Isoperimetric Inequalities for Non-Local Dirichlet Forms

Let $(E,\text{F},\mu )$ be a $\text{si}$-finite measure space. For a non-negative symmetric measure $J(\text{d}x,\text{d}y):=J(x,y)\mu (\text{d}x)\mu (\text{d}y)$ on $E\times E,$considerthequadraticform$$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 , J(\d x,\d y)$$in$L^2(\mu)$. We characterize the relationship between the isoperimetric inequality and the super Poincar'e inequality associated with $\text{E}$. In particular, sharp Orlicz-Sobolev type and Poincar'e type isoperimetric inequalities are derived for stable-like Dirichlet forms on ${\mathbb{R}}^n$, which include the existing fractional isoperimetric inequality as a special example.