  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 Schr\"odinger equation, $i \partial_t u + L u=0 $ and show that
for all $f\in L^p(X), 1<p<\infty,$
  \begin{eqnarray*}
  \left\| e^{itL} (I+L)^{-{\sigma n}} f\right\|_{p} \leq C(1+|t|)^{\sigma n}
\|f\|_{p},
  \ \ \ t\in{\mathbb R}, \ \ \ \sigma\geq \big|{1\over 2}-{1\over p}\big|,
\end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a
Poisson type upper bound. This extends the previous result in \cite{CDLY1} in
which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.
