Analogs of Korn's inequality on Heisenberg groups

We give two analogs of Korn's inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn's inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.