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arXiv · 2207.03658

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

Aparajita Dasgupta·Santosh Kumar Nayak
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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 $\mathcal{P}_{aff}$.

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Aparajita Dasgupta, Santosh Kumar Nayak. 2022-07-08. Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincar\'e Group. https://arxiv.org/abs/2207.03658

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