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−itΛgσ with σ∈(0,∞)\\{1} and Λg:=−Δg where Δg is the Laplace–Beltrami operator on asymptotically Euclidean manifolds (Rd,g). Let f0∈C0∞(R) be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part (1−f0)(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥2 inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption (1.12), the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part f0(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥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 (Rd,g), d≥3 under non-trapping condition.