Functional Inequalities for Stable-Like Dirichlet Forms

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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.