Derivative formulas and Poincaré 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 $$\L: = \frac{1}{2}\sum\nolimits_{i = 1}^m {X_i^2on{\kern 1pt} {\mathbb{R}^m} \times {\mathbb{R}^d}} $$ is investigated, where $${X_i}\left( {x,y} \right) = \sum\limits_{k = 1}^m {{\sigma _{ki}}{\partial _{xk}} + } \sum\limits_{l = 1}^d {{{\left( {{A_l}x} \right)}_i}{\partial _{yl}}} ,\left( {x,y} \right) \in {\mathbb{R}^{m + d}},1 \leqslant i \leqslant m$$ for σ an invertible m×m-matrix and {Al}1≤ l≤ d some m×m-matrices such that the Hörmander condition holds. We first establish Bismut-type and Driver-type derivative formulas 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. [ABSTRACT FROM AUTHOR]