Clifford Analysis, Dirac Operator and the Fourier Transform



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.


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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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