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

TL;DR

This paper establishes explicit criteria for Poincaré and super Poincaré inequalities for non-local Dirichlet forms with finite and large jumps, revealing the influence of jump sizes on these inequalities.

Contribution

It provides new sharp criteria for functional inequalities in non-local Dirichlet forms with finite and large jumps, using novel approaches like local Poincaré inequalities and Lyapunov conditions.

Findings

Super Poincaré inequality holds for finite range jumps under certain conditions.

Large jumps do not satisfy super Poincaré inequality, highlighting jump size effects.

Finite range jumps share properties with local Dirichlet forms regarding inequalities.

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 $\alpha\in(0,2)$ 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.