Functional inequalities for Nonlocal Dirichlet Forms With Finite Range Jumps or Large Jumps

Abstract

The paper is a continuation of our paper [12,2], and it studies functional inequalities for non-local Dirichlet forms with finite range jumps or large jumps. Let $α\in(0,2)$ and $μ_V(dx)=C_Ve^{-V(x)}\,dx$ be a probability measure. We present explicit and sharp criteria for the Poincaré inequality and the super Poincaré inequality of the following non-local Dirichlet form with finite range jump $$\mathscr{E}_{α, V}(f,f):= (1/2)\iint_{|x-y|\le 1}\frac{(f(x)-f(y))^2}{|x-y|^{d+α}} dy μ_V(dx);$$ on the other hand, we give sharp criteria for the Poincaré inequality of the non-local Dirichlet form with large jump as follows $$\mathscr{D}_{α, V}(f,f):= (1/2)\iint_{|x-y|> 1}\frac{(f(x)-f(y))^2}{|x-y|^{d+α}} dy μ_V(dx),$$ and also derive that the super Poincaré inequality does not hold for $\mathscr{D}_{α, V}$. To obtain these results above, some new approaches and ideas completely different from \cite{WW, CW} are required, e.g. local Poincaré inequality for $\mathscr{E}_{α, V}$ and $\mathscr{D}_{α, V}$, and the Lyapunov condition for $\mathscr{E}_{α, V}$. In particular, the results about $\mathscr{E}_{α, V}$ show that the probability measure fulfilling Poincaré inequality and super Poincaré inequality for non-local Dirichlet form with finite range jump and that for local Dirichlet form enjoy some similar properties; on the other hand, the assertions for $\mathscr{D}_{α, V}$ indicate that even if functional inequalities for non-local Dirichlet form heavily depend on the density of large jump in the associated Lévy measure, the corresponding small jump plays an important role for local super Poincaré inequality, which is inevitable to derive super Poincaré inequality.