  This work introduces a new approach to velocity averaging lemmas in kinetic
theory. This approach -- based upon the classical energy method -- provides a
powerful duality principle in kinetic transport equations which allows for a
natural extension of classical averaging lemmas to previously unknown cases
where the density and the source term belong to dual spaces. More generally,
this kinetic duality principle produces regularity results where one can trade
a loss of regularity or integrability somewhere in the kinetic transport
equation for a suitable opposite gain elsewhere. Also, it looks simpler and
more robust to rely on proving inequalities instead of constructing exact
parametrices.
  The results in this article are introduced from a functional analytic point
of view. They are motivated by the abstract regularity theory of kinetic
transport equations. However, we may recall that velocity averaging lemmas have
profound implications in kinetic theory and its related physical models. In
particular, the precise formulation of such results has the potential to lead
to important applications to the regularity of renormalizations of
Boltzmann-type equations, as well as kinetic formulations of gas dynamics, for
instance.
