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 L^{p}(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

@article{PS_2014__18__503_0, author = {Wang, Jian}, title = {A simple approach to functional inequalities for non-local {Dirichlet} forms}, journal = {ESAIM: Probability and Statistics}, pages = {503--513}, publisher = {EDP-Sciences}, volume = {18}, year = {2014}, doi = {10.1051/ps/2013048}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2013048/} }

TY - JOUR AU - Wang, Jian TI - A simple approach to functional inequalities for non-local Dirichlet forms JO - ESAIM: Probability and Statistics PY - 2014 SP - 503 EP - 513 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2013048/ DO - 10.1051/ps/2013048 LA - en ID - PS_2014__18__503_0 ER -

Wang, Jian. A simple approach to functional inequalities for non-local Dirichlet forms. ESAIM: Probability and Statistics, Volume 18 (2014), pp. 503-513. doi : 10.1051/ps/2013048. http://www.numdam.org/articles/10.1051/ps/2013048/

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