  We prove sharp pointwise $t^{-3}$ decay for scalar linear perturbations of a
Schwarzschild black hole without symmetry assumptions on the data. We also
consider electromagnetic and gravitational perturbations for which we obtain
decay rates $t^{-4}$, and $t^{-6}$, respectively. We proceed by decomposition
into angular momentum $\ell$ and summation of the decay estimates on the
Regge-Wheeler equation for fixed $\ell$. We encounter a dichotomy: the decay
law in time is entirely determined by the asymptotic behavior of the
Regge-Wheeler potential in the far field, whereas the growth of the constants
in $\ell$ is dictated by the behavior of the Regge-Wheeler potential in a small
neighborhood around its maximum. In other words, the tails are controlled by
small energies, whereas the number of angular derivatives needed on the data is
determined by energies close to the top of the Regge-Wheeler potential. This
dichotomy corresponds to the well-known principle that for initial times the
decay reflects the presence of complex resonances generated by the potential
maximum, whereas for later times the tails are determined by the far field.
However, we do not invoke complex resonances at all, but rely instead on
semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound
near the top energy.
