  The massive wave equation $\Box_g \psi - \alpha\frac{\Lambda}{3} \psi = 0$ is
studied on a fixed Kerr-anti de Sitter background
$(\mathcal{M},g_{M,a,\Lambda})$. We first prove that in the Schwarzschild case
(a=0), $\psi$ remains uniformly bounded on the black hole exterior provided
that $\alpha < {9/4}$, i.e. the Breitenlohner-Freedman bound holds. Our proof
is based on vectorfield multipliers and commutators: The usual energy current
arising from the timelike Killing vector field $T$ (which fails to be
non-negative pointwise) is shown to be non-negative with the help of a Hardy
inequality after integration over a spacelike slice. In addition to $T$, we
construct a vectorfield whose energy identity captures the redshift producing
good estimates close to the horizon. The argument is finally generalized to
slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing
vectorfield $T=\partial_t$ with $K=\partial_t + \lambda \partial_\phi$ for an
appropriate $\lambda \sim a$, which is also Killing and--in contrast to the
asymptotically flat case--everywhere causal on the black hole exterior. The
separability properties of the wave equation on Kerr-AdS are not used. As a
consequence, the theorem also applies to spacetimes sufficiently close to the
Kerr-AdS spacetime, as long as they admit a causal Killing field $K$ which is
null on the horizon.
