We consider solutions of the massless scalar wave equation \(\Box _g\psi =0\), without symmetry, on fixed subextremal Kerr backgrounds \(({{\mathcal {M}}}, g)\). 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 \({{\mathcal {H}}}^+\). Using the derived decay rate, we show that \(\psi \) is in fact uniformly bounded, \(|\psi |\le C\), in the black hole interior up to and including the bifurcate Cauchy horizon \({{\mathcal {C}}}{{\mathcal {H}}}^+\), to which \(\psi \) 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–Nordström backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and an application of Sobolev embedding. In contrast to the Reissner–Nordström case, the commutation leads to additional error terms that have to be controlled.

中文翻译：

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Kerr室内背景上无质量标量波的有界性

我们考虑在固定的极端克尔背景\（（{{\\ mathcal {M}}}，g）\）下无对称的无质量标量波动方程\（\ Box _g \ psi = 0 \）的解。从先前在Kerr外部进行的分析得出，对于在两端柯西超曲面上的足够规则数据产生的解\（\ psi \），该解及其导数沿事件视界\（{{\ mathcal { H}}} ^ + \）。使用衍生衰减速度，我们表明，\（\ PSI \）实际上是一致有界，\（| \ PSI | \勒Ç\） ，在黑洞内部直至并包括分叉柯西视界\（{{ \ mathcal {C}}} {{\ mathcal {H}}} ^ + \）到\（\ psi \）实际上是连续扩展的。与我们以前的论文[31]类似，在固定的极值下Reissner–Nordström背景下无质量标量波方程解的有界性，该分析取决于加权能量估计，角动量算符的交换和Sobolev嵌入的应用。与Reissner–Nordström情况相反，换向导致必须控制其他误差项。