Sharp estimates for screened Vlasov-Poisson system around Penrose-stable equilibria in $\mathbb{R}^d $, $ d\geq3$

In this paper, we study the asymptotic stability of Penrose-stable equilibria among solutions of the screened Vlasov-Poisson system in $\mathbb{R}^d$ with $d\geq 3$ that was first established by Bedrossian, Masmoudi, and Mouhot in \cite{JBedrossian2018} with smooth initial data. More precisely, we prove the sharp decay estimates for the density of the perturbed system, exactly like the free transport with only H\"older (i.e., $C^{a}$ for $0<a<1$) perturbed initial data. This improves the recent works in \cite{HanKwanD2021} by Han-Kwan, Nguyen, and Rousset for lower derivatives of the density and in \cite{NguyenTT2020} by T. Nguyen for higher derivatives with a logarithmic correction in time. Furthermore, we establish new estimates and cancellations of the kernel to the linearized problem to obtain this result. Moreover, we also prove this result for the Vlasov-Poisson system in which the electric field obeys a general nonlinear Poisson equation containing massless electrons/ions case.