A simple approach to functional inequalities for non-local Dirichlet forms
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 503-513.

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for Lp(p> 1) settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in [P.T. Gressman, J. Funct. Anal. 265 (2013) 867-889. C. Mouhot, E. Russ and Y. Sire, J. Math. Pures Appl. 95 (2011) 72-84.] To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than Lévy measures.

DOI : 10.1051/ps/2013048
Classification : 60G51, 60G52, 60J25, 60J75
Mots-clés : non-local dirichelt forms, Poincaré type inequalities, entropy inequalities, Beckner-type inequalities 
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Wang, Jian. A simple approach to functional inequalities for non-local Dirichlet forms. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 503-513. doi : 10.1051/ps/2013048. http://www.numdam.org/articles/10.1051/ps/2013048/

[1] S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | MR | Zbl

[2] D. Chafaï, Entropies, converxity, and functional inequalities. J. Math. Kyoto Univ. 44 (2004) 325-363. | MR | Zbl

[3] X. Chen and J. Wang, Weighted Poincaré inequalities for non-local Dirichlet forms. Preprint arXiv:1207.7140v1

[4] J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, Lq-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008) 35-59. | MR | Zbl

[5] W. Hebish and B. Zegarliński, Coercive inequalities on metric measure spaces. J. Funct. Anal. 258 (2010) 814-851. | Zbl

[6] P.T. Gressman, Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces. J. Funct. Anal. 265 (2013) 867-889. | MR | Zbl

[7] C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures. J. Math. Pures Appl. 95 (2011) 72-84. | MR | Zbl

[8] F.-Y. Wang, Orlicz-Poincaré inequalities. Proc. of Edinburgh Math. Soc. 51 (2008) 529-543. | MR | Zbl

[9] F.-Y. Wang and J. Wang, Functional inequalities for stable-like Dirichlet forms. To appear in J. Theoret. Probab. (2013).

[10] L.M. Wu, A new modified logarithmic Sobolev inequalities for Poisson point processes and serveral applications. Probab. Theoret. Relat. Fields 118 (2000) 427-438. | MR | Zbl

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