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Clifford Analysis, Dirac Operator and the Fourier Transform

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Abstract

In this chapter, we state basic knowledge, notations and terminologies in Clifford analysis and some related results. These preliminaries will be used to establish the theory of convolution singular integrals and Fourier multipliers on Lipschitz surfaces. In Sect. 3.1, we give a brief survey on basics of Clifford analysis. In Sect. 3.2, we state the monogenic functions on sectors introduced by Li, McIntosh, Qian [1]. Section 3.3 is devoted to the Fourier transform theory on sectors established by [1]. Section 3.4 is based on the Möbius covarian of iterated Dirac operators by Peeter and Qian in [2]. In Sect. 3.5, we give a generalization of the Fueter theorem in the setting of Clifford algebras [3]. In Chaps.  6 and  7, this generalization will be used to estimate the kernels of holomorphic Fourier multiplier operators on closed Lipschitz surfaces.

References

  1. 1.
    Li C, McIntosh A, Qian T. Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces. Rev Mat Iberoam. 1994;10:665–721.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Peeter J, Qian T. Möbious covariance of iterated Dirac operators. J Aust Math Soc. 1994;56:1–12.CrossRefGoogle Scholar
  3. 3.
    Qian T. Generalization of Fueter’s result to \(R^{n+1}\). Rend Mat Acc Lincei. 1997;8:111–7.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Li C, McIntosh A, Semmes S. Convolution singular integrals on Lipschitz surfaces. J Am Math Soc. 1992;5:455–81.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Sommen F. An extension of the Radon transform to Clifford analysis. Complex Var Theory Appl. 1987;8:243–66.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ryan J. Plemelj formula and transformations associated to plane wave decomposition in complex Clifford analysis. Proc Lond Math Soc. 1992;60:70–94.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ahlfors LV. Möbius transforms and Clifford numbers. Differential geometry and complex analysis: H.E. Rauch memorial volume. Berlin: Springer; 1985. p. 65–73.Google Scholar
  8. 8.
    Sce M. Osservazioni sulle serie di potenze nei moduli quadratici. Atti Acc Lincei Rend Fis. 1957;8:220–5.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kou K, Qian T, Sommen F. Generalizations of Fueter’s theorem. Method Appl Anal. 2002;9:273–90.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Stein E-M. Singular integrals and differentiability properties of functions. Princeton: Princeton University Press; 1970.Google Scholar
  11. 11.
    Qian T. Singular integrals on star-shaped Lipschitz surfaces in the quaternionic spaces. Math Ann. 1998;310:601–30.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sommen F. On a generalization of Fueter’s theorem. Z Anal Anwend. 2000;19:899–902.MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. and Science Press 2019

Authors and Affiliations

  1. 1.Macau Institute of Systems EngineeringMacau University of Science and TechnologyMacaoChina
  2. 2.School of Mathematics and StatisticsQingdao UniversityQingdaoChina

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