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 P _ 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.
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