# Global in time Strichartz estimates for the fractional Schr\"odinger equations on asymptotically Euclidean manifolds

## TL;DR

This paper establishes global in time Strichartz estimates for fractional Schr"odinger equations on asymptotically Euclidean manifolds, covering high and low frequency regimes under various geometric trapping conditions.

## Contribution

It extends Strichartz estimates to fractional Schr"odinger operators on asymptotically Euclidean manifolds, with new results for high and low frequency parts under different trapping assumptions.

## Key findings

- High frequency part satisfies global Strichartz estimates under non-trapping conditions.
- Low frequency part satisfies global Strichartz estimates without geometric assumptions.
- Results apply to fractional Schr"odinger and wave equations in higher dimensions.

## Abstract

In this paper, we prove global in time Strichartz estimates for the fractional Schr\"odinger operators, namely $e^{-it\Lambda_g^\sigma}$ with $\sigma \in (0,\infty)\backslash \{1\}$ and $\Lambda_g:=\sqrt{-\Delta_g}$ where $\Delta_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-it\Lambda_g^\sigma}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption, the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part $f_0(P)e^{-it\Lambda_g^\sigma}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 3$ without using any geometric assumption on $g$. As a byproduct, we prove global in time Strichartz estimates for the fractional Schr\"odinger and wave equations on $(\mathbb{R}^d, g), d\geq 3$ under non-trapping condition.