  For the spatially homogeneous Boltzmann equation with cutoff hard potentials
it is shown that solutions remain bounded from above, uniformly in time, by a
Maxwellian distribution, provided the initial data have a Maxwellian upper
bound. The main technique is based on a comparison principle that uses a
certain dissipative property of the linear Boltzmann equation. Implications of
the technique to propagation of upper Maxwellian bounds in the
spatially-inhomogeneous case are discussed.
