Coercive Inequalities on Carnot Groups: Taming Singularities

TL;DR

This paper develops a method to handle singularities in Carnot groups to establish coercive inequalities, leading to new probability measures satisfying Poincaré and Logarithmic Sobolev inequalities, with applications to spectral analysis.

Contribution

It introduces a novel technique for taming singularities in Carnot groups to derive coercive inequalities and construct measures satisfying key functional inequalities.

Findings

Constructed explicit probability measures on Carnot groups.

Established Poincaré and Logarithmic Sobolev inequalities for these measures.

Analyzed spectral properties of associated Markov generators.

Abstract

In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function $U$ in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincar\'e and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson-Ornstein-Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators.