Journal of Lie Theory 35 (2025), No. 3, 629--650
Copyright Heldermann Verlag 2025
Poincaré Inequalities on Carnot Groups and Spectral Gap of Schrödinger Operators
Marianna Chatzakou
Department of Mathematics, Ghent University, Belgium
marianna.chatzakou@ugent.be
Serena Federico
Department of Mathematics, University of Bologna, Italy
serena.federico2@unibo.it
Boguslaw Zegarlinski
Institut de Mathématiques, UMR5219, CNRS, Université de Toulouse, France
b.zegarlinski@math.univ-toulouse.fr
We give a sufficient condition under which the global Poincaré inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. Additionally, we show that the global Poincaré inequality holds true on any Carnot group for a certain choice of a probability measure adapted to the structure of each Carnot group, and whose formula is explicitly given. Consequently, we extend the results of a previous work by the authors [q-Poincaré inequalities on Carnot groups with a filiform Lie algebra, Potential Analysis 60/3 (2024) 1067--1092] targeted on filiform Carnot groups to any Carnot group. As a result, the Schrödinger operators associated with the density of the considered probability measure have a spectral gap.
Keywords: Poincaré inequalities, Carnot groups, sub-gradient, spectral gap.
MSC: 35R03, 35A23, 26D10.
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