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

TL;DR

This paper establishes explicit criteria for various Poincaré inequalities for a class of singular stable-like Dirichlet forms on , which are generated by independent symmetric lpha-stable processes with jump kernels on coordinate axes, and explores entropy inequalities with tensorization.

Contribution

It provides sharp, explicit criteria for Poincare9, super Poincare9, and weak Poincare9 inequalities for singular stable-like Dirichlet forms, including entropy inequalities with tensorization.

Findings

Sharp criteria for Poincare9 inequalities established.

Entropy inequality similar to log-Sobolev derived for product measures.

Results applicable to Dirichlet forms generated by stable processes on coordinate axes.

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.