# 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 & 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., (1982). Google Scholar [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. doi: 10.3934/cpaa.2007.6.549. Google Scholar [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. doi: 10.3934/cpaa.2007.6.619. Google Scholar [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. doi: 10.1016/j.aim.2009.06.015. Google Scholar [5] F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences,, in, (2009), 101. Google Scholar [6] F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus,, J. Geom. Anal., 19 (2009), 601. doi: 10.1007/s12220-009-9075-x. Google Scholar [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. doi: 10.1016/j.jmaa.2010.08.016. Google Scholar [8] 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. doi: 10.1002/mma.1315. Google Scholar [9] F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra,", Progress in Mathematical Physics, (2004). Google Scholar [10] F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators,, J. Funct. Anal., 254 (2008), 2255. doi: 10.1016/j.jfa.2007.12.008. Google Scholar [11] F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions,, Israel J. Math., 171 (2009), 385. doi: 10.1007/s11856-009-0055-4. Google Scholar [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. doi: 10.1007/s11856-010-0051-8. Google Scholar [13] F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions,, J. Reine Angew. Math., 645 (2010), 85. doi: 10.1515/CRELLE.2010.060. Google Scholar [14] F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions,", Progress in Mathematics Vol. 289, (2011). Google Scholar [15] A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions,, J. Math. Anal. Appl., 179 (1993), 610. doi: 10.1006/jmaa.1993.1372. Google Scholar [16] C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions,, Duke Math. J., 32 (1965), 139. doi: 10.1215/S0012-7094-65-03212-6. Google Scholar [17] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions,", Mathematics and Its Applications 53, (1992). Google Scholar [18] G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable,, Adv. Math., 216 (2007), 279. Google Scholar [19] R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras,, Adv. Math., 226 (2011), 1662. Google Scholar [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). Google Scholar [21] I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products,", Accademic Press LTD, (2000). Google Scholar [22] K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space,", Birkh\, (2008). Google Scholar [23] H. Hochstadt, "The Functions of Mathematical Physics,", Pure Appl. Math., (1971). Google Scholar [24] K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem,, Meth. Appl. Anal., 9 (2002), 273. Google Scholar [25] D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis,", PhD Dissertation, (2008). Google Scholar [26] D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem,, Complex Var. Elliptic Equ., 51 (2006), 913. doi: 10.1080/17476930600667650. Google Scholar [27] T. Qian, Generalization of Fueter's result to $R^{n+1}$,, Rend. Mat. Acc. Lincei, 8 (1997), 111. Google Scholar [28] T. Qian, Fourier analysis on a starlike Lipschitz aurfaces,, J. Funct. Anal., 183 (2001), 370. doi: 10.1006/jfan.2001.3750. Google Scholar [29] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici,, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220. Google Scholar [30] F. Sommen, On a generalization of Fueter's theorem,, Zeit. Anal. Anwen., 19 (2000), 899. Google Scholar

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##### References:
 [1] F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis,", Pitman Res. Notes in Math., (1982). Google Scholar [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. doi: 10.3934/cpaa.2007.6.549. Google Scholar [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. doi: 10.3934/cpaa.2007.6.619. Google Scholar [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. doi: 10.1016/j.aim.2009.06.015. Google Scholar [5] F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences,, in, (2009), 101. Google Scholar [6] F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus,, J. Geom. Anal., 19 (2009), 601. doi: 10.1007/s12220-009-9075-x. Google Scholar [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. doi: 10.1016/j.jmaa.2010.08.016. Google Scholar [8] 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. doi: 10.1002/mma.1315. Google Scholar [9] F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra,", Progress in Mathematical Physics, (2004). Google Scholar [10] F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators,, J. Funct. Anal., 254 (2008), 2255. doi: 10.1016/j.jfa.2007.12.008. Google Scholar [11] F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions,, Israel J. Math., 171 (2009), 385. doi: 10.1007/s11856-009-0055-4. Google Scholar [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. doi: 10.1007/s11856-010-0051-8. Google Scholar [13] F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions,, J. Reine Angew. Math., 645 (2010), 85. doi: 10.1515/CRELLE.2010.060. Google Scholar [14] F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions,", Progress in Mathematics Vol. 289, (2011). Google Scholar [15] A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions,, J. Math. Anal. Appl., 179 (1993), 610. doi: 10.1006/jmaa.1993.1372. Google Scholar [16] C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions,, Duke Math. J., 32 (1965), 139. doi: 10.1215/S0012-7094-65-03212-6. Google Scholar [17] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions,", Mathematics and Its Applications 53, (1992). Google Scholar [18] G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable,, Adv. Math., 216 (2007), 279. Google Scholar [19] R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras,, Adv. Math., 226 (2011), 1662. Google Scholar [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). Google Scholar [21] I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products,", Accademic Press LTD, (2000). Google Scholar [22] K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space,", Birkh\, (2008). Google Scholar [23] H. Hochstadt, "The Functions of Mathematical Physics,", Pure Appl. Math., (1971). Google Scholar [24] K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem,, Meth. Appl. Anal., 9 (2002), 273. Google Scholar [25] D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis,", PhD Dissertation, (2008). Google Scholar [26] D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem,, Complex Var. Elliptic Equ., 51 (2006), 913. doi: 10.1080/17476930600667650. Google Scholar [27] T. Qian, Generalization of Fueter's result to $R^{n+1}$,, Rend. Mat. Acc. Lincei, 8 (1997), 111. Google Scholar [28] T. Qian, Fourier analysis on a starlike Lipschitz aurfaces,, J. Funct. Anal., 183 (2001), 370. doi: 10.1006/jfan.2001.3750. Google Scholar [29] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici,, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220. Google Scholar [30] F. Sommen, On a generalization of Fueter's theorem,, Zeit. Anal. Anwen., 19 (2000), 899. Google Scholar
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