Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling

In this paper, we obtain the reverse Bakry-\'Emery type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincar\'e inequalities and the (right and reverse) logarithmic Sobolev inequalities are presented as consequences of such estimates. Wang-Harnack inequality, Hamilton's gradient estimate and Liouville property are also presented by reverse logarithmic Sobolev inequality.
Comment: Modify minor errors after acceptance by Forum Math