This question is about harmonic functions of subordinate Brownian motions.

We write $B=(\{B_t\}_{t \ge 0}, \{P_x\}_{x \in \mathbb{R}^d})$ for the $d$-dimensional Brownian motion. Let $\{S_t\}_{t \ge 0}$ be a subordinator, which is an increasing pure-jump Lévy process starting at zero independent of $B$. We set $Y_t=X_{S_t}$, $t \ge 0$. Then, $Y=(\{Y_t\}_{t \ge 0},\{P_x\}_{x \in \mathbb{R}^d})$ is a subordinate Brownian motion.

I could find some sufficient conditions for nonnegative harmonic functions for $Y$ to satisfy the Harnack inequality (perhaps, KM is a standard reference). As a consequence, we can prove that the harmonic functions are Hölder continuous.

However, this method will generally not allow us to obtain the exact value of the index of Hölder continuity. **Is there any preceeding research on the quantitative estimates of the index of the Hölder continuity?**