  We prove global existence and decay for small-data solutions to a class of
quasilinear wave equations on a wide variety of asymptotically flat spacetime
backgrounds, allowing in particular for the presence of horizons, ergoregions
and trapped null geodesics, and including as a special case the Schwarzschild
and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in
general relativity. There are two distinguishing aspects of our approach. The
first aspect is its dyadically localised nature: The nontrivial part of the
analysis is reduced entirely to time-translation invariant $r^p$-weighted
estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which
for large $r$ are outgoing. Global existence and decay then both immediately
follow by elementary iteration on consecutive such time-slabs, without further
global bootstrap. The second, and more fundamental, aspect is our direct use of
a "blackbox" linear inhomogeneous energy estimate on exactly stationary
metrics, together with a novel but elementary physical space top order identity
that need not capture the structure of trapping and is robust to perturbation.
In the specific example of Kerr black holes, the required linear inhomogeneous
estimate can then be quoted directly from the literature [DRSR16], while the
additional top order physical space identity can be shown easily in many cases
(we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$,
which can in fact be understood in this context simply as a perturbation of
Schwarzschild). In particular, the approach circumvents the need either for
producing a purely physical space identity capturing trapping or for a careful
analysis of the commutation properties of frequency projections with the wave
operator of time-dependent metrics.
