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](https://projecteuclid.org/download/pdf_1/euclid.ejp/1465062359) 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?**



Harmonic functions for subordinate Brownian motions and the Hölder continuity
Smoothness of harmonic functions for subordinate Brownian motions is proved in my paper with Tomasz Grzywny:



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> T. Grzywny, M. Kwaśnicki, *Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes*, Stoch. Proc. Appl. 128(1) (2018): 1–38, [DOI:j.spa.2017.04.004](https://doi.org/10.1016/j.spa.2017.04.004)
> 
> 
> 


See Theorem 1.7 and Remark 1.8(b) there.


Let me stress that the work of Ante Mimica and Panki Kim (and their co-authors) had a huge impact on the area. It is a great pity that Ante died so young.


