  We study the Cauchy problem for the wave equation on extreme Kerr backgrounds
under axisymmetry. Specifically, we consider regular axisymmetric initial data
prescribed on a Cauchy hypersurface S which connects the future event horizon
with spacelike or null infinity, and we solve the linear wave equation on the
domain of dependence of S. 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 solutions to the
wave equation. Our estimates hold up to and including the event horizon. We
remark that these results do not yield decay for the derivatives transversal to
the horizon, and this is suggestive that these derivatives may satisfy
instability properties analogous to those shown in our previous work on extreme
Reissner-Nordstrom backgrounds.
