Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes

In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection V + , on the tangent space of the null-shell, of the associated symplectic gradient turns out to enjoy good commutation properties with the massless Vlasov operator. This implies that V + f remains bounded, for any smooth solution f to the massless Vlasov equation. By identifying a well-chosen modification of V + , we are able to construct a W x , p 1 , 1 weighted norm for which any smooth solution to the massless Vlasov equation verifies an integrated local energy decay estimate without relative degeneration. Together with the r p -weighted energy method of Dafermos–Rodnianski, we establish time decay for the energy norm. This norm allows for the control of the energy-momentum tensor T [ f ] as well as all its first order derivatives. The method developed in this paper is in particular compatible with the approach of [Mav24, DHRT22] used to study quasi-linear wave equations on black hole spacetimes.