Poincar\'e-type Inequalities for Singular Stable-Like Dirichlet Forms

Abstract

This paper is concerned with a class of singular stable-like Dirichlet forms on $\R^d$, which are generated by $d$ independent copies of a one-dimensional symmetric $\alpha$-stable process, and whose L\'evy jump kernel measure is concentrated on the union of the coordinate axes. Explicit and sharp criteria for Poincar\'e inequality, super Poincar\'e inequality and weak Poincar\'e inequality of such singular Dirichlet forms are presented. When the reference measure is a product measure on $\R^d$, we also consider the entropy inequality for the associated Dirichlet forms, which is similar to the log-Sobolev inequality for local Dirichlet forms, and enjoys the tensorisation property.