Relative Fatou's Theorem for $(\Delta)^{\alpha/2}$harmonic Functions in Bounded $\kappa$fat Open Set
Abstract
We give a probabilistic proof of relative Fatou's theorem for $(\Delta)^{\alpha/2}$harmonic functions (equivalently for symmetric $\alpha$stable processes) in bounded $\kappa$fat open set where $\alpha \in (0,2)$. That is, if $u$ is positive $(\Delta)^{\alpha/2}$harmonic function in a bounded $\kappa$fat open set $D$ and $h$ is singular positive $(\Delta)^{\alpha/2}$harmonic function in $D$, then nontangential limits of $u/h$ exist almost everywhere with respect to the Martinrepresenting measure of $h$. It is also shown that, under the gaugeability assumption, relative Fatou's theorem is true for operators obtained from the generator of the killed $\alpha$stable process in bounded $\kappa$fat open set $D$ through nonlocal FeynmanKac transforms. As an application, relative Fatou's theorem for relativistic stable processes is also true if $D$ is bounded $C^{1,1}$open set.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2004
 arXiv:
 arXiv:math/0401309
 Bibcode:
 2004math......1309K
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis;
 31B25;
 60J75
 EPrint:
 This paper will appear in Journal of Functional Analysis