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

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 $L^p (p>1)$ settings. Our results yield a new sufficient condition for fractional Poincar\'{e} 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 L\'{e}vy measures.Comment: 12 page

Convex Sobolev inequalities and spectral gap

This note is devoted to the proof of convex Sobolev (or generalized Poincar\'{e}) inequalities which interpolate between spectral gap (or Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case

Poincar\'e inequality for non euclidean metrics and transportation cost inequalities on Rd\mathbb{R}^d

In this paper, we consider Poincar\'e inequalities for non euclidean metrics on $\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincar\'e type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities

Height inequalities and canonical class inequalities

An expository lecture on the analogy between the subjects of the title. Delivered at the International Conference on Number Theory at the Korea Institute for Advanced Study in December 1997.Comment: Not for separate publicatio