  We extend the $L^p$-theory of the Boltzmann collision operator by using
classical techniques based in the Carleman representation and Fourier analysis,
allied to new ideas that exploit the radial symmetry of this operator. We are
then able to greatly simplify existent technical proofs in this theory, extend
the range, and obtain explicit sharp constants in some convolution-like
inequalities for the gain part of the Boltzmann collision operator.
