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 the regularization and the convolution properties of the gain term and have the following features. The regularizing exponent is sharp both in the L-2 based inhomogeneous Sobolev spaces and the homogeneous Sobolev spaces, which is the exact exponent of the kinetic part of the collision kernel. The functions in these estimates belong to a wider scope of (weighted) Lebesgue spaces than the previous regularizing estimates. For the estimates in homogeneous Sobolev spaces, never seen before, we only need functions lying in Lebesgue spaces instead of weighted Lebesgue spaces; i.e., no loss of weight occurs in this case.