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Hilbert Transforms on the Sphere with the Clifford Algebra Setting

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Abstract

Through a double-layer potential argument the inner and outer Poisson kernels, the Cauchy-type conjugate inner and outer Poisson kernels, and the kernels of the Cauchy-type inner and outer Hilbert transformations on the sphere are deduced. We also obtain Abel sum expansions of the kernels and prove the L p-boundedness of the inner and outer Hilbert transformations for 1<p<∞.

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Authors and Affiliations

  1. Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira, S.J., Taipa, Macao, China

    Tao Qian

  2. School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, China

    Yan Yang

Authors
  1. Tao Qian
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  2. Yan Yang
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Correspondence to Tao Qian.

Additional information

Communicated by Hans G. Feichtinger.

The work was supported by research grant of the University of Macau No. RG079/04-05S/QT/FST.

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Qian, T., Yang, Y. Hilbert Transforms on the Sphere with the Clifford Algebra Setting. J Fourier Anal Appl 15, 753–774 (2009). https://doi.org/10.1007/s00041-009-9062-4

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  • DOI: https://doi.org/10.1007/s00041-009-9062-4

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