Abstract
The aim of this paper is to analyze the heat semigroup \({(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}\) generated by the usual Laplacian operator Δ on \({\mathbb{R}^{d}}\) 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 Poincare and reverse local Poincare inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup.
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