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Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria

In this paper, we establish derivative estimates for the Vlasov-Poisson system with screening interactions around Penrose-stable equilibria on the phase space $\mathbb{R}^d_x\times \mathbb{R}_v^d$, with dimension $d\ge 3$. In particular, we establish the optimal decay estimates for higher derivatives of the density of the perturbed system, precisely like the free transport, up to a log correction in time. This extends the recent work \cite{T-R-HK} by Han-Kwan, Nguyen and Rousset to higher derivatives of the density. The proof makes use of several key observations from \cite{T-R-HK} on the structure of the forcing term in the linear problem, with induction arguments to classify all the terms appearing in the derivative estimates.