Weyl pseudo-differential operator and Wigner transform on the Poincaré disk

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 S¹-invariant symbol in that space, we obtain a formula to reproduce the symbol from the kernel of the Weyl pseudo-differential operator.