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

TL;DR

This paper establishes Fatou and relative Fatou theorems for nonnegative harmonic functions related to subordinate Brownian motions with Gaussian components in smooth domains, including existence of nontangential limits and optimality of results.

Contribution

It extends Fatou theorems to a broad class of subordinate Brownian motions with Gaussian components in smooth domains, including new limit existence results and optimality analysis.

Findings

Proved relative Fatou's theorem for harmonic functions in bounded $C^{1,1}$ domains.

Established Fatou theorem for harmonic functions in balls.

Demonstrated the optimality of the results by showing non-tangential limits may not exist in some cases.

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 $\mathbb{R}^{d}$, $d\geq 2$, 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 $D=B(0,1)$.