I wrote with Luis Silvestre (University of Chicago) a series of articles about the Boltzmann equation without cut-off in the inhomogeneous case. The goal was to prove C∞ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities.
The final paper is entitled Global regularity estimates for the Boltzmann equation without cut-off.
It relies on the L∞ estimate derived by Luis Silvestre (see this paper), the local Hölder estimate derived in this paper, the Schauder estimate for kinetic equations with integral diffusion derived in this paper and the pointwise decay estimates for large velocities derived in this paper with Clément Mouhot and Luis Silvestre.