Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincaré Group

In this paper, we study harmonic analysis on the affine Poincaré group ${\mathcal{P}}_{aff}$, which is a non-unimodular group, and obtain pseudo-differential operators with operator valued symbols. More precisely, we study the boundedness properties of pseudo-differential operators on ${\mathcal{P}}_{aff}$. We also provide a necessary and sufficient condition on the operator-valued symbols such that the corresponding pseudo-differential operators are in the class of Hilbert–Schmidt operators. Consequently, we obtain a characterization of the trace class pseudo-differential operators on the Poincaré affine group ${\mathcal{P}}_{aff}$, and provide a trace formula for these trace class operators. Finally, we study the Wigner transform, and Weyl transform associated with the operator valued symbol on the Poincaré affine group ${\mathcal{P}}_{aff}$.