Coupled gravitational and electromagnetic perturbations of Reissner-Nordstr\"om spacetime in a polarized setting II - Combined estimates for the system of wave equations

In a previous paper on coupled gravitational and electromagnetic perturbations of Reissner-Nordstr\"om spacetime in a polarized setting, we derived a system of wave equations for two independent quantities, one related to the Weyl curvature and one related to the Ricci curvature of the perturbed spacetime. We analyze here the system of coupled wave equations, deriving combined energy-Morawetz and $r^p$-estimates for the system, in the case of small charge.