  We prove the Hardy-Littlewood-Sobolev type $L^p$ estimates for the gain term
of the Boltzmann collision operator including Maxwellian molecule, hard
potential and hard sphere models. Combining with the results of Alonso et al.
[2] for the soft potential and Maxwellian molecule models, we provide an
unified form of $L^p$ estimates for all cutoff models which are sharp in the
sense of scaling. The most striking feature of our new estimates for the hard
potential and hard sphere models is that they do not increase the moment, the
same as Maxwellian molecule and soft potential models. Based on these
novelties, we prove the global existence and scattering of the non-negative
unique mild solution for the Cauchy problem of the Boltzmann equation when the
positive initial data is small in the weighted $L^3_{x,v}$ space.
