A note on energy currents and decay for the wave equation on a Schwarzschild background

In recent work, we have proven uniform decay bounds for solutions of the wave equation ${\mathrm{\square }}_g\varphi =0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $\mathrm{\mid }\varphi \mathrm{\mid }\le C{v}_{+}^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_{+}=\max \{v,1\}$, and $C$ is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition.