Decay of Solutions of the Teukolsky Equation for Higher Spin in the Schwarzschild Geometry

We prove that the Schwarzschild black hole is linearly stable under electromagnetic and gravitational perturbations. Our method is to show that for spin $s=1$ or $s=2$, solutions of the Teukolsky equation with smooth, compactly supported initial data outside the event horizon, decay in $L_{loc}^{\mathrm{\infty }}$.