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We prove boundedness and polynomial decay statements for solutions of thespin $\pm2$ Teukolsky equation on a Kerr exterior background with parameterssatisfying $|a|\ll M$. The bounds are obtained by introducing generalisationsof the higher order quantities $P$ and $\underline{P}$ used in our previouswork on the linear stability of Schwarzschild. The existence of thesequantities in the Schwarzschild case is related to the transformation theory ofChandrasekhar. In a followup paper, we shall extend this result to the generalsub-extremal range of parameters $|a|<M$. As in the Schwarzschild case, thesebounds provide the first step in proving the full linear stability of the Kerrmetric to gravitational perturbations.

Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\ll M$