Strong Cosmic Censorship analysed on the level of scalar waves in Kerr interiors

We consider solutions of the massless scalar wave equation, without symmetry, on fixed subextremal Kerr backgrounds $(M, g)$.
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 $H^+$. Using the derived decay rate, we show here, that $\psi$ is in fact uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $CH^+$, to which $\psi$ in fact extends continuously.
In analogy to our previous paper, on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner-Nordström backgrounds, the analysis depends on weighted energy estimates,commutation of angular momentum operators and application of Sobolev embedding. In contrast to the Reissner-Nordström case the commutation leads to additional error terms that have to be controlled.