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

Propagation of $L^{1}$ and $L^{\infty}$ Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation