Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces
Volume 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 257-299
MSC: Primary 60J35; Secondary 46E35.
DOI: 10.4064/ba56-3-8
Abstract
Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms $\|f\|_{W^{\sigma,2}}$ of a function $f\in L^2(E,\mu)$ have the property $$\eqalign{ \frac{1}{C} \, {\cal E} (f,f)&\leq\liminf_{\sigma\nearrow 1}\, (1-\sigma )\|f\|_{W^{\sigma,2}} \leq \limsup_{\sigma\nearrow 1}\, (1-\sigma )\|f\|_{W^{\sigma,2}}\cr&\leq C {\cal E} (f,f), } $$ where ${\cal E}$ is the Dirichlet form relative to the fractional diffusion.
