Decay properties for massive Vlasov fields on Schwarzschild spacetime

In this paper, we obtain pointwise decay estimates in time for massive Vlasov fields on the exterior of Schwarzschild spacetime. We consider massive Vlasov fields supported on the closure of the largest domain of the mass-shell where timelike geodesics either cross $\mathcal{H}^+$, or escape to infinity. For this class of Vlasov fields, we prove that the components of the energy-momentum tensor decay like $v^{-\frac{1}{3}}$ in the bounded region $\{r\leq R\}$, and like $u^{-\frac{1}{3}}r^{-2}$ in the far-away region $\{r\geq R\}$, where $R>2M$ is sufficiently large. Here, $(u,v)$ denotes the standard Eddington--Finkelstein double null coordinate pair.