  We consider the $n$-dimensional space homogeneous Boltzmann equation for
elastic collisions for variable hard potentials with Grad (angular) cutoff. We
prove sharp moment inequalities, the propagation of $L^1$-Maxwellian weighted
estimates, and consequently, the propagation $L^\infty$-Maxwellian weighted
estimates to all derivatives of the initial value problem associated to the
afore mentioned problem.
  More specifically, we extend to all derivatives of the initial value problem
associated to this class of Boltzmann equations corresponding sharp moment
(Povzner) inequalities and time propagation of $L^1$-Maxwellian weighted
estimates as originally developed A.V. Bobylev in the case of hard spheres in 3
dimensions; an improved sharp moments inequalities to a larger class of angular
cross sections and $L^1$-exponential bounds in the case of stationary states to
Boltzmann equations for inelastic interaction problems with `heating' sources,
by A.V. Bobylev, I.M. Gamba and V.Panferov, where high energy tail decay rates
depend on the inelasticity coefficient and the the type of `heating' source;
and more recently, extended to variable hard potentials with angular cutoff by
I.M. Gamba, V. Panferov and C. Villani in the elastic case collision case and
so $L^1$-Maxwellian weighted estimated were shown to propagate if initial
states have such property. In addition, we also extend to all derivatives the
propagation of $L^\infty$-Maxwellian weighted estimates to solutions of the
initial value problem to the Boltzmann equations for elastic collisions for
variable hard potentials with Grad (angular) cutoff.
