Communications in Analysis and Geometry

Volume 28 (2020)

Number 3

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

Pages: 565 – 606



Der-Chen Chang (Department of Mathematics and Statistics, Georgetown University, Washington, District of Columbia, U.S.A.; and Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City, Taiwan)

Shu-Cheng Chang (Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei, Taiwan)

Chien Lin (Yau Mathematical Science Center, Tsinghua University, Beijing, China)


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