Positive Curvature Property for Sub-Laplacian on Nilpotent Lie Group of Rank Two

Abstract

In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ℍ. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely a CD(0, ∞ ) inequality), see Theorem 4.5, whereas the operator itself does not satisfy any CD(r, ∞ ) 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 (Qian, Bull Sci Math 135:262–278, 2011).