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Gradient estimates of harmonic functions and transition densities for Lévy processes


Authors: Tadeusz Kulczycki and Michał Ryznar
Journal: Trans. Amer. Math. Soc. 368 (2016), 281-318
MSC (2010): Primary 31B05, 60J45; Secondary 60J50, 60J75
DOI: https://doi.org/10.1090/tran/6333
Published electronically: May 13, 2015
MathSciNet review: 3413864
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Abstract: We prove gradient estimates for harmonic functions with respect to a $ d$-dimensional unimodal pure-jump Lévy process under some mild assumptions on the density of its Lévy measure. These assumptions allow for a construction of an unimodal Lévy process in $ \mathbb{R}^{d+2}$ with the same characteristic exponent as the original process. The relationship between the two processes provides a fruitful source of gradient estimates of transition densities. We also construct another process called a difference process which is very useful in the analysis of differential properties of harmonic functions. Our results extend the gradient estimates known for isotropic stable processes to a wide family of isotropic pure-jump processes, including a large class of subordinate Brownian motions.


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Additional Information

Tadeusz Kulczycki
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: Tadeusz.Kulczycki@pwr.wroc.pl

Michał Ryznar
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: Michal.Ryznar@pwr.wroc.pl

DOI: https://doi.org/10.1090/tran/6333
Received by editor(s): August 1, 2013
Received by editor(s) in revised form: October 31, 2013
Published electronically: May 13, 2015
Additional Notes: This research was supported in part by NCN grant no. 2011/03/B/ST1/00423.
Article copyright: © Copyright 2015 American Mathematical Society