  We introduce a novel framework for the analysis of linear wave equations on
nonstationary asymptotically flat spacetimes, under the assumptions of mode
stability and absence of zero energy resonances for a stationary model
operator. Our methods apply in all spacetime dimensions and to tensorial
equations, and they do not require any symmetry or almost-symmetry assumptions
on the spacetime metrics or on the wave type operators. Moreover, we allow for
the presence of terms which are asymptotically scaling critical at infinity,
such as inverse square potentials. For simplicity of presentation, we do not
allow for normally hyperbolic trapping or horizons.
  In the first part of the paper, we study stationary wave type equations, i.e.
equations with time-translation symmetry, and prove pointwise upper bounds for
their solutions. We establish a relationship between pointwise decay rates and
weights related to the mapping properties of the zero energy operator. Under a
nondegeneracy assumption, we prove that this relationship is sharp by
extracting leading order asymptotic profiles at late times. The main tool is
the analysis of the resolvent at low energies.
  In the second part, we consider a class of wave operators without
time-translation symmetry which settle down to stationary operators at a rate
$t_*^{-\delta}$ as an appropriate hyperboloidal time function $t_*$ tends to
infinity. The main result is a sharp solvability theory for forward problems on
a scale of polynomially weighted spacetime $L^2$-Sobolev spaces. The proof
combines a regularity theory for the nonstationary operator with the
invertibility of the stationary model established in the first part. The
regularity theory is fully microlocal and utilizes edge-b-analysis near null
infinity, as developed in joint work with Vasy, and 3b-analysis in the forward
cone.
