Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces

Katarzyna Pietruska-Pałuba¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 3, pp. 257-299

Résumé

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.

Détail

DOI : 10.4064/ba56-3-8

Keywords: supposing metric space question supports fractional diffusion prove after introducing appropriate multiplicative factor gagliardo seminorms sigma function have property eqalign frac cal leq liminf sigma nearrow sigma sigma leq limsup sigma nearrow sigma sigma leq cal where cal dirichlet form relative fractional diffusion