Tangential limits for harmonic functions with respect to $\varphi (\mathrm{\Delta })$ : stable and beyond

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