The wave equation on subextremal Kerr spacetimes with small non-decaying first order terms

We consider the perturbed covariant wave equation $\Box_{g_{M,a}} \Psi = \varepsilon \mathbf{B} \Psi$ on the exterior of a fixed subextremal Kerr spacetime $\left(\mathcal{M},g_{M,a}\right)$. Here $\mathbf{B}$ is a suitably regular first order differential operator respecting the symmetries of Kerr whose coefficients are assumed to decay in space but not in time. We establish integrated decay estimates for solutions of the associated Cauchy problem. The proof adapts the framework introduced by Dafermos--Rodnianski--Shlapentokh-Rothman \cite{DRSR} in the $\varepsilon=0$ case. We combine their estimates with a new global pseudodifferential commutator estimate, which generalises our previous work in the Schwarzschild case. The construction of the commutator exploits the central observation of \cite{DRSR} that superradiant frequencies are not trapped. A further major technical ingredient of the proof consists in establishing appropriate convolution estimates for expressions arising from the interplay of the $\mathbf{B}$-term and the time cutoffs in the microlocal framework.