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 [11]. The composition formula for the class of the operators defined in [11] 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 [11]. In the present paper, an asymptotic formula for the Wigner transform of the kernel of a Weyl pseudo-differential operator as ħ → 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.
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Tate, T. Weyl Pseudo-Differential Operator and Wigner Transform on the Poincaré Disk. Annals of Global Analysis and Geometry 22, 29–48 (2002). https://doi.org/10.1023/A:1016253829938
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DOI: https://doi.org/10.1023/A:1016253829938