Boundedness of massless scalar waves on Reissner-Nordstr\"om interior backgrounds

We consider solutions of the scalar wave equation ${\mathrm{\square }}_g\varphi =0$, without symmetry, on fixed subextremal Reissner-Nordstr"om backgrounds $({\mathcal M}, g)$withnonvanishingcharge\mathrm{.}Previously,ithasbeenshownthatfor$\phi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon ${\mathcal{H}}^{+}$. Using this, we show here that $\varphi$ is in fact uniformly bounded, $\mathrm{\mid }\varphi \mathrm{\mid }\le C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal{C}{\mathcal{H}}^{+}$, to which $\varphi$ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.