Quantitative De Giorgi methods in kinetic theory for non-local operators

Abstract

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies H\"older continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. This permits to quantitatively prove the intermediate value lemma for the full range of non-locality parameter $s \in (0, 1)$. Our results recover the results from Imbert and Silvestre [22] for the inhomogeneous Boltzmann equation in the non-cutoff case. The paper is self-contained.