Functional Inequalities for Stable-Like Dirichlet Forms

Abstract

Let $V\in C^2({\mathbb{R}}^d)$ such that $μ_V(\d x):= \e^{-V(x)}\,\d x$ is a probability measure, and let $\mathring{a}\in (0,2)$. Explicit criteria are presented for the $\mathring{a}$-stable-like Dirichlet form $$\E_{å,V}(f,f):= \int_{\R^d\times\R^d} \ff{|f(x)-f(y)|^2}{|x-y|^{d+α}}\,\d y\,\e^{-V(x)}\,\d x$$ 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 $V$ having some typical growths. Finally, the main result of \cite{MRS} on the Poincaré inequality is strengthened

24 pages

References in corpus (3)

• Perturbations of Functional Inequalities for Lévy Type Dirichlet Forms
• Functional Inequalities and Subordination: Stability of Nash and Poincaré inequalities
• Super-Poincaré and Nash-type inequalities for Subordinated Semigroups

Cited by in corpus (3)

• Weighted Poincaré Inequalities for Nonlocal Dirichlet Forms
• Ergodicity for Time Changed Symmetric Stable Processes
• Functional inequalities for Nonlocal Dirichlet Forms With Finite Range Jumps or Large Jumps