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Regularity of fundamental solutions for Lévy-type operators

Szczypkowski Karol

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

169--191

For a class of non-symmetric non-local Lévy-type operators Lκ, which include those of the form Lκf(x) := Rd(f(x + z) − f(x) − 1|z|<1⟨z, ∇f(x)⟩)κ(x, z)J(z) dz, e prove regularity of the fundamental solution pκ to the equation ∂t = Lκ.

Green function for gradient perturbation of unimodal Lévy processes

Grzywny T., Jakubowski T., Żurek G.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 1

119--143

We prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order greater than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.

Some gradient estimates on covering manifolds

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

437-443

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.