  We study integrability properties of a general version of the Boltzmann
collision operator for hard and soft potentials in $n$-dimensions. A
reformulation of the collisional integrals allows us to write the weak form of
the collision operator as a weighted convolution, where the weight is given by
an operator invariant under rotations. Using a symmetrization technique in
$L^p$ we prove a Young's inequality for hard potentials, which is sharp for
Maxwell molecules in the $L^2$ case. Further, we find a new
Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals
with soft potentials. The same method extends to radially symmetric,
non-increasing potentials that lie in some $L^{s}_{weak}$ or $L^{s}$. The
method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear
weighted convolution inequalities and follows a weak formulation setting.
Consequently, it is closely connected to the classical analysis of Young and
Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants
are explicitly given by formulas depending on integrability conditions of the
angular cross section (in the spirit of Grad cut-off). As an additional
application of the technique we also obtain estimates with exponential weights
for hard potentials in both conservative and dissipative interactions.
