Functional Inequalities for Stable-Like Dirichlet Forms

Abstract

Let $V\in C^2(\mathbb{R }^d)$ such that $\mu _V(\text{ d }x):= \text{ e }^{-V(x)}\,\text{ d }x$ is a probability measure, and let $\alpha \in (0,2)$. Explicit criteria are presented for the $\alpha$-stable-like Dirichlet form

$$\begin{aligned} {\fancyscript{E}}_{\alpha ,V}(f,f):= \int \!\!\!\!\!\!\!\int \limits _{\mathbb{R }^d\times \mathbb{R }^d} \frac{|f(x)-f(y)|^2}{|x-y|^{d+\alpha }}\,\text{ d }y\,\text{ e }^{-V(x)}\,\text{ d }x \end{aligned}$$

to satisfy Poincaré-type (i.e., Poincaré, weak Poincaré and super Poincaré) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with $V$ having some typical growths. Finally, the main result of [15] on the Poincaré inequality is strengthened.