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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ v $\end{document}</tex-math></inline-formula>-weighted <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> integrability for the initial data and extend the result to the Vlasov-Yukawa system.</p>