In plasma physics and also in general relativity, the dynamics of collisionless particles can be described by Vlasov-wave type coupled systems of equations, e.g., Vlasov-Maxwell system and the Einstein-Vlasov system. A widely assumption used in the literature for the study of Vlasov equation is compact support assumption.

By combining the strength of Klainerman vector field method and Fourier method, we show global regularity and sharp decay estimates for the $3D$ massive relativistic Vlasov-Nordstr\"om system and the $3D$ massive relativistic Vlasov-Maxwell system for small initial data without any compact support assumption.