Propagation of $L^p$ estimates for the Spatially Homogeneous Relativistic Boltzmann Equation

In this paper, we prove the propagation of $L^p$ upper bounds for the spatially homogeneous relativistic Boltzmann equation for any $1<p<\mathrm{\infty }$. We consider the case of relativistic $\mathit{\text{hard}}\mathit{\text{ball}}$ with Grad’s angular cutoff. Our proof is based on a detailed study of the interrelationship between the relative momenta, the regularity and the $L^p$ estimates for the gain operator, the development of the relativistic Carleman representation, and several estimates on the relativistic hypersurface $E_{v^{\mathrm{\prime }}-v}^{v_{\ast }}$. We also derive a Pythagorean theorem for the $\mathit{\text{relative}}\mathit{\text{momenta}}$ $g(v,v_{\ast }),$ $g(v,v^{\mathrm{\prime }})$, and $g(v^{\mathrm{\prime }},v_{\ast })$, which has a crucial role in the reduction of the momentum singularity.