In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These polynomial and exponential \(L^\infty \) 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 counterpart 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: another proof of the Boltzmann H-theorem, and the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.

在本文中，我们证明了空间齐次相对论玻尔兹曼方程的一致上限的传播。这些多项式和指数\（L ^ \ infty \）在相对论动力学理论中，已知边界是一个具有挑战性的开放性问题。为此，我们为碰撞算子的增益部分建立了两种类型的估计。首先，我们证明了潜在的类型估计和相对论的超表面积分估计。然后，我们使用Carleman表示形式的相对论对应物组合这些估计，以推导相对论碰撞算符的增益项的统一控制。这使我们能够证明解的均匀边界的理想传播。我们进一步介绍了均匀上限的传播的两个应用：玻尔兹曼H-定理的另一证明，以及相对论麦克斯韦平衡的解的渐近收敛性。