Harnack inequality and H\" older regularity estimates for a L\' evy process with small jumps of high intensity

TL;DR

This paper establishes a scale-invariant Harnack inequality and Hölder regularity estimates for a specific Lévy process with a characteristic exponent involving a logarithmic correction, advancing understanding of its harmonic functions.

Contribution

It introduces new Harnack inequalities and Hölder regularity results for a Lévy process with a novel characteristic exponent involving logarithmic terms.

Findings

Proved scale-invariant Harnack inequality for the process.

Established Hölder regularity estimates for harmonic functions.

Extended regularity theory to Lévy processes with high-intensity small jumps.

Abstract

We consider a L\' evy process in $\R^d$ $ (d\geq 3)$ with the characteristic exponent \[ \Phi(\xi)=\frac{|\xi|^2}{\ln(1+|\xi|^2)}-1. \] The scale invariant Harnack inequality and apriori estimates of harmonic functions in H\" older spaces are proved.