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

We consider the Laplacian on an asymptotically hyperbolic manifold X, as defined by Mazzeo and Melrose. We give pointwise bounds on the Schwartz kernel of the spectral measure for the operator (− n²/4)^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, Melrose, Sá Barreto and Vasy, the present authors, and Wang. 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) → L^p(X) operator norm bound on the spectral measure. The second is a spectral multiplier result under the additional assumption that X has negative curvature everywhere, that is, a bound on functions of the Laplacian of the form F((− n²/4)^1/2₊), 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.