Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $\mathrm{\mid }a\mathrm{\mid }\ll M$

We prove boundedness and polynomial decay statements for solutions of the spin $\pm 2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $\mathrm{\mid }a\mathrm{\mid }\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underset{‾}{P}$ 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 $\mathrm{\mid }a\mathrm{\mid }<M$. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.