  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.
