We prove Landau damping for the collisionless Vlasov equation with a class of $L1$ interaction potentials (including the physical case of screened Coulomb interactions) on $Rx3\times Rv3$ for localized disturbances of an infinite, homogeneous background. Unlike the confined case $Tx3\times Rv3$, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from zero, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an additional dispersive mechanism available on $Rx3$ which reduces the strength of the plasma echo resonance.