Local smoothing estimates for Schr\"odinger equations on hyperbolic space

TL;DR

This paper proves global-in-time local smoothing estimates for Schrödinger equations on hyperbolic space, incorporating potentials and spectral conditions, with applications to stability of nonlinear Schrödinger equations and development of a heat flow based Littlewood-Paley theory.

Contribution

It introduces a novel Littlewood-Paley framework on hyperbolic space and establishes local smoothing estimates under broad potential conditions, advancing analysis on hyperbolic manifolds.

Findings

Established frequency localized smoothing estimates on hyperbolic space.

Proved full smoothing estimates under spectral conditions without error.

Developed a heat flow based Littlewood-Paley machinery for hyperbolic space.

Abstract

We establish global-in-time frequency localized local smoothing estimates for Schr\"odinger equations on hyperbolic space $\mathbb{H}^d$. 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 Schr\"odinger-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 Schr\"odinger maps evolution on the hyperbolic plane; see [29]. 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.