Abstract
We prove gradient estimates for harmonic functions with respect to a d 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 R d + 2 \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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