  We prove the sharp regularizing estimates for the gain term of the Boltzmann
collision operator including hard sphere, hard potential and Maxwell molecule
models. Our new estimates characterize both regularization and convolution
properties of the gain term and have the following features. The regularizing
exponent is sharp both in the $L^2$ based inhomogeneous and homogeneous Sobolev
spaces which is exact the exponent of the kinetic part of collision kernel. The
functions in these estimates belong to a wider scope of (weighted) Lebesgue
spaces than the previous regularizing estimates. Furthermore, for the estimates
in homogeneous Sobolev spaces, we only need functions lying in Lebesgue spaces
instead of weighted Lebesgue spaces, i.e., no loss of weight occurs in this
case.
