  In a recent paper with J.-P. Nicolas [J.-P. Nicolas and P.T. Xuan, Annales
Henri Poincare 2019], we studied the peeling for scalar fields on Kerr metrics.
The present work extends these results to Dirac fields on the same geometrical
background. We follow the approach initiated by L.J. Mason and J.-P. Nicolas
[L. Mason and J.-P. Nicolas, J.Inst.Math.Jussieu 2009; L. Mason and J.-P.
Nicolas, J.Geom.Phys 2012] on the Schwarzschild spacetime and extended to Kerr
metrics for scalar fields. The method combines the Penrose conformal
compactification and geometric energy estimates in order to work out a
definition of the peeling at all orders in terms of Sobolev regularity near
$\mathscr{I}$, instead of ${\mathcal C}^k$ regularity at $\mathscr{I}$, then
provides the optimal spaces of initial data such that the associated solution
satisfies the peeling at a given order. The results confirm that the analogous
decay and regularity assumptions on initial data in Minkowski and in Kerr
produce the same regularity across null infinity. 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.
