TY - JOUR

T1 - Weyl pseudo-differential operator and Wigner transform on the Poincaré disk

AU - Tate, Tatsuya

N1 - Funding Information:
Research partially supported by JSPS.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Poincaré disk

KW - Weyl pseudo-differential operators

KW - Wigner transform

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U2 - 10.1023/A:1016253829938

DO - 10.1023/A:1016253829938

M3 - Article

AN - SCOPUS:0141688320

SN - 0232-704X

VL - 22

SP - 29

EP - 48

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 1

ER -