## Abstract

The purpose of this paper is to investigate some relations between the kernel of a Weyl pseudo-differential operator and the Wigner transform on Poincaré disk defined in our previous paper [II]. The composition formula for the class of the operators defined in [II] has not been proved yet. However, some properties and relations, which are analogous to the Euclidean case, between the Weyl pseudo-differential operator and the Wigner transform have been investigated in [II]. In the present paper, an asymptotic formula for the Wigner transform of the kernel of a Weyl pseudo-differential operator as h → 0 is given. We also introduce a space of functions on the cotangent bundle T*D whose definition is based on the notion of the Schwartz space on the Poincaré disk. For an S^{1}-invariant symbol in that space, we obtain a formula to reproduce the symbol from the kernel of the Weyl pseudo-differential operator.

Original language | English |
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Pages (from-to) | 29-48 |

Number of pages | 20 |

Journal | Annals of Global Analysis and Geometry |

Volume | 22 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Dec 1 |

Externally published | Yes |

## Keywords

- Poincaré disk
- Weyl pseudo-differential operators
- Wigner transform

## ASJC Scopus subject areas

- Analysis
- Political Science and International Relations
- Geometry and Topology