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 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.
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D. Bakry , L’hypercontractivité et son utilisation en théorie de semi-groupes, in P. Bernard, edotor, Lecture in Probability Theory. École d’été de Probabilités de Saint-Flour 1992, Lecture Note in Math, Vol 1581, 1994, p. 1–114, Spring-Verlag, New-York/Berlin.
D. Bakry, F. Bolley, and I. Gentil , Dimension dependent hypercontractivity for Gaussian kernels, to appear in Probability Theory and related Fields (2012).
Beckner W.: A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105, 397–400 (1989)
Bentaleb A.: Développement de la moyenne d’une fonction pour la mesure ultrasphérique, C. R. Acad. Sci. Paris Série. I Math. 317, 781–784 (1993)
Bolley F., Gentil I.: Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl. 93, 449–473 (2010)
Carlen E.: Superadditivity of Fisher information and logarithmic Sobolev inequalities, J. Funct. Anal. 101, 194–211 (1991)
Gross L.: Logarithmic Sobolev inequalities, Amer. J. Math. 97, 1061–1083 (1975)
Houdré C., Kagan A.: Variances inequalities for functions of Gaussian variables, J. Theoretical Prob. 8, 23–30 (1995)
Ledoux M.: On an integral criterion for hypercontractivity of diffusion semigroups, J. Funct. Anal. 105, 444–465 (1992)
Ledoux M.: L’algèbre de Lie des gradients itérés d’un générateur markovien-Développements de moyennes et entropies, Ann. scient. Ec. Norm. Sup. 28, 435–460 (1995)
Nelson E.: The free Markov field, J. Funct. Anal. 12, 211–227 (1973)
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This paper was written while the first author was visiting the Institute of Mathematics, Paul Sabatier University (Toulouse) in June 2011.
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Bentaleb, A., Fahlaoui, S. & Hafidi, A. On extension and refinement of the Poincaré inequality. Arch. Math. 100, 63–74 (2013). https://doi.org/10.1007/s00013-012-0464-1
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DOI: https://doi.org/10.1007/s00013-012-0464-1