Family of Functional Inequalities for the Uniform Measure

Khalid Boutahir¹ and Ali Hafidi^2,

¹Département de Mathématiques & Informatique, Université My Ismail, B. P. 11 201 Zitoune, Meknès, MAROC

²Faculté des Sciences et Techniques, B.P.509, Boutalamine Errachidia, MAROC

Cite this paper:
Khalid Boutahir and Ali Hafidi. Family of Functional Inequalities for the Uniform Measure. Journal of Mathematical Sciences and Applications. 2017; 5(1):19-23. doi: 10.12691/jmsa-5-1-3

Abstract

We consider on the interval [-1,1] the heat semigroup generated by the Legendre operator acting on the Hilbert space with respect to the uniform measure By means of a simple method involving some semigroup techniques, we describe a large family of optimal integral inequalities with the Poincaré and logarithmic Sobolev inequalities as particular cases.

Keywords:
heat semigroup legendre operator spectral gap poincaré inequality sobolev inequality logarithmic sobolev inequality φ-entropy inequality

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	D.Bakry, M. Émery, Diffusions hypercontractives. Séminaire de probabilities de Strasbourg 19 (1985): 177-206.
[2]	W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Am. Math. Soc. 1989; 105 (2): 397-400.
[3]	A.Bentaleb,S. Fahlaoui, A.Hafidi Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup,Semigroup Forum, 2012; 85 (2): 361-368.
[4]	A.Bentaleb,S.Fahlaoui, a family integral inequalities on the circle S1, Proc. Jpn. Acad., Ser. A 2010; 86: 55-59.
[5]	F. Bolley, I. Gentil, Phi-entropy inequalities for diffusion semigroups. J. Math.Pures Appl. 2010; 93 (5): 449-473.
[6]	J. Doulbeault, I. Gentil, and A. Jüngel, A logarithmic fourth-order parabolic equation and related logarithmic sobolev inequalitiesn Comm. Math. Sci. 2006; 4 (2): 275-290.
[7]	L.Gross,logarithmic Sobolev inequalities, Amer.J.Math. 1975; 97 (4), 1061-1083.
[8]	M. Ledoux, The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. 2000; 6(9)(2): 305-366.
[9]	E. Nelson, The free Markov field. J. Funct. Anal. 1973; 12: 211-227.
[10]	F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites, Potential Anal.22(2005) no 1, 1-15.