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