  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.
