A simple approach to functional inequalities for non-local Dirichlet forms. (10th October 2014)

• Record Type: Journal Article
• Authors: Wang, Jian
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 503
• Page End: 513
• Publication Date: 2014-10-10
• Subjects: Non-local Dirichelt forms; Poincaré type inequalities; entropy inequalities; Beckner-type inequalities; Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013048
• Languages: English
• ISSNs: 1292-8100

Abstract

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 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.