  We consider solutions to the massless Vlasov equation on the domain of outer
communications of the Schwarschild black hole. By adapting the r^p-weighted
energy method of Dafermos and Rodnianski, used extensively in order to study
wave equations, we prove arbitrary decay for a non-degenerate energy flux of
the Vlasov field f through a well-chosen foliation. An essential step of this
methodology consists in proving a non-degenerate integrated local energy decay.
For this, we take in particular advantage of the red-shift effect near the
event horizon. The trapping at the photon sphere requires however to lose an
epsilon of integrability in the velocity variable. Pointwise decay estimates on
the velocity average of f are then obtained by functional inequalities, adapted
to the study of Vlasov fields, which allow us to deal with the lack of
conservation law for the radial derivative.
