## On harmonic functions of symmetric Lévy processes.(English. French summary)Zbl 1298.60054

The author considers subordinate Brownian motions $$X_t:= B(S_t)$$ such that:
(i)
The Brownian motion $$B$$ and the subordinator $$S$$ are independent.
(ii)
The Lévy and potential measures of $$S$$ have decreasing densities.
(iii)
The Laplace exponent $$\phi(\lambda):=-\log(\operatorname{E} e^{-\lambda S_1})$$ of $$S$$ satisfies $\lim_{\lambda\to +\infty} \frac{\phi'(\lambda x)}{\phi'(\lambda)} = x^{\alpha/2 -1} \quad (x>0)$ for some $$\alpha\in [0,2]$$.
For this class of Lévy processes, the author proves the existence of a constant $$c>0$$ such that for any $$r\in (0,\frac{1}{4})$$ and any bounded function $$f:\mathbb{R}^d \rightarrow \mathbb{R}$$ which is harmonic in $$B_{4r}(0):=\{x: |x|<4r\}$$, $|f(x)-f(y)|\leq c \|f\|_{\infty} \frac{\phi(r^{-2})}{\phi(|x-y|^{-2})} \quad (x,y\in B_{4r}(0)).$ With such a priori regularity estimates for harmonic functions, the author extends, in particular, results of Krylov and Safonov (as in [R. F. Bass and D. A. Levin, Potential Anal. 17, No. 4, 375–388 (2002; Zbl 0997.60089)]) and H. Šikić et al. [Probab. Theory Relat. Fields 135, No. 4, 547–575 (2006; Zbl 1099.60051)].

### MSC:

 60G51 Processes with independent increments; Lévy processes 60J45 Probabilistic potential theory 60J75 Jump processes (MSC2010) 60J25 Continuous-time Markov processes on general state spaces

### Citations:

Zbl 0997.60089; Zbl 1099.60051
Full Text:

### References:

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