  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.
