  Let $L$ be a one-to-one operator of type $\omega$ having a bounded $H_\infty$
functional calculus and satisfying the $k$-Davies-Gaffney estimates with
$k\in{\mathbb N}$. In this paper, the authors introduce the weak Hardy space
$WH_L^p(\mathbb{R}^n)$ associated to $L$ for $p\in (0,\,1]$ via the
non-tangential square function $S_L$ and establish a weak molecular
characterization of $WH_L^p(\mathbb{R}^n)$. Typical examples of such operators
include the $2k$-order divergence form homogeneous elliptic operator
$L_1:=(-1)^k\sum_{|\alpha|=k=|\beta|}\partial^\beta(a_{\alpha,\beta}\partial^\alpha)$,
where $\{a_{\alpha,\beta}\}_{|\alpha|=k=|\beta|}$ are complex bounded
measurable functions, and the $2k$-order Schr\"odinger type operator $L_2:=
(-\Delta)^k+V^k$, where $\Delta$ is the Laplacian operator and $0\le V\in
L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$. As applications, for $i\in\{1,\,2\}$
and $p\in(\frac{n}{n+k},\,1]$, the authors prove that the associated Riesz
transform $\nabla^k (L_i^{-1/2})$ is bounded from $WH^p_{L_i}(\mathbb{R}^n)$ to
the classical weak Hardy space $WH^p(\mathbb{R}^n)$ and, for all $0<p<r\le1$
and $\alpha=n(\frac{1}{p}-\frac{1}{r})$, the fractional power
$L_i^{-\frac{\alpha}{2k}}$ is bounded from $WH_{L_i}^p(\mathbb{R}^n)$ to
$WH_{L_i}^r(\mathbb{R}^n)$. Furthermore, the authors find the dual space of
$WH_L^p(\mathbb{R}^n)$ for $p\in(0,\,1]$, which can be defined via mean
oscillations based on some subtle coverings of bounded open sets and, even when
$L:=-\Delta$, are also previously unknown. In particular, if $L$ is a
nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the
Davies-Gaffney estimates, the authors further establish the weak atomic
characterization of $WH_L^p(\mathbb{R}^n)$.
