Frequency multiplier estimates for the linearized relativistic Boltzmann operator without angular cutoff

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. The non-cutoff theory for the relativistic Boltzmann equation has been rarely studied even under a smallness assumption on the initial data due to the lack of understanding of the spectrum and the need for coercivity estimates on the linearized collision operator. Namely, it is crucial to obtain the sharp asymptotics for the frequency multiplier to obtain this coercivity that has never been established before. In this paper, we prove the sharp asymptotics for the frequency multiplier for a general relativistic scattering kernel without angular cutoff. As a consequence of our calculations, we further explain how the well-known change of variables $p' \to p$ is not well defined in the special relativistic context.