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

2022/07/08 by Dasgupta, Aparajita, Nayak, Santosh Kumar
#47G30 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47G10 #Secondary 42C40

paper · doi:10.48550/arxiv.2207.03658

Abstract

In this paper, we study harmonic analysis on the affine Poincaré group 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 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 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 P_aff.

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