  We justify the Villani conjecture on the smoothing effect for measure value
solutions to the space homogeneous Boltzmann equation of Maxwellian type cross
sections. This is the first rigorous proof of the smoothing effect for any
measure value initial data except the single Dirac mass, which gives the
optimal description on the regularity of solutions for positive time, caused by
the singularity in the cross section. The main new ingredient in the proof is
the introduction of a time degenerate coercivity estimate by using the
microlocal analysis.
