  We prove smoothing estimates for velocity averages of the kinetic transport
equation in hyperbolic Sobolev spaces at the critical regularity, leading to a
complete characterisation of the allowable regularity exponents. Such estimates
will be deduced from some mixed-norm estimates for the cone multiplier operator
at a certain critical index. Our argument is not particular to the geometry of
the cone and we illustrate this by establishing analogous estimates for the
paraboloid.
