L²-L^p estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

Abstract

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate $L^2$–$L^p$ boundedness of pseudo-differential operators on the Heisenberg motion group for the range $2\le p\le \mathrm{\infty }$. We also provide a necessary and sufficient condition on the operator-valued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators.

Keywords:

• Pseudo-differential operators
• Heisenberg motion group
• λ-Weyl transforms
• trace class operators
• Hilbert–Schmidt operators

2010 Mathematics Subject Classifications:

• Primary 35S05
• 43A99
• Secondary 22E45
• 43A80

Acknowledgments

The authors would like to thank an anonymous referee for valuable suggestions which improved the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Additional information

Funding

Vishvesh Kumar is supported by the FWO Odysseus 1 [grant number G.0H94.18N]: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) [grant number 01M01021]. Shyam Swarup Mondal gratefully acknowledges the support provided by IIT Guwahati, Government of India.