Low regularity Sobolev well-posedness for Vlasov--Poisson

We consider the Vlasov–Poisson equation on ${\mathbb{R}}^n\times {\mathbb{R}}^n$ with $n\ge 3$. We prove local well-posedness in $H^s({\mathbb{R}}^n\times {\mathbb{R}}^n)$ with $s>n\mathrm{/}2-1\mathrm{/}4$, for initial distribution $f_0\in H^s({\mathbb{R}}^n\times {\mathbb{R}}^n)$ having compact support in $v$. In particular, data not belonging to $L^p({\mathbb{R}}^n\times {\mathbb{R}}^n)$ for large $p$ are allowed.