  We consider Kerr spacetimes with parameters a and M such that |a|<< M,
Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally,
stationary axisymmetric black hole exterior spacetimes which are sufficiently
close to a Schwarzschild metric with parameter M>0, with appropriate geometric
assumptions on the plane spanned by the Killing fields. We show uniform
boundedness on the exterior for sufficiently regular solutions to the scalar
homogeneous wave equation. In particular, the bound holds up to and including
the event horizon. No unphysical restrictions are imposed on the behaviour of
the solution near the bifurcation surface of the event horizon. The pointwise
estimate derives in fact from the uniform boundedness of a positive definite
energy flux. Note that in view of the very general assumptions, the
separability properties of the wave equation on the Kerr background are not
used.
