Positive curvature property for sub-Laplace on nilpotent Lie group of rank two

TL;DR

This paper investigates the positive curvature properties of the sub-Laplace operator on a rank-two nilpotent Lie group, extending known results from the Heisenberg group to derive sharp gradient bounds for the heat kernel.

Contribution

It demonstrates that the sub-Laplace on the rank-two nilpotent Lie group satisfies a $CD(0, abla)$ inequality on radial functions, generalizing previous results from simpler groups.

Findings

The sub-Laplace operator satisfies a $CD(0, abla)$ inequality on radial functions.

Sharp gradient bounds for the heat kernel are derived.

The operator does not satisfy any $CD(r, abla)$ inequality for general functions.

Abstract

In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by $n$ Brownian motions and their $\frac{n(n-1)}2$ L\'evy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$. In order to study contraction properties of the heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motion model, the restriction of the sub-Laplace acting on radial functions (see Definition \ref{radial fun}) satisfies a positive Ricci curvature condition (more precisely a $CD(0, \infty)$ inequality, see Theorem \ref{positive}, whereas the operator itself does not satisfy any $CD(r,\infty)$ inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper \cite{Qian2}.