The spectral measure on non-trapping asymptotically hyperbolic manifolds

Abstract

In this thesis we systematically discuss the spectral measure on non-trapping asymptotically hyperbolic manifolds and related applications in spectral multipliers and Schrodinger equations. Spectral measure is a notion from spectral theory and can be studied via resolvent by Stone's formula. Following the works, due to Mazzeo and Melrose, Melrose, Sa Barreto and Vasy, we construct high energy resolvent by techniques such as the 0-calculus, semiclassical Fourier integral operators, semiclassical intersecting Lagrangian distributions. Borrowing the pseudo differential operator microlocalization tricks, formulated by Guillarmou, Hassell and Sikora, we then prove the spectral measure estimates for large spectral parameters. From the perspective of harmonic analysis, spectral measure is also tied to restriction theorem, which plays a key role in the theory of spectral multipliers. This trilateral relationship is formulated by Guillarmou, Hassell and Sikora. We apply their theory and get restriction theorem for high energy and weak restriction theorem for low energy, together with a crude Lp + L2 boundedness of spectral multipliers, though the spectral measure on asymptotically hyperbolic manifolds is not so ideal as it requires. From dispersive equations' point of view, spectral measure is a cornerstone of Schrodinger propagator. Our spectral measure estimates apply to dispersive estimates for microlocalized high energy truncated propagators for short time as in the work of Hassell and Zhang. Noting the discrepancy of the spectral measure between low and high energy, we also prove the long time dispersive estimates and low energy truncated estimates for short time. By modified Keel-Tao bilinear arguments, due to Anker and Pierfelice, we obtain Strichartz estimates.