Communications in Analysis and Geometry
Volume 28 (2020)
On Li–Yau gradient estimate for sum of squares of vector fields up to higher step
Pages: 565 – 606
In this paper, we generalize 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.
The first author is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. The second and the third author are partially supported by the M.O.S.T. of Taiwan.
Received 5 March 2016
Accepted 6 February 2018
Published 6 July 2020