Poincaré Inequalities on Carnot Groups and Spectral Gap of Schrödinger Operators

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.