  We study the peeling on Kerr spacetime for fields satisfying conformally
invariant linear and nonlinear scalar wave equations. We follow an approach
initiated by L.J. Mason and the first author for the Schwarzschild metric,
based on a Penrose compactification and energy estimates. This approach
provides a definition of the peeling at all orders in terms of Sobolev
regularity near ${\mathscr I}$ instead of ${\cal C}^k$ regularity at ${\mathscr
I}$, allowing to characterise completely and without loss the classes of
initial data ensuring a certain order of peeling at ${\mathscr I}$. This paper
extends the construction to the Kerr metric, confirms the validity and
optimality of the flat spacetime model (in the sense that the same regularity
and fall-off assumptions on the data guarantee the peeling behaviour in flat
spacetime and on the Kerr metric) and does so for the first time for a
nonlinear equation. Our results are local near spacelike infinity and are valid
for all values of the angular momentum of the spacetime, including for fast
Kerr metrics.
