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A simple approach to functional inequalities for non-local Dirichlet 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. jianwang@fjnu.edu.cn
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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> 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.

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

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