A Simple Approach to Functional Inequalities for Non-local Dirichlet Forms
Abstract
With direct and simple proofs, we establish Poincar\'e type inequalities (including Poincar\'e inequalities, weak Poincar\'e inequalities and super Poincar\'e 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 Poincar\'e inequalities, which were recently studied in MRS,Gre. To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than L\'evy measures.
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