Relativistic collision geometry and global near-vacuum solutions at critical transverse regularity

We establish global well-posedness near vacuum for the relativistic Boltzmann equation with soft and hard potentials in the range $γ\in(-2,1]$. The initial data have critical Besov regularity of order $2/r$ in the transverse spatial variables for an admissible range of $r$ depending on $γ$, together with weighted integrability in the longitudinal and momentum variables. The proof combines a fixed-direction div--curl lemma with bilinear collision estimates. In particular, to control the gain term, we combine localization to spherical caps with energy-weighted angular estimates. The key relativistic estimate controls the singularity associated with the first component of the post-collisional relative velocity. These bounds yield the required time decay.