Abstract: In this talk, we discuss remainder term representation formulae for the higher-order Poincare inequality for vector fields which imply short and direct proofs of the sharp (classical) Poincare inequalities. The obtained results directly imply sharp Poincare type inequalities for some vector fields satisfying Hormander’s condition, for example. As an application, we obtain a blow-up result for solutions to the Dirichlet initial-boundary value problem for the Baouendi-Grushin heat operator.