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: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013048
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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 <italic>L</italic>^{<italic>p</italic>}(<italic>p</italic>&gt; 1) settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in [P.T. Gressman, <italic>J. Funct. Anal. </italic><bold>265 </bold>(2013) 867â€"889. C. Mouhot, E. Russ and Y. Sire, <italic>J. Math. Pures Appl. </italic><bold>95 </bold>(2011) 72â€"84.] To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than Lévy measures.</p> </abstract>