  We introduce a new, physical-space-based method for deriving the precise
leading-order late-time behaviour of solutions to geometric wave equations on
asymptotically flat spacetime backgrounds and apply it to the setting of wave
equations with asymptotically inverse-square potentials on Schwarzschild black
holes. This provides a useful toy model setting for introducing methods that
are applicable to more general linear and nonlinear geometric wave equations,
such as wave equations for electromagnetically charged scalar fields, wave
equations on extremal Kerr black holes and geometric wave equations in even
space dimensions, where existing proofs for deriving precise late-time
asymptotics might not apply. The method we introduce relies on exploiting the
spatial decay properties of time integrals of solutions to derive the existence
and precise genericity properties of asymptotic late-time tails and obtain
sharp, uniform decay estimates in time.
