Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds

Abstract

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger 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 $f_0\in C_0^{\infty }(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-it{\Lambda }_g^{\sigma }}$ 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 $f_0(P)e^{-it{\Lambda }_g^{\sigma }}$ 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 Schrödinger and wave equations on $({\mathbb{R}}^d,g),d\ge 3$ under non-trapping condition.

50 pages, to appear in Journal Functional Analysis

References in corpus (4)

• Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary
• Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces
• Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping
• Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

Cited by in corpus (1)

• Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds