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This paper contains the first two parts (I-II) of a three-part seriesconcerning the scalar wave equation \Box_g{\psi} = 0 on a fixed Kerrbackground. We here restrict to two cases: (II1) |a| \ll M, general {\psi} or(II2) |a| < M, {\psi} axisymmetric. In either case, we prove a version of'integrated local energy decay', specifically, that the 4-integral of anenergy-type density (degenerating in a neighborhood of the Schwarzschild photonsphere and at infinity), integrated over the domain of dependence of aspacelike hypersurface {\Sigma} connecting the future event horizon withspacelike infinity or a sphere on null infinity, is bounded by a natural(non-degenerate) energy flux of {\psi} through {\Sigma}. (The case (II1) has infact been treated previously in our Clay Lecture notes: Lectures on black holesand linear waves, arXiv:0811.0354.) In our forthcoming Part III, therestriction to axisymmetry for the general |a| < M case is removed. Thecomplete proof is surveyed in our companion paper The black hole stabilityproblem for linear scalar perturbations, which includes the essential detailsof our forthcoming Part III. Together with previous work (see our: A newphysical-space approach to decay for the wave equation with applications toblack hole spacetimes, in XVIth International Congress on Mathematical Physics,Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arxiv:0910.4957), this resultleads, under suitable assumptions on initial data of {\psi}, to polynomialdecay bounds for the energy flux of {\psi} through the foliation of the blackhole exterior defined by the time translates of a spacelike hypersurface{\Sigma} terminating on null infinity, as well as to pointwise decay estimates,of a definitive form useful for nonlinear applications.

Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry