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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?

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Smoothness of harmonic functions for subordinate Brownian motions is proved in my paper with Tomasz Grzywny:

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

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.

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  • $\begingroup$ Thank you for your very helpful comment. I will read your joint paper immediately. I'm not familiar with the research area and may have asked some basic questions. I'm sorry. $\endgroup$ – sharpe 2 days ago

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