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 $L^2(X)$ 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 Schrodinger equation $i \partial_{t} u + L u = 0$ and show that for all $f \in L^{p}(X)$, $1 < p < \infty$, $\| e^{itL} (I+L)^{-{\sigma n}} f\|_{p} \leq C(1 + |t|)^{\sigma n} \| f \|_{p}$, $t\in{\mathbb R}$, $\sigma \geq |1/2-1/p|$, where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound.