  In this paper we prove integrated energy and pointwise decay estimates for
solutions of the vacuum linearized Einstein equation on the domain of outer
communication of the Kerr black hole spacetime. The estimates are valid for the
full subextreme range of Kerr black holes, provided integrated energy estimates
for the Teukolsky Master Equation holds. For slowly rotating Kerr backgrounds,
such estimates are known to hold, due to the work of one of the authors. The
results in this paper thus provide the first stability results for linearized
gravity on the Kerr background, in the slowly rotating case, and reduce the
linearized stability problem for the full subextreme range to proving
integrated energy estimates for the Teukolsky equation. This constitutes an
essential step towards a proof of the black hole stability conjecture, i.e. the
statement that the Kerr family is dynamically stable, one of the central open
problems in general relativity.
