Semi-classical analysis on H-type groups
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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.
KeywordsH-type groups semi-classical pseudodifferential operators semi-classical measures Wigner transform asymptotic analysis microlocal analysis
MSC(2010)35S05 22E30 46L89
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.
- 1.Anantharaman N, Faure F, Fermanian Kammerer C. Chaos Quantique. Palaiseau: École Polytechnique, 2014Google Scholar
- 5.Bahouri H, Chemin J-Y, Danchin R. A frequency space for the Heisenberg group. ArXiv:1609.03850, 2016Google Scholar
- 6.Bahouri H, Chemin J-Y, Danchin R. Tempered distributions and Fourier transform on the Heisenberg group. ArXiv: 1705.02195, 2017Google Scholar
- 14.Beals R, Greiner P. Calculus on Heisenberg Manifolds. Annals of Mathematics Studies, vol. 119. Princeton: Princeton University Press, 1988Google Scholar
- 15.Christ M, Geller D, Glowacki P, et al. Pseudodifferential operators on groups with dilations. Duke Math J, 1992, 68: 31–65Google Scholar
- 21.Fermanian Kammerer C, Fischer V. Defect measures on graded lie groups. ArXiv:1707.04002, 2017Google Scholar
- 34.Korányi A. Some applications of Gelfand pairs in classical analysis. In: Harmonic Analysis and Group Representations. Naples: Liguori, 1982, 333–348Google Scholar
- 35.Lévy G. The Fourier transform on 2-step Lie groups. ArXiv:1712.09880, 2017Google Scholar