Entropy dissipation estimates for the Boltzmann equation without cut-off

We prove a bound on the entropy dissipation for the Boltzmann collision operator from below by a weighted $L^p$-Norm. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. As an application, we study weak solutions to the spatially homogeneous Boltzmann equation and prove a weighted $L_t^1(L_v^p)$ estimate.