Averaging lemmas and hypoellipticity

We use the methods of commutator and fundamental solutions to establish averaging lemmas and hypoelliptic estimates for purely kinetic transport equations. Assuming certain amount of velocity regularity for solutions, we extend our analysis using the commutator method to derive the averaging and hypoelliptic regularity properties for kinetic equations in the presence of general inhomogeneous fluxes. These results find applications in the study of hypoelliptic advection-diffusion equations and kinetic formulations of hyperbolic conservation laws including Burgers’ equation with transport and isentropic gas dynamics.