On extension and refinement of the Poincaré inequality

摘要

The aim of this paper is to analyze the heat semigroup (N_t)_t>0 = {e~(tΔ)}_(t>0) generated by the usual Laplacian operator Δ on ?~d equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein-Uhlenbeck semigroup.