From the Teukolsky equation to a system of wave equations on Schwarzschild

We review the proof of energy boundeness and decay for solutions of the Teukolsky equation on the Schwarzschild geometry. This result was first shown by Dafermos, Holzegel and Rodniasnki (Acta. Math. 22(1):1–214, 2019). The proof here is based on an analysis of a transformed system of wave equations obtained by appropriately differentiating the Teukolsky equation. This approach was developed in collaboration with Shlapentokh-Rothman in a series of joint works (arXiv: 2007.07211, 2302.08916) concerning the Teukolsky equation on the more general Kerr family of rotating black holes.