This is a joint work with Cl\'{e}ment Mouhot (Universit\'{e} Paris-IX Dauphine). We prove $L^1_s\cap L^{\infty}_s$ regularities of iterated Boltzmann collision integrals of $L^1_s$ functions for hard potentials with angular
cutoff, and give an application to the exponential convergence
to equilibrium for measure-valued solutions of the specially homogeneous Boltzmann
equation: We show that if $F_0\ge 0$ is a Borel measure on ${\bf R}^N$
with finite moments up to order 2, then the unique conservative
solution $F_t$ with the initial data $F_0$ converges to the Maxwellian distribution $M$ in the
exponential rate: $\|F_t-M\|\le C e^{-\lambda t} (t\ge 0)$, where
$\|\mu\|$ stands for the total variation of a finite measure $\mu$
on ${\bf R}^N$ and $\lambda>0$ is the spectral gap for the
corresponding linearized collision operator.
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