Functional Inequalities for Stable-Like Dirichlet Forms

Let $$V\in C^2(\mathbb{R }^d)$$ V ∈ C 2 ( R d ) such that $$\mu _V(\text{ d }x):= \text{ e }^{-V(x)}\,\text{ d }x$$ μ V ( d x ) : = e − V ( x ) d x is a probability measure, and let $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) . Explicit criteria are presented for the $$\alpha $$ α -stable-like Dirichlet form $$\begin{aligned} {\fancyscript{E}}_{\alpha ,V}(f,f):= \int \!\!\!\!\!\!\!\int \limits _{\mathbb{R }^d\times \mathbb{R }^d} \frac{|f(x)-f(y)|^2}{|x-y|^{d+\alpha }}\,\text{ d }y\,\text{ e }^{-V(x)}\,\text{ d }x \end{aligned}$$ E α , V ( f , f ) : = ∫ ∫ R d × R d | f ( x ) − f ( y ) | 2 | x − y | d + α d y e − V ( x ) d x 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 $$V$$ V having some typical growths. Finally, the main result of [15] on the Poincaré inequality is strengthened.