Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case |a|<<M

We prove boundedness and polynomial decay statements for solutions of the spin +/- 2 Teukolsky equation on a Kerr exterior background with parameters satisfying |a|<< M. The bounds are obtained by introducing generalisations of the higher order quantities P and <mml:munder>P_</mml:munder> 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 |a|<M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.