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

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