Improved Decay for Solutions to the Linear Wave Equation on a Schwarzschild Black Hole

We prove that sufficiently regular solutions to the wave equation ${\square_g\phi=0}We prove that sufficiently regular solutions to the wave equation \square_gf = 0{\square_g\phi=0} on the exterior of the Schwarzschild black hole obey the estimates |f| ￡ C_d v₊^-\frac32+d{|\phi|\leq C_\delta v_+^{-\frac{3}{2}+\delta}} and |?_tf| ￡ C_d v₊^-2+d{|\partial_t\phi|\leq C_{\delta} v_+^{-2+\delta}} on a compact region of r, including inside the black hole region. 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.