Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

Abstract

Abstract. This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we construct the high energy resolvent on general non-trapping asymptotically hyperbolic manifolds, using semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, together with the 0-calculus of Mazzeo-Melrose.Our results generalize recent work of Melrose, Sá Barreto and Vasy [23], which applies to metrics close to the exact hyperbolic metric. We note that there is an independent work by Y. Wang [30] which also constructs the high-energy resolvent.

Resolvent and Spectral Measure on Non-Trapping Asymptotically Hyperbolic Manifolds II: Spectral Measure, Restriction Theorem, Spectral Multipliers

“…In Section 6 we prove Theorem 6 for high energy. This uses, in a crucial way, the semiclassical Lagrangian structure of the high-energy spectral measure proved in [10] and [45]. Finally, in Section 8, we prove the spectral multiplier result, Theorem 8.…”

“…In Section 2, we show how the main results in Section 1.6 follow in the simple case of hyperbolic 3-space H 3 . In Section 3, we review the geometry and analysis of asymptotically hyperbolic manifolds, recalling the main results of [34] and [10]. In Section 4 we prove the restriction estimate, Theorem 6, for low energy, which exploits, in some sense, the Kunze-Stein phenomenon on H n+1 .…”

“…As in [10], we write down coordinate systems in various regions of X 2 0 , in terms of coordinates (x, y) = (x, y 1 , . .…”

“…This paper, following [10], is the second in a series of three devoted to the analysis of the resolvent family and spectral measure for the Laplacian on an asymptotically hyperbolic, nontrapping manifold. The third paper, by the first author alone, will establish global-in-time Strichartz estimates on such a manifold.…”

“…In this paper, we analyze the spectral measure dE P (λ) of the operator P , under the assumption that (X • , g) is nontrapping (that is, every geodesic reaches infinity both forward and backward) and that there is no resonance at the bottom of the continuous spectrum, n 2 /4. To do this, we express the spectral measure dE P (λ) in terms of the boundary values of the resolvent (∆ − n 2 /4 − (λ ± i0) 2 ) −1 just 'above' and 'below' the spectrum in C. We then use the construction of the resolvent given by Mazzeo and Melrose [34] (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy [36] (high energy estimates for a perturbation of the hyperbolic metric) and the present authors [10] (and, independently, [45]) in the general high-energy case to get precise information about the Schwartz kernel of the spectral measure. In particular, following the work of the second author with Guillarmou and Sikora [19] in the asymptotically conic setting, this will allow us to obtain precise pointwise bounds on the Schwartz kernel, when (micro)localized near the diagonal in a certain sense.…”