Semi-local behaviour of non-local hypoelliptic equations: Boltzmann

The purpose of this note is to demonstrate the announced result in Loher, The Strong Harnack inequality for the Boltzmann equation, S\'eminaire Laurent Schwartz proceeding by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in arXiv:2404.05612 does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation.