Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincar\'e Group

TL;DR

This paper develops harmonic analysis tools on the affine Poincaré group, including pseudo-differential operators, trace formulas, and transforms, expanding the mathematical framework for non-unimodular groups.

Contribution

It introduces a comprehensive analysis of pseudo-differential operators with operator-valued symbols on the affine Poincaré group, including boundedness, Hilbert-Schmidt criteria, and trace class characterizations.

Findings

Characterization of bounded pseudo-differential operators on the group

Necessary and sufficient conditions for Hilbert-Schmidt operators

Trace formula for trace class pseudo-differential operators

Abstract

In this paper, we study harmonic analysis on the affine Poincar\'e 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 Poincar\'e 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 Poincar\'e affine group $\mathcal{P}_{aff}$.