The Vlasov-Poisson equation in R^3 with infinite charge and velocities

We consider the Vlasov–Poisson equation in R3 with initial data which are not L1 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 the existence and uniqueness of the solution, thus generalizing the analogous result by the same authors for data compactly supported in the velocities