The velocity averaging lemma to the relativistic free transport equation

In this paper, we revisit a velocity averaging lemma for the relativistic free transport equation using a modified vector field method. After averaging with respect to the velocity of the solution by certain weight functions $\phi$, we demonstrate that the averaged quantity ${\rho }_{\phi }(t,x)$ belongs to the Sobolev space $W_x^{1,p}$ for $p\in [1,+\mathrm{\infty }]$. This result reveals the regularizing effect of the velocity averaging of the solution. Furthermore, we also show the quantitative effects of both the particle mass and the speed of light. The proof relies on the key observation that the differential operator $t{\mathrm{\nabla }}_x+[{\mathrm{\nabla }}_v(\hat{v}){]}^{-1}{\mathrm{\nabla }}_v$ commutes with the operator ${\mathrm{\partial }}_t+\hat{v}\cdot {\mathrm{\nabla }}_x$.