Complex Analysis and Operator Theory

, Volume 12, Issue 3, pp 811–833 | Cite as

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



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.


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 


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Copyright information

© 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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