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Local Smoothing Estimates for Schrödinger Equations on Hyperbolic Space
About this Title
Andrew Lawrie, Jonas Lührmann, Sung-Jin Oh and Sohrab Shahshahani
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 291, Number 1447
ISBNs: 978-1-4704-6697-8 (print); 978-1-4704-7680-9 (online)
DOI: https://doi.org/10.1090/memo/1447
Published electronically: November 8, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries, Function Spaces and Multipliers
- 3. Overview of the scheme and outline of the paper
- 4. Regularity theory of the heat flow: Littlewood–Paley theory
- 5. Smoothing for Low Frequencies
- 6. Smoothing for High Frequencies
- 7. Transitioning Estimate and Proof of Theorems 1.5 and 1.15
- 8. Local smoothing estimate in the stationary, symmetric case and its perturbations
- 9. Proof of Corollaries 1.18 and 1.19
Abstract
We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb {H}^d$, $d \geq 2$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood–Paley machinery on $\mathbb {H}^d$ based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on $\mathbb {H}^{d}$. Specifically, some of the estimates established in this paper play a crucial role in the authors’ proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see Lawrie, Lührmann, Oh, and Shahshahani, 2023.
As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.
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