Boundedness of massless scalar waves on Kerr interior backgrounds

We consider solutions of the massless scalar wave equation $\Box_g\psi=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $\psi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $\psi$ is in fact uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $\psi$ in fact extends continuously. In analogy to our previous paper, [30], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner--Nordstr\"om backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner--Nordstr\"om case the commutation leads to additional error terms that have to be controlled.