Gain of Integrability for the Boltzmann Collisional Operator

In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the W l,r k regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gain of regularity estimates for the gain collisional integral. In addition our method calculates explicit constants involved in the estimates.