A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes

We present short proofs of integrated local energy decay estimates on Schwarzschild, extremal Reissner-Nordstr\"om, and Schwarzschild-de Sitter spacetimes. The proofs employ novel global physical space multipliers, which, besides their remarkable simplicity, (a) are directly derivable from the geodesic flow, (b) do not require decomposition into spherical harmonics, and (c) whose boundary terms can be controlled by the conserved $T$-energy alone. We also elaborate on the intimate connection between the multipliers of the present paper and the globally good commutators introduced by the first two authors.