Local Well-Posedness for Vlasov--Poisson with $L^{d+}$ Initial Density and Fractional Velocity Regularity

We prove a local well-posedness criterion for the Vlasov–Poisson system on ${\mathbb{R}}_x^d\times {\mathbb{R}}_v^d$, $d\ge 2$, under an anisotropic assumption on the initial distribution. The datum has finite mass, its weighted velocity supremum belongs to $L_x^p$ for some $p>d$, and it has an arbitrarily small positive H"older regularity in the velocity variable, uniformly with respect to velocity and with the same spatial $L^p$ control. The main estimate is a nonlinear mixing bound $$[\rho (t){]}_{C_x^{\alpha }}\lesssim t^{-d\mathrm{/}p-ϵ}C(f_0),ϵ>0\text{ small}\mathrm{.}$$ Thus the density is integrable in time with values in a positive spatial H"older class, and the corresponding electric field belongs to $L_t^1{C}_x^{1,\alpha }$. We construct a solution by a Schauder fixed point on the density and prove uniqueness by a Loeper-type stability estimate.