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We consider the Laplacian $\Delta$ on an asymptotically hyperbolic manifold$X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectralmeasure for the operator $(\Delta - n^2/4)_+^{1/2}$ on such manifolds, underthe assumptions that $X$ is nontrapping and there is no resonance at the bottomof the spectrum. This uses the construction of the resolvent given by Mazzeoand Melrose (valid when the spectral parameter lies in a compact set), Melrose,S\'a Barreto and Vasy (high energy estimates for a perturbation of thehyperbolic metric) and the present authors (see also the work by Yiran Wang) inthe 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 theasymptotically conic case, is a restriction theorem, that is, a $L^p(X) \toL^{p'}(X)$ operator norm bound on the spectral measure. The second is aspectral 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 thefunction $F$. Compared to the asymptotically conic case, our spectralmultiplier result is weaker, but the restriction estimate is stronger. In bothcases, the difference can be traced to the exponential volume growth atinfinity for asymptotically hyperbolic manifolds, as opposed to polynomialgrowth in the asymptotically conic setting.  The pointwise bounds on the spectral measure established here will also beapplied to Strichartz estimates in another work by the first author.

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: spectral measure, restriction theorem, spectral multiplier