  We consider the $3D$ spatially homogeneous Boltzmann equation for (true) hard
and moderately soft potentials. We assume that the initial condition is a
probability measure with finite energy and is not a Dirac mass. For hard
potentials, we prove that any reasonable weak solution immediately belongs to
some Besov space. For moderately soft potentials, we assume additionally that
the initial condition has a moment of sufficiently high order ($8$ is enough)
and prove the existence of a solution that immediately belongs to some Besov
space. The considered solutions thus instantaneously become functions with a
finite entropy. We also prove that in any case, any weak solution is
immediately supported by ${\mathbb {R}}^3$.
