Regge-Wheeler Type Equations

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.