  In this paper we prove propagation in time of weighted $L^\infty$ bounds for
solutions to the non-cutoff homogeneous Boltzmann equation that satisfy
propagation in time of weighted $L^1$ bounds. To emphasize that the propagation
in time of weighted $L^{\infty}$ bounds relies on the propagation in time of
weighted $L^1$ bounds, we express our main result using certain general
weights. Consequently we apply the main result to cases of exponential and
Mittag-Leffler weights, for which propagation in time of weighted $L^1$ bounds
holds.
