In this talk, we focus on the study of a class of kinetic equations, whose prototype is the following Fokker-Planck equation

      

           $\Delta_v f(v,x,t) + v \cdot \nabla_x u(v,x,t) - \partial_t f(v,x,t) = s(v,x,t)$,     

      

where $(v,x,t) \in \mathbb{R}^{2n+1}$. Through a “trajectory" approach, we prove a Poincaré inequality for weak solutions, a keystone for the study of the weak regularity theory and the consequent proof of a weak Harnack inequality. The talk is based on a joint work in collaboration with Dietert, Guerand, Loher, Mouhot and Rebucci.