A simple approach to functional inequalities for non-local Dirichlet forms

Jian Wang

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 503-513
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Wang, Jian. "A simple approach to functional inequalities for non-local Dirichlet forms." ESAIM: Probability and Statistics 18 (2014): 503-513. <http://eudml.org/doc/273647>.

@article{Wang2014,
abstract = {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&gt; 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.},
author = {Wang, Jian},
journal = {ESAIM: Probability and Statistics},
keywords = {non-local dirichelt forms; Poincaré type inequalities; entropy inequalities; Beckner-type inequalities; non-local Dirichelt forms Poincaré type inequalities entropy inequalities},
language = {eng},
pages = {503-513},
publisher = {EDP-Sciences},
title = {A simple approach to functional inequalities for non-local Dirichlet forms},
url = {http://eudml.org/doc/273647},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Wang, Jian
TI - A simple approach to functional inequalities for non-local Dirichlet forms
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 503
EP - 513
AB - 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&gt; 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.
LA - eng
KW - non-local dirichelt forms; Poincaré type inequalities; entropy inequalities; Beckner-type inequalities; non-local Dirichelt forms Poincaré type inequalities entropy inequalities
UR - http://eudml.org/doc/273647
ER -

References

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  1. [1] S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal.156 (1998) 347–365. Zbl0920.60002MR1636948
  2. [2] D. Chafaï, Entropies, converxity, and functional inequalities. J. Math. Kyoto Univ.44 (2004) 325–363. Zbl1079.26009MR2081075
  3. [3] X. Chen and J. Wang, Weighted Poincaré inequalities for non-local Dirichlet forms. Preprint arXiv:1207.7140v1 
  4. [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. Zbl1148.26018MR2366398
  5. [5] W. Hebish and B. Zegarliński, Coercive inequalities on metric measure spaces. J. Funct. Anal.258 (2010) 814–851. Zbl1189.26032
  6. [6] P.T. Gressman, Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces. J. Funct. Anal.265 (2013) 867–889. Zbl1282.26024MR3067789
  7. [7] C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures. J. Math. Pures Appl.95 (2011) 72–84. Zbl1208.26025MR2746437
  8. [8] F.-Y. Wang, Orlicz-Poincaré inequalities. Proc. of Edinburgh Math. Soc.51 (2008) 529–543. Zbl1159.47023MR2465923
  9. [9] F.-Y. Wang and J. Wang, Functional inequalities for stable-like Dirichlet forms. To appear in J. Theoret. Probab. (2013). Zbl06482261
  10. [10] L.M. Wu, A new modified logarithmic Sobolev inequalities for Poisson point processes and serveral applications. Probab. Theoret. Relat. Fields118 (2000) 427–438. Zbl0970.60093MR1800540

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