The Schrödinger equation in 𝐿^𝑝 spaces for operators with heat kernel satisfying Poisson type bounds

Abstract

Let 𝐿 be a non-negative self-adjoint operator acting on 𝐿²(𝑋) where 𝑋 is a space of homogeneous type with a dimension 𝑛. In this paper, we study sharp endpoint 𝐿^𝑝-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation 𝑖𝜕_𝑡𝑢 + 𝐿𝑢 = 0 and show that for all 𝑓 ∈ 𝐿^𝑝(𝑋), 1 < 𝑝 < ∞, ‖𝑒^𝑖𝑡𝐿 (𝐼 + 𝐿)^−𝜎𝑛 𝑓‖_𝑝 ≤ 𝐶(1 + |𝑡|)^𝜎𝑛 ‖ 𝑓 ‖_𝑝, 𝑡 ∈ ℝ, 𝜎 ≥ |1/2 −1/𝑝|, where the semigroup 𝑒^−𝑡𝐿 generated by 𝐿 satisfies a Poisson type upper bound.