Project: Functional estimates and evolution equations on Carnot groups

2022-10-01 – 2025-09-30

Abstract

The project is devoted to the actively developing areas of noncommutative analysis fuelled by strong recent advances in the global quantization theories and functional analysis on nilpotent Lie groups. We will mainly work in the setting of (homogeneous) Carnot groups (or stratified groups). These are the nilpotent Lie groups allowing for a collection of vector fields whose iterated commutators span the whole of the Lie algebra. Such groups play a central role in the sub-Riemannian analysis, allowing for the use of techniques from analysis, geometry, algebra, and the theory of partial differential equations. In the project, we will concentrate on three very important related problems: Log-Sobolev inequalities, lower bounds, and hypoellitpic evolution PDEs, all in the setting of Carnot groups. Earlier we proved higher order Log-Sobolev inequalities for general Carnot groups, and here we aim to establish them in abstract Hilbert spaces, and to find the best constant in the case of the Heisenberg group, where the necessary machinery is available, and subsequently pass to the infinite (with respect to the first stratum)-dimensional version of it. At the same time, the lower bounds such as Gårding, sharp Gårding and Fefferman-Phong inequalities on Carnot groups would provide effective ways for dealing with hypoelliptic partial differential operators and their evolutions. These are open and challenging problems that we are planning to address in our work on this project.