Gradient estimates of harmonic functions and transition densities for Lévy processes

Kulczycki, Tadeusz; Ryznar, Michał

doi:10.1090/tran/6333

Gradient estimates of harmonic functions and transition densities for Lévy processes
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by Tadeusz Kulczycki and Michał Ryznar PDF

Trans. Amer. Math. Soc. 368 (2016), 281-318 Request permission

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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