Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Abstract

Let V be a locally bounded measurable function on ${\mathbb {R}}^d$ such that $\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x$ is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form

$$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$

Taking $\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}$ with $0<\alpha <2$ and $\delta \geqslant 0$, we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).