A Simple Approach to Functional Inequalities for Non-local Dirichlet Forms

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 \cite{MRS,Gre}. To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than Lévy measures.

12 pages

References in corpus (1)

• Weighted Poincaré Inequalities for Nonlocal Dirichlet Forms

Cited by in corpus (1)

• Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods