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In this paper, we establish the asymptotic behavior along outgoing and incoming radial geodesics, i.e. the peeling property for the tensorial Fackerell–Ipser and spin [Formula: see text] Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity [Formula: see text] and the initial Cauchy hypersurface [Formula: see text] in a neighborhood of spacelike infinity [Formula: see text] far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

In this paper, we study the Cauchy and Goursat problems of the spin-$n/2$ zero rest-mass equations on Minkowski spacetime by using the conformal geometric method. In our strategy, we prove the wellposedness of the Cauchy problem in Einstein's cylinder. Then we establish pointwise decays of the fields and prove the energy equalities of the conformal fields between the null conformal boundaries $\scri^\pm$ and the hypersurface $\Sigma_0=\left\{ t=0 \right\}$. Finally, we prove the wellposedness of the Goursat problem in the partial conformal compactification by using the energy equalities and the generalisation of H\"ormander's result.