Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T19:36:36.554Z Has data issue: false hasContentIssue false
  • English
  • Français

A simple approach to functional inequalities for non-localDirichlet forms

Published online by Cambridge University Press:  10 October 2014

Jian Wang
Jian Wang*
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, 350007 Fuzhou, P.R. China. [email protected]
Get access

Abstract

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

Keywords

Non-local Dirichelt forms; Poincaré type inequalities; entropy inequalities;Beckner-type inequalities
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 503 - 513
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bobkov, S.G. and Ledoux, M., On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347–365. Google Scholar
Chafaï, D., Entropies, converxity, and functional inequalities. J. Math. Kyoto Univ. 44 (2004) 325–363. Google Scholar
X. Chen and J. Wang, Weighted Poincaré inequalities for non-local Dirichlet forms. Preprint arXiv:1207.7140v1
Dolbeault, J., Gentil, I., Guillin, A. and Wang, F.-Y., L q-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008) 35–59. Google Scholar
Hebish, W. and Zegarliński, B., Coercive inequalities on metric measure spaces. J. Funct. Anal. 258 (2010) 814–851. Google Scholar
Gressman, P.T., Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces. J. Funct. Anal. 265 (2013) 867–889. Google Scholar
Mouhot, C., Russ, E. and Sire, Y., Fractional Poincaré inequalities for general measures. J. Math. Pures Appl. 95 (2011) 72–84. Google Scholar
Wang, F.-Y., Orlicz-Poincaré inequalities. Proc. of Edinburgh Math. Soc. 51 (2008) 529–543. Google Scholar
F.-Y. Wang and J. Wang, Functional inequalities for stable-like Dirichlet forms. To appear in J. Theoret. Probab. (2013).
Wu, L.M., A new modified logarithmic Sobolev inequalities for Poisson point processes and serveral applications. Probab. Theoret. Relat. Fields 118 (2000) 427–438. Google Scholar