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

Understand

We consider solutions of the scalar wave equation $\Box_g\phi=0$, without symmetry, on fixed subextremal Reissner-Nordstr\"om backgrounds $({\mathcal M}, g)$ with nonvanishing charge.

• Previously, it has been shown that for $\phi$ 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 $\phi$ is in fact uniformly bounded, $|\phi| \leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon ${\mathcal C}{\mathcal H}^+$, to which $\phi$ 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.