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

In this paper, we prove global in time Strichartz estimates for the fractional Schr"odinger operators, namely $e^{-it{\mathrm{\Lambda }}_g^{\sigma }}$ with $\sigma \in (0,\mathrm{\infty })\mathrm{\backslash }\{1\}$ and ${\mathrm{\Lambda }}_g:=\sqrt{-{\mathrm{\Delta }}_g}$ where ${\mathrm{\Delta }}_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $({\mathbb{R}}^d,g)$. Let $f_0\in C_0^{\mathrm{\infty }}(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-it{\mathrm{\Lambda }}_g^{\sigma }}$ satisfies global in time Strichartz estimates as on ${\mathbb{R}}^d$ of dimension $d\ge 2$ 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 $f_0(P)e^{-it{\mathrm{\Lambda }}_g^{\sigma }}$ satisfies global in time Strichartz estimates as on ${\mathbb{R}}^d$ of dimension $d\ge 3$ 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 $({\mathbb{R}}^d,g),d\ge 3$ under non-trapping condition.