The weak Harnack inequality for the Boltzmann equation without cut-off

Abstract

In this paper, we obtain the weak Harnack inequality and H\"older estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local H\"older estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off.