### 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.

Original language | English |
---|---|

Pages (from-to) | 1057-1086 |

Number of pages | 30 |

Journal | Science China Mathematics |

Volume | 62 |

Issue number | 6 |

Early online date | 15 Apr 2019 |

DOIs | |

Publication status | Published - 1 Jun 2019 |

### Keywords

- asymptotic analysis
- H-type groups
- microlocal analysis
- semi-classical measures
- semi-classical pseudodifferential operators
- Wigner transform
- 22E30
- 46L89
- 35S05

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Science China Mathematics*,

*62*(6), 1057-1086. https://doi.org/10.1007/s11425-018-9515-6

**Semi-classical analysis on H-type groups.** / Kammerer, Clotilde Fermanian; Fischer, Véronique.

Research output: Contribution to journal › Article

*Science China Mathematics*, vol. 62, no. 6, pp. 1057-1086. https://doi.org/10.1007/s11425-018-9515-6

}

TY - JOUR

T1 - Semi-classical analysis on H-type groups

AU - Kammerer, Clotilde Fermanian

AU - Fischer, Véronique

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

KW - asymptotic analysis

KW - H-type groups

KW - microlocal analysis

KW - semi-classical measures

KW - semi-classical pseudodifferential operators

KW - Wigner transform

KW - 22E30

KW - 46L89

KW - 35S05

UR - http://www.scopus.com/inward/record.url?scp=85064716913&partnerID=8YFLogxK

U2 - 10.1007/s11425-018-9515-6

DO - 10.1007/s11425-018-9515-6

M3 - Article

VL - 62

SP - 1057

EP - 1086

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 6

ER -