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

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)^−σnf∥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.