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We prove that sufficiently regular solutions to the wave equation$\Box_g\phi=0$ on the exterior of the Schwarzschild black hole obey theestimates $|\phi|\leq C_\delta v_+^{-{3/2}+\delta}$ and $|\partial_t\phi|\leqC_{\delta} v_+^{-2+\delta}$ on a compact region of $r$ and along the eventhorizon. This is proved with the help of a new vector field commutator that isanalogous to the scaling vector field on Minkowski spacetime. This resultimproves the known decay rates in the region of finite $r$ and along the eventhorizon.

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