Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case

We prove boundedness and polynomial decay statements for solutions of the spin $$ Teukolsky equation on a Kerr exterior background with parameters satisfying $$ . The bounds are obtained by introducing generalisations of the higher order quantities P and $$ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters $$ . As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.