Fatou and relative Fatou theorem for subordinate Brownian motions with Gaussian components on smooth domains

Abstract

We prove relative Fatou's theorem for nonnegative harmonic functions with respect to a large class of killed subordinate Brownian motions with Gaussian components in bounded $C^{1,1}$ open sets in ${\mathbb{R}}^d$, $d\ge 2$, which asserts the existence of nontangential limit of the ratio of two harmonic functions with respect to the killed processes. When $D=B(x_0,r)$ is a ball we prove Fatou theorem. That is, we establish the existence of nontangential limit of a single nonnegative harmonic function. We also prove this is the best result possible by showing that there is a nonnegative harmonic function which does not have a tangential limit a.e. when $d=2$ and $D=B(0,1)$.

arXiv admin note: text overlap with arXiv:1106.5858 by other authors