Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds

We study the Cauchy problem for the wave equation □gψ=0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ0. We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H+.