Sharp regularizing estimates for the gain term of the Boltzmann collision operator

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.