The Vlasov-Poisson equation in ${\mathbb{R}}^3$ with infinite charge and velocities

We consider the Vlasov-Poisson equation in ${\mathbb{R}}^3$ with initial data which are not $L^1$ in space and have unbounded support in the velocities. Assuming for the density a slight decay in space and a strong decay in velocities, we prove existence and uniqueness of the solution, thus generalizing the analogous result given in [5] for data compactly supported in the velocities.