An extension of the $r^p$ method for wave equations with scale-critical potentials and first-order terms

The $r^p$ method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Gr\"{o}nwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.