  We consider the Laplacian $\Delta$ on an asymptotically hyperbolic manifold
$X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectral
measure for the operator $(\Delta - n^2/4)_+^{1/2}$ on such manifolds, under
the assumptions that $X$ is nontrapping and there is no resonance at the bottom
of the spectrum. This uses the construction of the resolvent given by Mazzeo
and Melrose (valid when the spectral parameter lies in a compact set), Melrose,
S\'a Barreto and Vasy (high energy estimates for a perturbation of the
hyperbolic metric) and the present authors (see also the work by Yiran Wang) in
the general high-energy case.
  Under the additional assumption that $X$ has negative curvature everywhere,
we give two applications of the spectral measure estimates. The first,
following work due to Guillarmou and Sikora with the second author in the
asymptotically conic case, is a restriction theorem, that is, a $L^p(X) \to
L^{p'}(X)$ operator norm bound on the spectral measure. The second is a
spectral multiplier result, that is, a bound on functions $F((\Delta -
n^2/4)_+^{1/2})$ of the square root of the Laplacian, in terms of norms of the
function $F$. Compared to the asymptotically conic case, our spectral
multiplier result is weaker, but the restriction estimate is stronger. In both
cases, the difference can be traced to the exponential volume growth at
infinity for asymptotically hyperbolic manifolds, as opposed to polynomial
growth in the asymptotically conic setting.
  The pointwise bounds on the spectral measure established here will also be
applied to Strichartz estimates in another work by the first author.
