A simple approach to functional inequalities for non-local Dirichlet forms. (10th October 2014) - Wang, Jian
<x content-type="archive" xml:space="preserve">Abstract</x> With direct and simple proofs, we establish PoincarÃ© type inequalities (including PoincarÃ© inequalities, weak PoincarÃ© inequalities and super PoincarÃ© 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 PoincarÃ© inequalities, which were recently studied in [P.T. Gressman, J. Funct. Anal. 265 (2013) 867â€"889. C. Mouhot, E. Russ and Y. Sire, J. Math. Pures Appl. 95 (2011) 72â€"84.] To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than LÃ©vy measures.