Boundedness and Decay for the Teukolsky Equation on Kerr in the Full Subextremal Range $$|a|

Abstract

Abstract In this paper, and in the companion ([152]), we study the gauge-invariant components of linear perturbations of Kerr black hole spacetimes in the full subextremal range of parameters $$|a| | a | < M . We show that, for all $$|a| | a | < M , solutions of the Teukolsky equation of spin $$\pm 1$$ ± 1 and spin $$\pm 2$$ ± 2 arising from regular initial data remain bounded and decay in time. Our result makes completely explicit the dependence of these bounds on the Kerr black hole parameters, while being entirely self-contained. For completeness, we also include the case of the scalar wave equation, $$s=0$$ s = 0 . The present paper, which is an abridged version of our preprint ([151]), considers fixed frequency solutions of the transformed system of equations introduced by Dafermos et al. (Acta Math 222(1):1–214, 2019). We obtain estimates which are uniform in the separation parameters for subextremal $$|a| | a | < M , as well as estimates away from the superradiant frequency threshold up to and including the extremal $$|a|=M$$ | a | = M . In the subextremal $$|a| | a | < M case, these estimates are summed in our companion paper ([152]) to yield the conclusion of the proof of our main result.