Geometric Invariance of the Semi-Classical Calculus on Nilpotent Graded Lie Groups

Clotilde Fermanian-Kammerer; Veronique Fischer; Steven Flynn

doi:10.1007/s12220-022-01163-z

Geometric Invariance of the Semi-Classical Calculus on Nilpotent Graded Lie Groups

Clotilde Fermanian-Kammerer, Veronique Fischer, Steven Flynn

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (SciVal)

Abstract

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are uniformly Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds.

Original language	English
Article number	127
Journal	The Journal of Geometric Analysis
Volume	33
Issue number	4
Early online date	2 Feb 2023
DOIs	https://doi.org/10.1007/s12220-022-01163-z
Publication status	Published - 30 Apr 2023

Bibliographical note

Funding Information:
The authors thank Antoine Julia and Pierre Pansu for inspiring discussions. The authors acknowledge the support of The Leverhulme Trust for this work via Research Project Grant 2020-037, Quantum limits for sub-elliptic operators.

Keywords

Abstract harmonic analysis
Analysis on nilpotent Lie groups
Semi-classical analysis on nilpotent Lie groups and on filtered manifolds

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-022-01163-zLicence: CC BY

Cite this

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abstract = "In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are uniformly Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds. ",

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N2 - In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are uniformly Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds.

AB - In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are uniformly Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds.

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Geometric Invariance of the Semi-Classical Calculus on Nilpotent Graded Lie Groups

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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