  We establish smoothing estimates in the framework of hyperbolic Sobolev
spaces for the velocity averaging operator $\rho$ of the solution of the
kinetic transport equation. If the velocity domain is either the unit sphere or
the unit ball, then, for any exponents $q$ and $r$, we find a characterisation
of the exponents $\beta_+$ and $\beta_-$, except possibly for an endpoint case,
for which $D_+^{\beta_+}D_-^{\beta_-} \rho$ is bounded from space-velocity
$L^2_{x,v}$ to space-time $L^q_tL^r_x$. Here, $D_+$ and $D_-$ are the classical
and hyperbolic derivative operators, respectively. In fact, we shall provide an
argument which unifies these velocity domains and the velocity averaging
estimates in either case are shown to be equivalent to mixed-norm bounds on the
cone multiplier operator acting on $L^2$. We develop our ideas further in
several ways, including estimates for initial data lying in certain Besov
spaces, for which a key tool in the proof is the sharp $\ell^p$ decoupling
theorem recently established by Bourgain and Demeter. We also show that the
level of permissible smoothness increases significantly if we restrict
attention to initial data which are radially symmetric in the spatial variable.
