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

Boundedness of massless scalar waves on Kerr interior backgrounds