Sharp estimates for Schrödinger groups on non-doubling manifolds with ends

Let Δ be the Laplace–Beltrami operator acting on a non-doubling manifold with two ends M=R^m♯Rⁿ, m>n≥3. In this paper, we will prove the following estimate ‖(I+Δ)^−m/2e^iτΔ f‖_{L 1,∞ (M)} ≤C(1 + |τ|)^m/2‖f‖_L 1( M),∀τ∈R. Hence, by interpolation, for 1<p<∞ and s=m|1/2 − 1/p|, ‖(I+Δ)^−se^irΔf‖ _Lp (M) ≤ C(1+|τ|)^s‖f‖_Lp (M) ,∀τ∈R. These can be viewed as sharp estimates for Schrödinger flows associated with the Laplace–Beltrami operator Δ. We note that these results also hold for more general second order differential operator L whose heat kernel satisfies the same upper bound as the Laplace–Beltrami operator Δ, such as the Schrödinger operator 𝔏 = Δ+V with non-negative potential V.