Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis exclusively in physical-space. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis [13] through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [63] in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.