Tangential limits for harmonic functions with respect to $ϕ(Δ)$ : stable and beyond

Abstract

In this paper, we discuss tangential limits for regular harmonic functions with respect to $ϕ(Δ):=-ϕ(-Δ)$ in the $C^{1,1}$ open set $D$ in $\mathbb{R}^d$, where $ϕ$ is the complete Bernstein function and $d \ge 2$. When the exterior function $f$ is local $L^p$-Hölder continuous of order $β$ on $D^c$ with $ p\in(1,\infty]$ and $β>1/p$, for a large class of Bernstein function $ϕ$, we show that the regular harmonic function $u_f$ with respect to $ϕ(Δ)$, whose value is $f$ on $D^c$, converges a.e. through a certain parabola that depends on $ϕ$ and $ϕ'$. Our result includes the case $ϕ(λ)=\log(1+λ^{α/2})$. Our proofs use both the probabilistic and analytic methods.