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

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.