Limiting Behaviour of Dirichlet Forms for Stable Processes on Metric Spaces

Volume 56 / 2008

Katarzyna Pietruska-Pałuba 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.

Authors

  • Katarzyna Pietruska-PałubaInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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