Our goal consists in developing a commutation vector field approach for the solutions to the massless Vlasov equation on the exterior of a Schwarzschild black hole. We would like an approach that verifies two properties. First, it has to provide decay estimates for the energy flux induced by the energy-momentum tensor of Vlasov fields and their derivatives. Secondly, we would like an approach that is compatible with the ones used to study wave equations on black hole spacetimes. For this, we make use of a weight function that captures the concentration phenomenon of future-trapped null geodesics in phase space. Moreover, we consider the associated symplectic gradient $V_-$, which turns out to enjoy good commutation properties with the massless Vlasov operator. By using a well-chosen modification of $V_-$, we construct a $W_{x,p}^{1,1}$ norm for which any smooth solution to the massless Vlasov equation verifies an integrated energy decay estimate without relative degeneration. This is joint work with Renato Velozo. The lecture will take place as part of the Imperial Day in Muenster.(https://www.uni-muenster.de/MathematicsMuenster/events/2024/imperialday.shtml).