  We investigate the Boltzmann equation with spatial smearing, diffusive
boundary conditions, and Lions' collision kernel. Both, the physical as well as
the velocity space, are assumed to be bounded. Existence and uniqueness of a
stationary solution, which is a probability density, has been demonstrated in
[3] under a certain smallness assumption on the collision term. We prove that
whenever there is a stationary solution then it is a.e. positively bounded from
below and above.
