The wave equation on the extreme Reissner–Nordström black hole

We study the scalar wave equation on the open exterior region of an extreme Reissner–Nordström black hole and prove that, given compactly supported data on a Cauchy surface orthogonal to the timelike Killing vector field, the solution, together with its ( t , s , θ, ϕ) derivatives of arbitrary order, s a tortoise radial coordinate, is bounded by a constant that depends only on the initial data. Our technique does not allow studying transverse derivatives at the horizon, which is outside the coordinate patch that we use. However, using previous results that show that second and higher transverse derivatives at the horizon of a generic solution grow unbounded along horizon generators, we show that any such divergence, if present, would be milder for solutions with compact initial data.