symmetry and pseudoanalytic functions, Compl. Var. Theory Appl., 13 (1990), 195-212.  Google Scholar

[15]

K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.  Google Scholar

[16]

D. Pena-Pena, "Cauchy-Kowalevski Extensions, Fueter's Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent (2008). Google Scholar

[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.  Google Scholar

[18]

T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.  Google Scholar

[19]

T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.  Google Scholar

[20]

T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann., 310 (1998), 601-630.  Google Scholar

[21]

T. Qian and F. Sommen, Deriving harmonic functions in higher dimensional spaces, Zeit. Anal. Anwen., 2 (2003), 1-12.  Google Scholar

[22]

M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.  Google Scholar

[23]

F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902.  Google Scholar

show all references

References:
[1]

F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982.  Google Scholar

[2]

F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcalF$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066. doi: 10.1002/mma.1315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[11]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Academic Press LTD, Mathematics/Engineering, Sixth Edition, 2000.  Google Scholar

[12]

K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008.  Google Scholar

[13]

H. Hochstadt, "The functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971.  Google Scholar

[14]

G. Jank and F. Sommen, Clifford analysis, biaxial symmetry and pseudoanalytic functions, Compl. Var. Theory Appl., 13 (1990), 195-212.  Google Scholar

[15]

K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290.  Google Scholar

[16]

D. Pena-Pena, "Cauchy-Kowalevski Extensions, Fueter's Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent (2008). Google Scholar

[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.  Google Scholar

[18]

T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117.  Google Scholar

[19]

T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412. doi: 10.1006/jfan.2001.3750.  Google Scholar

[20]

T. Qian, Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space, Math. Ann., 310 (1998), 601-630.  Google Scholar

[21]

T. Qian and F. Sommen, Deriving harmonic functions in higher dimensional spaces, Zeit. Anal. Anwen., 2 (2003), 1-12.  Google Scholar

[22]

M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225.  Google Scholar

[23]

F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902.  Google Scholar

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