Derivative Formulae and Poincar\'e Inequality for Kohn-Laplacian Type Semigroups

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator $L:=\ff 1 2 \sum_{i=1}^m X_i^2$on$\R^{m+d}:= \R^m\times\R^d$isinvestigated,where$$X_i(x,y)= \sum_{k=1}^m \si_{ki} \pp_{x_k} + \sum_{l=1}^d (A_l x)i\pp{y_l},
\ (x,y)\in\R^{m+d}, 1\le i\le m$$for$\si$aninvertible$m\times m$-matrix and $\{A_l{\}}_{1\le l\le d}$ some $m\times m$-matrices such that the H"ormander condition holds. We first establish Bismut-type and Driver-type derivative formulae with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.