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Hilbert Transforms on the Sphere and Lipschitz Surfaces

  • Tao Qian
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Through a double-layer potential argument we define harmonic conjugates of the Cauchy type and prove their existence and uniqueness in Lipschitz domains. We further define inner and outer Hilbert transformations on Lipschitz surfaces and prove their boundedness in L p , where the range for the index p depends on the Lipschitz constant of the boundary surface. The inner and outer Poisson kernels, the Cauchy type conjugate inner and outer Poisson kernels, and the kernels of the inner and outer Hilbert transformations on the sphere are obtained. We also obtain Abel sum expansions of the kernels. The study serves as a justification of the methods in a series of papers of Brackx et al. based on their method for computation of a certain type of harmonic conjugates.

Keywords

Poisson kernel Conjugate Poisson kernel Schwarz kernel Hilbert transformation Cauchy integral double-layer potential Clifford algebra 

Mathematics Subject Classification (2000)

Primary 62D05, 30D10 Secondary 42B35 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Tao Qian
    • 1
  1. 1.Department of MathematicsUniversity of Macau MacaoChina SAR

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