On Li-Yau gradient estimate for sum of squares of vector fields up to higher step

TL;DR

This paper extends Li-Yau gradient estimates to higher-step sum of squares of vector fields under generalized curvature-dimension conditions, leading to new bounds for CR heat equations and kernels.

Contribution

It generalizes Cao-Yau's gradient estimate to higher-step vector fields using a curvature-dimension inequality in pseudohermitian manifolds.

Findings

Derived Li-Yau gradient estimate for CR heat equation

Established Harnack inequality for CR heat kernel

Provided upper bounds for the CR heat kernel

Abstract

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.