Improved decay for solutions to the linear wave equation on a Schwarzschild black hole

We prove that sufficiently regular solutions to the wave equation $\Box_gϕ=0$ on the exterior of the Schwarzschild black hole obey the estimates $|ϕ|\leq C_δv_+^{-{3/2}+δ}$ and $|\partial_tϕ|\leq C_δ v_+^{-2+δ}$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.

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