# Harmonic functions for subordinate Brownian motions and the Hölder continuity

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?