Decay Estimates for q (Theorem M1)

This chapter examines the proof for Theorem M1, deriving decay estimates for the quantity q for k ≤ k small + 20 derivatives. To this end, it uses the wave equation satisfied by q. The spacetime M is decomposed as M = (int)M u (ext)M and that u is an outgoing optical function on (ext)M while u is an ingoing optical function. The chapter relies on the global frame defined in section 3.5, and r and m denote the corresponding scalar functions associated to it. It also proves two theorems on improved weighted estimates, as well as flux decay estimates for q.