Integral estimates for a linear singular operator linked with the Boltzmann operator: Part I: Small singularities O"<" v "<"1

We give precise integral estimates for a linear operator linked with Boltzmann quadratic operator. This is done for singular cross-sections. We show Calderon-Zygmund and commutators type estimates. In particular, these estimates can be used to deduce some functional properties of Boltzmann operator itself.