# American Institute of Mathematical Sciences

July  2011, 10(4): 1165-1181. doi: 10.3934/cpaa.2011.10.1165

## 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, Italy 3 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.
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.

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##### 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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