Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

摘要

We consider solutions of the massless scalar wave equation g. = 0, without symmetry, on fixed subextremal Kerr backgrounds (M, g). It follows from previous analyses in the Kerr exterior that for solutions. 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 that. is in fact uniformly bounded, |.| = C, in the black hole interior up to and including the bifurcate Cauchy horizon CH+, to which. in fact extends continuously. In analogy to our previous paper [31], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner-Nordstrom backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and an application of Sobolev embedding. In contrast to the Reissner-Nordstrom case, the commutation leads to additional error terms that have to be controlled.