On Harnack inequality and H\"{o}lder regularity for isotropic unimodal L\'{e}vy processes

TL;DR

This paper establishes scale-invariant Harnack inequalities and regularity results for harmonic functions related to isotropic unimodal Lévy processes, providing sharp estimates of potential measures and kernels under certain scaling conditions.

Contribution

It introduces new sharp estimates for potential measures and kernels, and extends Harnack inequalities to a broader class of Lévy processes with scaling properties.

Findings

Proved scale-invariant Harnack inequality for isotropic unimodal Lévy processes.

Derived sharp estimates of potential measures and kernels.

Established Krylov-Safonov type estimates for these processes.

Abstract

We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal L\'{e}vy process with the characteristic exponent $\psi$ satisfying some scaling condition. We show sharp estimates of the potential measure and capacity of balls, and further, under the assumption of that $\psi$ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allow us to establish the Krylov-Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.