Pseudo-differential Operators, Wigner Transforms and Weyl Transforms on the Poincaré Unit Disk

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Abstract

Using the affine group and the Cayley transform from the unit disk \({{\mathbb {D}}}\) onto the upper half plane, we can turn \({{\mathbb {D}}}\) into a group, which we call the Poincaré unit disk. With this construction, \({{\mathbb {D}}}\) is a noncompact and nonunimodular Lie group. We characterize all infinite-dimensional, irreducible and unitary representations of \({{\mathbb {D}}}\). By means of these representations, the Fourier transform on \({{\mathbb {D}}}\) is defined. The Plancherel theorem and hence the Fourier inversion formula can be given. Then pseudo-differential operators with operator-valued symbols, operator-valued Wigner transforms, and Weyl transforms on \({{\mathbb {D}}}\) are defined.

Keywords

Affine group Cayley transform Poincaré unit disk Fourier transform Plancherel formula Fourier inversion formula Operator-valued symbol Pseudo-differential operator Hilbert–Schmidt operator Schatten–von Neumann class Wigner transform Moyal identity Weyl transform 

Mathematics Subject Classification

Primary 47G30 

References

  1. 1.
    Dasgupta, A., Wong, M.W.: Hilbert–Schmidt and trace class pseudo-differential operators on the Heisenberg group. J. Pseudo-Differ. Oper. Appl. 4, 345–359 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dasgupta, A., Wong, M.W.: Pseudo-differential operators on the affine group. In: Pseudo-differential Operators: Groups, Geometry and Applications, Trends in Mathematics, pp. 1–14. Birkhäuser, Basel (2017)Google Scholar
  3. 3.
    Delgado, J., Wong, M.W.: $L^{p}$ -nuclear pseudo-differential operators on $\mathbb{Z}$ and ${\mathbb{S}}^1$. Proc. Am. Math. Soc. 141, 3935–3942 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Duflo, M., Moore, C.C.: On the regular representation of a non-unimodular locally compact group. J. Funct. Anal. 21, 209–243 (1976)CrossRefMATHGoogle Scholar
  5. 5.
    Helgason, S.: Groups and Geometric Analysis, Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence (2000)CrossRefGoogle Scholar
  6. 6.
    Molahajloo, S.: Pseudo-differential operators on ${\mathbb{Z}}$. In: New Developments in Pseudo-differential Operators, Operator Theory: Advances and Applications, vol. 205, pp. 213–221. Birkhäuser, Basel (2009)Google Scholar
  7. 7.
    Molahajloo, S., Wong, M.W.: Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on ${\mathbb{S}}^1$. J. Pseudo-Differ. Oper. Appl. 1, 183–205 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Peng, L., Zhao, J.: Weyl transforms associated with the Heisenberg group. Bull. Sci. Math. 132, 78–86 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Peng, L., Zhao, J.: Weyl transforms on the upper half plane. Rev. Mat. Complut. 23, 77–95 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Segal, I.E.: An extension of Plancherels formula to separable unimodular groups. Ann. Math. 52, 272–292 (1950)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Springer, New York (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Tate, T.: Weyl pseudo-differential operator and Wigner transform on the Poincaré disk. Ann. Glob. Anal. Geom 22, 29–48 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wong, M.W.: Complex Analysis. World Scientific, Singapore (2008)CrossRefMATHGoogle Scholar
  14. 14.
    Wong, M.W.: Weyl Transforms. Springer, New York (1998)MATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsInstitute for Advanced Studies in Basic SciencesZanjanIran

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