Smoothing effect for Boltzmann equation with full-range interactions

In this work, we are concerned with the regularities of the solutions to Boltzmann equation with the physical collision kernels for the full range of intermolecular repulsive potentials, $r^{-(p-1)}$ with $p>2$. We give the new and constructive upper and lower bounds for the collision operator in terms of standard fractional Sobolev norm. As an application, we prove that the strong solutions obtained by Desvillettes \& Mouhot \cite{dm} to homogeneous Boltzmann equation and classical solutions obtained by Gressman-Strain \cite{gs1,gs2} or Alexandre-Morimoto-Ukai-Xu-Yang \cite{amuxy3,amuxy5} for the inhomogeneous Boltzmann equation become immediately smooth with respect to all variables. And as another application, we obtain the global entropy dissipation estimate which is a little stronger than the one of Alexandre-Desvillettes-Villani-Wennberg \cite{advw}.