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

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