  This paper contains the first two parts (I-II) of a three-part series
concerning the scalar wave equation \Box_g{\psi} = 0 on a fixed Kerr
background. 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 an
energy-type density (degenerating in a neighborhood of the Schwarzschild photon
sphere and at infinity), integrated over the domain of dependence of a
spacelike hypersurface {\Sigma} connecting the future event horizon with
spacelike infinity or a sphere on null infinity, is bounded by a natural
(non-degenerate) energy flux of {\psi} through {\Sigma}. (The case (II1) has in
fact been treated previously in our Clay Lecture notes: Lectures on black holes
and linear waves, arXiv:0811.0354.) In our forthcoming Part III, the
restriction to axisymmetry for the general |a| < M case is removed. The
complete proof is surveyed in our companion paper The black hole stability
problem for linear scalar perturbations, which includes the essential details
of our forthcoming Part III. Together with previous work (see our: A new
physical-space approach to decay for the wave equation with applications to
black hole spacetimes, in XVIth International Congress on Mathematical Physics,
Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arxiv:0910.4957), this result
leads, under suitable assumptions on initial data of {\psi}, to polynomial
decay bounds for the energy flux of {\psi} through the foliation of the black
hole 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.
