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

Abstract

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 Poincare 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.