  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 $n$. Suppose that the heat
kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator
$(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s> n|{1/ 2}-{1/p}| $ and $
p\in (1, \infty)$ (see for example, \cite{CCO, H, Sj}). The index $s= n|{1/
2}-{1/p}|$ was only obtained recently in \cite{CDLY, CDLY2}, and this range of
$s$ is sharp since it is precisely the range known in the case when $L$ is the
Laplace operator $\Delta$ on $X=\mathbb R^n$ (\cite{Mi1}). In this paper we
establish that for $p=1,$ the operator $(1+L)^{-n/2}e^{itL}$ is of weak type
$(1, 1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that
\begin{eqnarray*}
  \mu\Big(\Big\{x: \big|(I+L)^{-n/2 }e^{itL} f(x)\big|>\lambda \Big\}\Big) \leq
C\lambda^{-1}(1+|t|)^{n/2} {\|f\|_{L^1(X)} }, \ \ \ t\in{\mathbb R}
\end{eqnarray*} (for $\lambda > 0$ when $\mu (X) = \infty$ and
$\lambda>\mu(X)^{-1}\|f\|_{L^1(X)}$ when $\mu (X) < \infty$). Moreover, we also
show the index $n/2$ is sharp when $L$ is the Laplacian on ${\mathbb R^n}$ by
providing an example.
  Our results are applicable to Schr\"odinger group for large classes of
operators including elliptic operators on compact manifolds, Schr\"odinger
operators with non-negative potentials and Laplace operators acting on Lie
groups of polynomial growth or irregular non-doubling domains of Euclidean
spaces.
