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Semi-classical analysis on H-type groups

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Abstract

In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodi fferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account thefinite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.

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Acknowledgements

This paper was written while Clotilde Fermanian Kammerer was visiting Technische Universität München and she thanks the members of the mathematics department of this institution for their kind hospitality, especially Caroline Lasser and Simone Warzel.

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Authors and Affiliations

  1. Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, Créteil, 94010, France

    Clotilde Fermanian Kammerer

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Véronique Fischer

Authors
  1. Clotilde Fermanian Kammerer
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  2. Véronique Fischer
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Correspondence to Véronique Fischer.

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Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

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Kammerer, C.F., Fischer, V. Semi-classical analysis on H-type groups. Sci. China Math. 62, 1057–1086 (2019). https://doi.org/10.1007/s11425-018-9515-6

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