\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

The inverse Fueter mapping theorem

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 30G35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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 $\mathcalF$-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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(245) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return