Linear Waves in the Interior of Extremal Black Holes II

We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr-Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717-770, 2017) for extremal Reissner-Nordstrom and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon.