  In this paper, we prove the propagation of uniform upper bounds for the
spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds
have been known to be a challenging open problem in relativistic kinetic
theory. To accomplish this, we establish two types of estimates for the gain
part of the collision operator: first, we prove a potential type estimate and a
relativistic hyper-surface integral estimate. We then combine those estimates
using the relativistic counter-part of the Carleman representation to derive
uniform control of the gain term for the relativistic collision operator. This
allows us to prove the desired propagation of the uniform bounds of the
solution. We further present two applications of the propagation of the uniform
upper bounds: first we give another proof of the Boltzmann $H$-theorem, and
second we prove the asymptotic convergence of solutions to the relativistic
Maxwellian equilibrium.
