Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data

In this paper, we present sharp decay estimates for small data solutions to the following two systems: the Vlasov-Poisson (V-P) system in dimension 3 or higher and the Vlasov-Yukawa (V-Y) system in dimension 2 or higher. We rely on a modification of the vector field method for transport equation as developed by Smulevici in 2016 for the Vlasov-Poisson system. Using the Green's function in particular to estimate the bilinear terms, we improve Smulevici's result by removing the requirement of some \begin{document}$ v $\end{document} -weighted \begin{document}$ L^p $\end{document} integrability for the initial data and extend the result to the Vlasov-Yukawa system.