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
Keywords: non-local dirichelt forms, Poincaré type inequalities, entropy inequalities, Beckner-type inequalities 
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     title = {A simple approach to functional inequalities for non-local {Dirichlet} forms},
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     url = {https://www.numdam.org/articles/10.1051/ps/2013048/}
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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

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