Title: A simple approach to functional inequalities for non-local Dirichlet forms
Abstract: With direct and simple proofs, we establish Poincare type inequalities (including Poincare inequalities, weak Poincare inequalities and super Poincare 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 Poincare 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 Levy measures.